https://rigidgeometricalgebra.org/wiki/api.php?action=feedcontributions&feedformat=atom&user=Eric+Lengyel 2026-07-29T23:01:15Z User contributions MediaWiki 1.40.0 https://rigidgeometricalgebra.org/wiki/index.php?title=Flector&diff=495 2026-07-19T04:32:41Z

<p>Eric Lengyel: /* Norm */</p> <hr /> <div>[[Image:improper_isom.svg|400px|thumb|right|'''Figure 1.''' A flector represents an improper Euclidean isometry, which can always be regarded as a rotation about a line $$\boldsymbol l$$ and a reflection across a plane perpendicular to the same line.]]<br /> A ''flector'' is an operator that performs an improper isometry in Euclidean space. Such isometries encompass all possible combinations of an odd number of [[reflections]], [[inversions]], [[transflections]], and rotoreflections. The name flector is a portmanteau of ''reflection operator''. Flectors cannot perform proper isometries that do not include [[reflections]]; those are instead performed by operators called [[motors]].<br /> <br /> In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a flector $$\mathbf F$$ has the general form<br /> <br /> :$$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$ ,<br /> <br /> which is the sum of a [[point]] $$\mathbf p$$ and a [[plane]] $$\mathbf g$$. To satisfy the [[geometric constraint]], the components of $$\mathbf F$$ must satisfy the equation<br /> <br /> :$$F_{px} F_{gx} + F_{py} F_{gy} + F_{pz} F_{gz} + F_{pw} F_{gw} = 0$$ ,<br /> <br /> and this means that the [[point]] $$\mathbf p$$ must lie in the [[plane]] $$\mathbf g$$.<br /> <br /> An element $$\mathbf x$$ is transformed by a flector $$\mathbf F$$ through the operation $$\mathbf x' = -\mathbf F \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}}$$, where $$\unicode{x27C7}$$ is the [[geometric antiproduct]].<br /> <br /> The set of all flectors is the coset of the set $$\unicode{x1D544}$$ of all [[motors]] in a geometric algebra. As such, the set of all flectors is $$\{\mathbf Q \mathbin{\unicode{x27C7}} \mathbf F \mid \mathbf Q \in \unicode{x1D544}\}$$, where $$\mathbf F$$ is any fixed flector. In particular, since $$\mathbf e_4$$ is a flector (representing [[inversion]] through the origin), any flector can be written as $$\mathbf Q \mathbin{\unicode{x27C7}} \mathbf e_4$$ for some motor $$\mathbf Q$$.<br /> <br /> == Norm ==<br /> <br /> The [[bulk norm]] of a flector $$\mathbf F$$ is given by<br /> <br /> :$$\left\Vert\mathbf F\right\Vert_\unicode[&quot;segoe ui symbol&quot;]{x25CF} = \sqrt{\mathbf F \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf{F}\vphantom{F_{py}^2}} = \sqrt{F_{px}^2 + F_{py}^2 + F_{pz}^2 + F_{gw}^2}$$ ,<br /> <br /> and its [[weight norm]] is given by<br /> <br /> :$$\left\Vert\mathbf F\right\Vert_\unicode[&quot;segoe ui symbol&quot;]{x25CB} = \sqrt{\mathbf F \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2218}} \mathbf{F}\vphantom{F_{gy}^2}} = {\large\unicode{x1D7D9}}\sqrt{F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2}$$ .<br /> <br /> The [[geometric norm]] of a flector $$\mathbf F$$ is thus<br /> <br /> :$$\widehat{\left\Vert\mathbf F\right\Vert} = \sqrt{\dfrac{F_{px}^2 + F_{py}^2 + F_{pz}^2 + F_{gw}^2}{F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2}}$$ ,<br /> <br /> and this is equal to half the distance that the origin is moved by the operator.<br /> <br /> A flector is [[unitized]] when $$F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2 = 1$$.<br /> <br /> == Trigonometric Form ==<br /> <br /> A general flector $$\mathbf F$$ can be expressed in terms of a [[unitized]] [[point]] $$\mathbf p$$ and a [[unitized]] [[plane]] $$\mathbf g$$ as<br /> <br /> :$$\mathbf F = \mathbf p\sin\phi + \mathbf g\cos\phi$$ .<br /> <br /> This can be interpreted as performing a rotoreflection consisting of a reflection through the plane $$\mathbf g$$ and a rotation by twice the angle $$\phi$$ about an axis perpendicular to $$\mathbf g$$ passing through the point $$\mathbf p$$. All combinations of a reflection, a rotation, and a translation, even if the original rotation axis is not perpendicular to the original reflection plane, can be formulated as a rotoreflection with respect to some plane.<br /> <br /> == Factorization ==<br /> <br /> Any [[unitized]] flector $$\mathbf F$$ for which $$F_{pw} \neq \pm1$$ can be factored into the product of a simple [[motor]] $$\mathbf Q$$ and a [[plane]] by calculating<br /> <br /> :$$\mathbf Q = \dfrac{1}{\sqrt{1 - F_{pw}^2}}\left(\mathbf F \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{g}}\right)$$ ,<br /> <br /> where the division unitizes the plane $$\mathbf g = F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$. The motor $$\mathbf Q$$ is then given by<br /> <br /> :$$\mathbf Q = \dfrac{1}{\sqrt{1 - F_{pw}^2}}\left[-F_{gx} F_{pw} \mathbf e_{41} - F_{gy} F_{pw} \mathbf e_{42} - F_{gz} F_{pw} \mathbf e_{43} + (F_{gy} F_{pz} - F_{gz} F_{py})\mathbf e_{23} + (F_{gz} F_{px} - F_{gx} F_{pz})\mathbf e_{31} + (F_{gx} F_{py} - F_{gy} F_{px})\mathbf e_{12} + (1 - F_{pw}^2){\large\unicode{x1d7d9}}\right]$$ ,<br /> <br /> which is always unitized. The original flector $$\mathbf F$$ can now be expressed as<br /> <br /> :$$\mathbf F = \mathbf Q \mathbin{\unicode{x27C7}} \dfrac{\mathbf g}{\sqrt{1 - F_{pw}^2}}$$ .<br /> <br /> == Conversion from Flector to Matrix ==<br /> <br /> Given a specific [[Unitization | unitized]] flector $$\mathbf G$$, define the matrices<br /> <br /> :$$\mathbf A = \begin{bmatrix}2(F_{gy}^2 + F_{gz}^2) - 1 &amp; -2F_{gx}F_{gy} &amp; -2F_{gz}F_{gx} &amp; 2(F_{px}F_{pw} - F_{gx}F_{gw}) \\ -2F_{gx}F_{gy} &amp; 2(F_{gz}^2 + F_{gx}^2) - 1 &amp; -2F_{gy}F_{gz} &amp; 2(F_{py}F_{pw} - F_{gy}F_{gw}) \\ -2F_{gz}F_{gx} &amp; -2F_{gy}F_{gz} &amp; 2(F_{gx}^2 + F_{gy}^2) - 1 &amp; 2(F_{pz}F_{pw} - F_{gz}F_{gw}) \\ 0 &amp; 0 &amp; 0 &amp; 1\end{bmatrix}$$<br /> <br /> and<br /> <br /> :$$\mathbf B = \begin{bmatrix}0 &amp; 2F_{gz}F_{pw} &amp; -2F_{gy}F_{pw} &amp; 2(F_{gy}F_{pz} - F_{gz}F_{py}) \\ -2F_{gz}F_{pw} &amp; 0 &amp; 2F_{gx}F_{pw} &amp; 2(F_{gz}F_{px} - F_{gx}F_{pz}) \\ 2F_{gy}F_{pw} &amp; -2F_{gx}F_{pw} &amp; 0 &amp; 2(F_{gx}F_{py} - F_{gy}F_{px}) \\ 0 &amp; 0 &amp; 0 &amp; 0\end{bmatrix}$$ .<br /> <br /> Then the corresponding 4&amp;times;4 matrix $$\mathbf M$$ that transforms a [[point]] $$\mathbf q$$, regarded as a column matrix, as $$\mathbf q' = \mathbf{Mq}$$ is given by<br /> <br /> :$$\mathbf M = \mathbf A + \mathbf B$$ .<br /> <br /> The inverse of $$\mathbf M$$, which transforms a [[plane]] $$\mathbf h$$, regarded as a row matrix, as $$\mathbf h' = \mathbf{hM^{-1}}$$ is given by<br /> <br /> :$$\mathbf M^{-1} = \mathbf A - \mathbf B$$ .<br /> <br /> == Conversion from Matrix to Flector ==<br /> <br /> Let $$\mathbf M$$ be an orthogonal 4&amp;times;4 matrix with determinant &amp;minus;1 having the form<br /> <br /> :$$\mathbf M = \begin{bmatrix} M_{00} &amp; M_{01} &amp; M_{02} &amp; M_{03} \\ M_{10} &amp; M_{11} &amp; M_{12} &amp; M_{13} \\ M_{20} &amp; M_{21} &amp; M_{22} &amp; M_{23} \\ 0 &amp; 0 &amp; 0 &amp; 1\end{bmatrix}$$ .<br /> <br /> Then, by equating the entries of $$\mathbf M$$ to the entries of $$\mathbf A + \mathbf B$$ from above, we have the following four relationships based on the diagonal entries of $$\mathbf M$$:<br /> <br /> :$$1 - M_{00} + M_{11} + M_{22} = 4F_{gx}^2$$<br /> <br /> :$$1 - M_{11} + M_{22} + M_{00} = 4F_{gy}^2$$<br /> <br /> :$$1 - M_{22} + M_{00} + M_{11} = 4F_{gz}^2$$<br /> <br /> :$$1 - M_{00} - M_{11} - M_{22} = 4(1 - F_{gx}^2 - F_{gy}^2 - F_{gz}^2) = 4F_{pw}^2$$<br /> <br /> And we have the following six relationships based on the off-diagonal entries of $$\mathbf M$$:<br /> <br /> :$$M_{21} + M_{12} = -4F_{gy}F_{gz}$$<br /> <br /> :$$M_{02} + M_{20} = -4F_{gz}F_{gx}$$<br /> <br /> :$$M_{10} + M_{01} = -4F_{gx}F_{gy}$$<br /> <br /> :$$M_{21} - M_{12} = -4F_{gx}F_{pw}$$<br /> <br /> :$$M_{02} - M_{20} = -4F_{gy}F_{pw}$$<br /> <br /> :$$M_{10} - M_{01} = -4F_{gz}F_{pw}$$<br /> <br /> If $$M_{00} + M_{11} + M_{22} \leq 0$$, then we calculate<br /> <br /> :$$F_{pw} = \pm \dfrac{1}{2}\sqrt{1 - M_{00} - M_{11} - M_{22}}$$ ,<br /> <br /> where either sign can be chosen. In this case, we know $$|F_{pw}|$$ is at least $$1/2$$, so we can safely divide by $$-4F_{pw}$$ in the last three off-diagonal relationships above to solve for $$F_{gx}$$, $$F_{gy}$$, and $$F_{gz}$$. Otherwise, if $$M_{00} + M_{11} + M_{22} &gt; 0$$, then we select one of the first three diagonal relationships based on the smallest diagonal entry $$M_{00}$$, $$M_{11}$$, or $$M_{22}$$. After calculating $$F_{gx}$$, $$F_{gy}$$, or $$F_{gz}$$, we plug its value into two of the first three off-diagonal relationships to solve for the other two values of $$F_{gx}$$, $$F_{gy}$$, and $$F_{gz}$$. Finally, we plug it into one of the last three off-diagonal relationships to solve for $$F_{pw}$$.<br /> <br /> Setting $$t_x = M_{03}$$, $$t_y = M_{13}$$, and $$t_z = M_{23}$$, the values of $$F_{px}$$, $$F_{py}$$, $$F_{pz}$$, and $$F_{gw}$$ are given by<br /> <br /> $$F_{px} = \dfrac{1}{2}(F_{pw}t_x + F_{gz}t_y - F_{gy}t_z)$$<br /> <br /> $$F_{py} = \dfrac{1}{2}(F_{pw}t_y + F_{gx}t_z - F_{gz}t_x)$$<br /> <br /> $$F_{pz} = \dfrac{1}{2}(F_{pw}t_z + F_{gy}t_x - F_{gx}t_y)$$<br /> <br /> $$F_{gw} = -\dfrac{1}{2}(F_{gx}t_x + F_{gy}t_y + F_{gz}t_z)$$ .<br /> <br /> == Flector Transformations ==<br /> <br /> [[Point | Points]], [[Line | lines]], and [[Plane | planes]] are transformed by a [[unitized]] flector $$\mathbf F$$ as shown in the following table, where $$\mathbf p = (F_{px}, F_{py}, F_{pz})$$ and $$\mathbf g = (F_{gx}, F_{gy}, F_{gz})$$.<br /> <br /> {| class=&quot;wikitable&quot;<br /> ! Type || Transformation<br /> |-<br /> | style=&quot;padding: 12px;&quot; | [[Point]]<br /> <br /> $$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf a = \mathbf g \times \mathbf q_{xyz} - q_w\mathbf p$$<br /> <br /> $$\mathbf q'_{xyz} = \mathbf q_{xyz} + 2(F_{pw}\mathbf a + \mathbf g \times \mathbf a + F_{gw}q_w\mathbf g)$$<br /> <br /> $$q'_w = -q_w$$<br /> |-<br /> | style=&quot;padding: 12px;&quot; | [[Line]]<br /> <br /> $$\begin{split}\mathbf k =\, &amp;k_{vx} \mathbf e_{41} + k_{vy} \mathbf e_{42} + k_{vz} \mathbf e_{43} \\ +\, &amp;k_{mx} \mathbf e_{23} + k_{my} \mathbf e_{31} + k_{mz} \mathbf e_{12}\end{split}$$<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf a = \mathbf g \times \mathbf k_{\mathbf v}$$<br /> <br /> $$\mathbf b = \mathbf g \times \mathbf k_{\mathbf m}$$<br /> <br /> $$\mathbf c = \mathbf p \times \mathbf k_{\mathbf v}$$<br /> <br /> $$\mathbf k'_{\mathbf v} = 2(\mathbf a \times \mathbf g - F_{pw}\mathbf a) - \mathbf k_{\mathbf v}$$<br /> <br /> $$\mathbf k'_{\mathbf m} = 2[F_{gw}\mathbf a + F_{pw}(\mathbf c - \mathbf b) + \mathbf g \times (\mathbf c - \mathbf b) + \mathbf p \times \mathbf a] - \mathbf k_{\mathbf m}$$<br /> |-<br /> | style=&quot;padding: 12px;&quot; | [[Plane]]<br /> <br /> $$\mathbf h = h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412} + h_w \mathbf e_{321}$$<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf a = \mathbf g \times \mathbf h_{xyz}$$<br /> <br /> $$\mathbf h'_{xyz} = 2(\mathbf a \times \mathbf g - F_{pw}\mathbf a) - \mathbf h_{xyz}$$<br /> <br /> $$h'_w = h_w + 2[(\mathbf h_{xyz} \times \mathbf p - F_{gw}\mathbf h_{xyz}) \cdot \mathbf g + F_{pw}(\mathbf p \cdot \mathbf h_{xyz})]$$<br /> |}<br /> <br /> == In the Book ==<br /> <br /> * General reflection operators (flectors) are discussed in Section 3.7.<br /> <br /> == See Also ==<br /> <br /> * [[Motor]]<br /> * [[Reflection]]<br /> * [[Inversion]]<br /> * [[Transflection]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Motor&diff=494 2026-07-19T04:31:45Z

<p>Eric Lengyel: /* Norm */</p> <hr /> <div>[[Image:proper_isom.svg|400px|thumb|right|'''Figure 1.''' A motor represents a proper Euclidean isometry, which can always be regarded as a rotation about a line $$\boldsymbol l$$ and a displacement along the same line.]]<br /> A ''motor'' is an operator that performs a proper isometry in Euclidean space. Such isometries encompass all possible combinations of any number of [[rotations]] and [[translations]]. The name motor is a portmanteau of ''motion operator'' or ''moment vector''. Motors are equivalent to the set of ''dual quaternions'' used in conventional theories, and the functionality is properly generalized in rigid geometric algebra. Motors cannot perform improper isometries that include an odd number of [[reflections]]; those are instead performed by operators called [[flectors]].<br /> <br /> In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a motor $$\mathbf Q$$ has the general form<br /> <br /> :$$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$ .<br /> <br /> To satisfy the [[geometric constraint]], the components of $$\mathbf Q$$ must satisfy the equation<br /> <br /> :$$Q_{vx} Q_{mx} + Q_{vy} Q_{my} + Q_{vz} Q_{mz} + Q_{vw} Q_{mw} = 0$$ .<br /> <br /> Motors are capable of representing any general screw motion consisting of a rotation about a line combined with a displacement along the same line. By Chasles' theorem, all proper rigid motions in 3D space are screw motions.<br /> <br /> An element $$\mathbf x$$ is transformed by a motor $$\mathbf Q$$ through the operation $$\mathbf x' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$, where $$\unicode{x27C7}$$ is the [[geometric antiproduct]].<br /> <br /> The set of all motors, sometimes denoted by $$\unicode{x1D544}$$, forms a subgroup of index 2 in a geometric algebra. Its coset is the set of [[flectors]].<br /> <br /> == Simple Motors ==<br /> <br /> If $$Q_{mw} = 0$$ (i.e., the scalar part $$Q_{\mathbf 1}$$ is zero), then the motor $$\mathbf Q$$ is called a ''simple motor''. Every simple motor represents either a pure [[translation]] or a pure [[rotation]] about a line without any displacement along that line.<br /> <br /> In the case of a pure translation, the motor $$\mathbf T$$ is given by<br /> <br /> :$$\mathbf T = t_x \mathbf e_{23} + t_y \mathbf e_{31} + t_z \mathbf e_{12} + {\large\unicode{x1d7d9}}$$ ,<br /> <br /> and this performs a translation by twice the displacement vector $$(t_x, t_y, t_z)$$.<br /> <br /> In the case of a pure rotation, the motor $$\mathbf R$$ is given by<br /> <br /> :$$\mathbf R = \boldsymbol l\sin\phi + {\large\unicode{x1d7d9}}\cos\phi$$ ,<br /> <br /> and this performs a rotation by twice the angle $$\phi$$ about the [[line]] $$\boldsymbol l$$.<br /> <br /> == Motors Built from Geometry ==<br /> <br /> Motors having specific geometric constructions can be built from [[Point | points]], [[Line | lines]], and [[Plane | planes]] as shown in the following table.<br /> <br /> {| class=&quot;wikitable&quot;<br /> ! Motor|| Description<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$\begin{split}\mathbf h \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{g}}} =\, &amp;(g_yh_z - g_zh_y)\mathbf e_{41} + (g_zh_x - g_xh_z)\mathbf e_{42} + (g_xh_y - g_yh_x)\mathbf e_{43} \\ +\, &amp;(h_xg_w - g_xh_w)\mathbf e_{23} + (h_yg_w - g_yh_w)\mathbf e_{31} + (h_zg_w - g_zh_w)\mathbf e_{12} \\ +\, &amp;(g_xh_x + g_yh_y + g_zh_z)\smash{\large\unicode{x1d7d9}}\end{split}$$<br /> | style=&quot;padding: 12px;&quot; | Rotation about the line where planes $$\mathbf g$$ and $$\mathbf h$$ intersect by twice the angle between them in the direction from $$\mathbf g$$ to $$\mathbf h$$.<br /> <br /> $$\mathbf g = g_x \mathbf e_{234} + g_y \mathbf e_{314} + g_z \mathbf e_{124} + g_w \mathbf e_{321}$$<br /> <br /> $$\mathbf h = h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$\begin{split}\boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{k}}}<br /> =\, &amp;(l_{vz} k_{vy} - l_{vy} k_{vz})\mathbf e_{41} + (l_{vx} k_{vz} - l_{vz} k_{vx})\mathbf e_{42} + (l_{vy} k_{vx} - l_{vx} k_{vy})\mathbf e_{43} \\<br /> +\, &amp;(l_{vz} k_{my} - l_{vy} k_{mz} + l_{mz} k_{vy} - l_{my} k_{vz})\mathbf e_{23} + (l_{vx} k_{mz} - l_{vz} k_{mx} + l_{mx} k_{vz} - l_{mz} k_{vx})\mathbf e_{31} + (l_{vy} k_{mx} - l_{vx} k_{my} + l_{my} k_{vx} - l_{mx} k_{vy})\mathbf e_{12} \\<br /> +\, &amp;(l_{vx} k_{mx} + l_{vy} k_{my} + l_{vz} k_{mz} + l_{mx} k_{vx} + l_{my} k_{vy} + l_{mz} k_{vz}) \\<br /> +\, &amp;(l_{vx} k_{vx} + l_{vy} k_{vy} + l_{vz} k_{vz})\smash{\large\unicode{x1d7d9}}\end{split}$$<br /> | style=&quot;padding: 12px;&quot; | Rotation about the line containing the closest points on lines $$\mathbf k$$ and $$\boldsymbol l$$ by twice the angle between the directions $$(l_{vx}, l_{vy}, l_{vz})$$ and $$(k_{vx}, k_{vy}, k_{vz})$$, and translation by twice the distance between the lines in the direction from $$\mathbf k$$ to $$\boldsymbol l$$.<br /> <br /> $$\mathbf k = k_{vx} \mathbf e_{41} + k_{vy} \mathbf e_{42} + k_{vz} \mathbf e_{43} + k_{mx} \mathbf e_{23} + k_{my} \mathbf e_{31} + k_{mz} \mathbf e_{12}$$<br /> <br /> $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf q \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{p}}} = (q_xp_w - p_xq_w)\mathbf e_{23} + (q_yp_w - p_yq_w)\mathbf e_{31} + (q_zp_w - p_zq_w)\mathbf e_{12} + (p_wq_w)\smash{\large\unicode{x1d7d9}}$$<br /> | style=&quot;padding: 12px;&quot; | Translation by twice the distance between points $$\mathbf p$$ and $$\mathbf q$$ in the direction from $$\mathbf p$$ to $$\mathbf q$$.<br /> <br /> $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$<br /> <br /> $$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$<br /> |}<br /> <br /> == Norm ==<br /> <br /> The [[bulk norm]] of a motor $$\mathbf Q$$ is given by<br /> <br /> :$$\left\Vert\mathbf Q\right\Vert_\unicode[&quot;segoe ui symbol&quot;]{x25CF} = \sqrt{\mathbf Q \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf{Q}\vphantom{Q_{my}^2}} = \sqrt{Q_{mx}^2 + Q_{my}^2 + Q_{mz}^2 + Q_{mw}^2}$$ ,<br /> <br /> and its [[weight norm]] is given by<br /> <br /> :$$\left\Vert\mathbf Q\right\Vert_\unicode[&quot;segoe ui symbol&quot;]{x25CB} = \sqrt{\mathbf Q \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2218}} \mathbf{Q}\vphantom{Q_{vy}^2}} = {\large\unicode{x1D7D9}}\sqrt{Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2}$$ .<br /> <br /> The [[geometric norm]] of a motor $$\mathbf Q$$ is thus<br /> <br /> :$$\widehat{\left\Vert\mathbf Q\right\Vert} = \sqrt{\dfrac{Q_{mx}^2 + Q_{my}^2 + Q_{mz}^2 + Q_{mw}^2}{Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2}}$$ ,<br /> <br /> and this is equal to half the distance that the origin is moved by the operator.<br /> <br /> A motor is [[unitized]] when $$Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2 = 1$$.<br /> <br /> == Exponential Form ==<br /> <br /> A motor $$\mathbf Q$$ can be expressed as the exponential of a [[unitized]] [[line]] $$\boldsymbol l$$ multiplied by $$d + \phi{\large\unicode{x1D7D9}}$$, where $$\phi$$ is half the angle of rotation about the line $$\boldsymbol l$$, and $$d$$ is half the displacement distance along the line $$\boldsymbol l$$. The exponential form can be written as<br /> <br /> :$$\mathbf Q = \exp_\unicode{x27C7}((d + \phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l) = \cos_\unicode{x27C7}(d + \phi{\large\unicode{x1D7D9}}) + \sin_\unicode{x27C7}(d + \phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l$$ .<br /> <br /> This expands to<br /> <br /> :$$\mathbf Q = \boldsymbol l\sin\phi + (d \mathbin{\unicode{x27C7}} \boldsymbol l)\cos\phi - d\sin\phi + {\large\unicode{x1D7D9}}\cos\phi$$ ,<br /> <br /> and replacing the [[line]] $$\boldsymbol l$$ with its components gives us<br /> <br /> :$$\mathbf Q = (l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12})\sin\phi + d(l_{vx} \mathbf e_{23} + l_{vy} \mathbf e_{31} + l_{vz} \mathbf e_{12})\cos\phi - d\sin\phi + {\large\unicode{x1D7D9}}\cos\phi$$ .<br /> <br /> A motor in exponential form is always [[unitized]], and its [[geometric norm]] can be written as<br /> <br /> :$$\widehat{\left\Vert\mathbf Q\right\Vert} = \sqrt{d^2 + (l_{mx}^2 + l_{my}^2 + l_{mz}^2)\sin^2\phi}$$ .<br /> <br /> The quantity $$d + \phi{\large\unicode{x1D7D9}}$$, as a homogeneous magnitude, is the ''pitch'' of the screw transformation performed by the motor, which is the amount of translation along the screw axis per radian of rotation.<br /> <br /> == Logarithm ==<br /> <br /> Given an arbitrary [[unitized]] motor $$\mathbf Q$$ with $$|Q_{vw}| &lt; 1$$, the values of $$\boldsymbol l$$, $$d$$, and $$\phi$$ can be determined from the components of $$\mathbf Q$$, essentially taking the logarithm of a motor. First, the [[scalar]] and [[antiscalar]] terms satisfy the equations<br /> <br /> :$$Q_{mw} = -d\sin\phi$$<br /> :$$Q_{vw} = \cos\phi$$ .<br /> <br /> Assuming that $$\phi &gt; 0$$, we define $$s = \sin\phi = \sqrt{1 - Q_{vw}^2}$$. Then,<br /> <br /> :$$d = -\dfrac{Q_{mw}}{s}$$<br /> :$$\phi = \tan^{-1}\left(\dfrac{s}{Q_{vw}}\right)$$ .<br /> <br /> If $$Q_{vw} = 0$$, then we assign $$\phi = \pi/2$$. If $$Q_{vw} &lt; 0$$, then we can add $$\pi$$ to $$\phi$$ to make the angle positive, but this is not required.<br /> <br /> The components of $$\boldsymbol l$$ are then given by<br /> <br /> :$$(l_{vx}, l_{vy}, l_{vz}) = \dfrac{1}{s}(Q_{vx}, Q_{vy}, Q_{vz})$$<br /> <br /> :$$(l_{mx}, l_{my}, l_{mz}) = \dfrac{1}{s}\left[(Q_{mx}, Q_{my}, Q_{mz}) - \dfrac{dQ_{vw}}{s}(Q_{vx}, Q_{vy}, Q_{vz})\right] = \dfrac{1}{s}\left[(Q_{mx}, Q_{my}, Q_{mz}) + \dfrac{Q_{vw} Q_{mw}}{s^2}(Q_{vx}, Q_{vy}, Q_{vz})\right]$$ .<br /> <br /> In the special case that $$Q_{vw} = \pm 1$$, the motor must be a pure [[translation]] with $$\phi = 0$$ and $$(Q_{vx}, Q_{vy}, Q_{vz}) = (0, 0, 0)$$.<br /> <br /> == Square Root ==<br /> <br /> The square root of a [[quaternion]] can be calculated by summing with the identity and renormalizing. If we attempt to do the same thing with a motor $$\mathbf Q$$, we find that it works only under certain conditions. However, such a calculation does give us a starting point from which we can make a correction for the general case.<br /> <br /> Using the exponential form of $$\mathbf Q$$, we first examine the [[weight norm]] of $$\mathbf Q + {\large\unicode{x1D7D9}}$$ and find that<br /> <br /> :$$\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode[&quot;segoe ui symbol&quot;]{x25CB} = \sqrt{\sin^2\phi + (\cos\phi + 1)^2} = \sqrt{\vphantom{\sin^2\phi}2 + 2Q_\smash{\large\unicode{x1D7D9}}}$$ ,<br /> <br /> where $$Q_{\large\unicode{x1D7D9}} = \cos\phi$$. Applying the trigonometric identity $$\cos^2(\phi/2) = (1 + \cos\phi)/2$$, we can rewrite this as<br /> <br /> :$$\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode[&quot;segoe ui symbol&quot;]{x25CB} = 2\cos(\phi/2)$$ .<br /> <br /> Applying several more trigonometric identities, we now observe<br /> <br /> :$$\dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode[&quot;segoe ui symbol&quot;]{x25CB}} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{2\cos(\phi/2)} = \boldsymbol l\sin(\phi/2) + \left(\dfrac{d}{2} \mathbin{\unicode{x27C7}} \boldsymbol l\right)\cos(\phi/2) - \dfrac{d}{2}\sin(\phi/2) + {\large\unicode{x1D7D9}}\cos(\phi/2) - \left(\dfrac{d}{2} \mathbin{\unicode{x27C7}} \boldsymbol l\right)\sin(\phi/2)\tan(\phi/2) - \dfrac{d}{2}\sin(\phi/2)$$ .<br /> <br /> The first four terms are exactly the square root of $$\mathbf Q$$ because the distance $$d$$ and angle $$\phi$$ have both been halved. But there are two additional terms that we need to eliminate. It just so happens that<br /> <br /> :$$\dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode[&quot;segoe ui symbol&quot;]{x25CB}} \mathbin{\unicode{x27C7}} \dfrac{d}{2}\tan(\phi/2) = \left(\dfrac{d}{2} \mathbin{\unicode{x27C7}} \boldsymbol l\right)\sin(\phi/2)\tan(\phi/2) + \dfrac{d}{2}\sin(\phi/2)$$ ,<br /> <br /> which means that<br /> <br /> :$$\sqrt{\mathbf Q} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\sqrt{2 + 2Q_\smash{\large\unicode{x1D7D9}}}} \mathbin{\unicode{x27C7}} \left({\large\unicode{x1D7D9}} + \dfrac{d}{2}\tan(\phi/2)\right)$$ .<br /> <br /> The scalar component of $$\mathbf Q$$ can be rewritten as $$Q_{\mathbf 1} = -d\sin\phi = -2d\sin(\phi/2)\cos(\phi/2) $$. Dividing this by $$2 + 2Q_{\large\unicode{x1D7D9}} = 4\cos^2(\phi/2)$$ yields<br /> <br /> :$$\dfrac{Q_\mathbf 1}{2 + 2Q_{\large\unicode{x1D7D9}}} = -\dfrac{d}{2}\tan(\phi/2)$$ .<br /> <br /> The square root of a general motor $$\mathbf Q$$ is thus given by<br /> <br /> :$$\sqrt{\mathbf Q} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\sqrt{2 + 2Q_\smash{\large\unicode{x1D7D9}}}} \mathbin{\unicode{x27C7}} \left({\large\unicode{x1D7D9}} - \dfrac{Q_\mathbf 1}{2 + 2Q_{\large\unicode{x1D7D9}}}\right)$$ .<br /> <br /> If $$\mathbf Q$$ is a simple motor, then $$Q_{\mathbf 1} = 0$$, and this reduces to<br /> <br /> :$$\sqrt{\mathbf Q} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\sqrt{2 + 2Q_\smash{\large\unicode{x1D7D9}}}} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode[&quot;segoe ui symbol&quot;]{x25CB}}$$ .<br /> <br /> == Factorization ==<br /> <br /> Any [[unitized]] motor $$\mathbf Q$$ can be factored into the product of a motor $$\mathbf R = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1D7D9}}$$ corresponding to a pure [[rotation]] about a line through the origin and a motor $$\mathbf T = {t_x \mathbf e_{23} + t_y \mathbf e_{31} + t_z \mathbf e_{12} + \large\unicode{x1d7d9}}$$ corresponding to a pure [[translation]] by calculating<br /> <br /> :$$\mathbf T = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{R}}$$ .<br /> <br /> The motor $$\mathbf T$$ is then given by<br /> <br /> :$$\mathbf T = (Q_{vy} Q_{mz} - Q_{vz} Q_{my} + Q_{mx} Q_{vw} - Q_{vx} Q_{mw})\mathbf e_{23} + (Q_{vz} Q_{mx} - Q_{vx} Q_{mz} + Q_{my} Q_{vw} - Q_{vy} Q_{mw})\mathbf e_{31} + (Q_{vx} Q_{my} - Q_{vy} Q_{mx} + Q_{mz} Q_{vw} - Q_{vz} Q_{mw})\mathbf e_{12} + {\large\unicode{x1d7d9}}$$ .<br /> <br /> The original motor $$\mathbf Q$$ can now be expressed as<br /> <br /> :$$\mathbf Q = \mathbf T \mathbin{\unicode{x27C7}} \mathbf R$$ ,<br /> <br /> where both $$\mathbf R$$ and $$\mathbf T$$ are [[unitized]] simple motors.<br /> <br /> == Conversion from Motor to Matrix ==<br /> <br /> Given a specific [[Unitization | unitized]] motor $$\mathbf Q$$, define the matrices<br /> <br /> :$$\mathbf A = \begin{bmatrix}1 - 2(Q_{vy}^2 + Q_{vz}^2) &amp; 2Q_{vx} Q_{vy} &amp; 2Q_{vz} Q_{vx} &amp; 2(Q_{vy} Q_{mz} - Q_{vz} Q_{my}) \\ 2Q_{vx} Q_{vy} &amp; 1 - 2(Q_{vz}^2 + Q_{vx}^2) &amp; 2Q_{vy} Q_{vz} &amp; 2(Q_{vz} Q_{mx} - Q_{vx} Q_{mz}) \\ 2Q_{vz} Q_{vx} &amp; 2Q_{vy} Q_{vz} &amp; 1 - 2(Q_{vx}^2 + Q_{vy}^2) &amp; 2(Q_{vx} Q_{my} - Q_{vy} Q_{mx}) \\ 0 &amp; 0 &amp; 0 &amp; 1\end{bmatrix}$$<br /> <br /> and<br /> <br /> :$$\mathbf B = \begin{bmatrix}0 &amp; -2Q_{vz} Q_{vw} &amp; 2Q_{vy} Q_{vw} &amp; 2(Q_{vw} Q_{mx} - Q_{vx} Q_{mw}) \\ 2Q_{vz} Q_{vw} &amp; 0 &amp; -2Q_{vx} Q_{vw} &amp; 2(Q_{vw} Q_{my} - Q_{vy} Q_{mw}) \\ -2Q_{vy} Q_{vw} &amp; 2Q_{vx} Q_{vw} &amp; 0 &amp; 2(Q_{vw} Q_{mz} - Q_{vz} Q_{mw}) \\ 0 &amp; 0 &amp; 0 &amp; 0\end{bmatrix}$$ .<br /> <br /> Then the corresponding 4&amp;times;4 matrix $$\mathbf M$$ that transforms a [[point]] $$\mathbf p$$, regarded as a column matrix, as $$\mathbf p' = \mathbf{Mp}$$ is given by<br /> <br /> :$$\mathbf M = \mathbf A + \mathbf B$$ .<br /> <br /> The inverse of $$\mathbf M$$, which transforms a [[plane]] $$\mathbf g$$, regarded as a row matrix, as $$\mathbf g' = \mathbf{gM^{-1}}$$ is given by<br /> <br /> :$$\mathbf M^{-1} = \mathbf A - \mathbf B$$ .<br /> <br /> == Conversion from Matrix to Motor ==<br /> <br /> Let $$\mathbf M$$ be an orthogonal 4&amp;times;4 matrix with determinant +1 having the form<br /> <br /> :$$\mathbf M = \begin{bmatrix} M_{00} &amp; M_{01} &amp; M_{02} &amp; M_{03} \\ M_{10} &amp; M_{11} &amp; M_{12} &amp; M_{13} \\ M_{20} &amp; M_{21} &amp; M_{22} &amp; M_{23} \\ 0 &amp; 0 &amp; 0 &amp; 1\end{bmatrix}$$ .<br /> <br /> Then, by equating the entries of $$\mathbf M$$ to the entries of $$\mathbf A + \mathbf B$$ from above, we have the following four relationships based on the diagonal entries of $$\mathbf M$$:<br /> <br /> :$$M_{00} - M_{11} - M_{22} + 1 = 4Q_{vx}^2$$<br /> <br /> :$$M_{11} - M_{22} - M_{00} + 1 = 4Q_{vy}^2$$<br /> <br /> :$$M_{22} - M_{00} - M_{11} + 1 = 4Q_{vz}^2$$<br /> <br /> :$$M_{00} + M_{11} + M_{22} + 1 = 4(1 - Q_{vx}^2 - Q_{vy}^2 - Q_{vz}^2) = 4Q_{vw}^2$$<br /> <br /> And we have the following six relationships based on the off-diagonal entries of $$\mathbf M$$:<br /> <br /> :$$M_{21} + M_{12} = 4Q_{vy} Q_{vz}$$<br /> <br /> :$$M_{02} + M_{20} = 4Q_{vz} Q_{vx}$$<br /> <br /> :$$M_{10} + M_{01} = 4Q_{vx} Q_{vy}$$<br /> <br /> :$$M_{21} - M_{12} = 4Q_{vx} Q_{vw}$$<br /> <br /> :$$M_{02} - M_{20} = 4Q_{vy} Q_{vw}$$<br /> <br /> :$$M_{10} - M_{01} = 4Q_{vz} Q_{vw}$$<br /> <br /> If $$M_{00} + M_{11} + M_{22} \geq 0$$, then we calculate<br /> <br /> :$$Q_{vw} = \pm \dfrac{1}{2}\sqrt{M_{00} + M_{11} + M_{22} + 1}$$ ,<br /> <br /> where either sign can be chosen. In this case, we know $$|Q_{vw}|$$ is at least $$1/2$$, so we can safely divide by $$4Q_{vw}$$ in the last three off-diagonal relationships above to solve for $$Q_{vx}$$, $$Q_{vy}$$, and $$Q_{vz}$$. Otherwise, if $$M_{00} + M_{11} + M_{22} &lt; 0$$, then we select one of the first three diagonal relationships based on the largest diagonal entry $$M_{00}$$, $$M_{11}$$, or $$M_{22}$$. After calculating $$Q_{vx}$$, $$Q_{vy}$$, or $$Q_{vz}$$, we plug its value into two of the first three off-diagonal relationships to solve for the other two values of $$Q_{vx}$$, $$Q_{vy}$$, and $$Q_{vz}$$. Finally, we plug it into one of the last three off-diagonal relationships to solve for $$Q_{vw}$$.<br /> <br /> Setting $$t_x = M_{03}$$, $$t_y = M_{13}$$, and $$t_z = M_{23}$$, the values of $$Q_{mx}$$, $$Q_{my}$$, $$Q_{mz}$$, and $$Q_{mw}$$ are given by<br /> <br /> $$Q_{mx} = \dfrac{1}{2}(Q_{vw} t_x + Q_{vz} t_y - Q_{vy} t_z)$$<br /> <br /> $$Q_{my} = \dfrac{1}{2}(Q_{vw} t_y + Q_{vx} t_z - Q_{vz} t_x)$$<br /> <br /> $$Q_{mz} = \dfrac{1}{2}(Q_{vw} t_z + Q_{vy} t_x - Q_{vx} t_y)$$<br /> <br /> $$Q_{mw} = -\dfrac{1}{2}(Q_{vx} t_x + Q_{vy} t_y + Q_{vz} t_z)$$ .<br /> <br /> == Motor Transformations ==<br /> <br /> [[Point | Points]], [[Line | lines]], and [[Plane | planes]] are transformed by a [[unitized]] motor $$\mathbf Q$$ as shown in the following table, where $$\mathbf v = (Q_{vx}, Q_{vy}, Q_{vz})$$ and $$\mathbf m = (Q_{mx}, Q_{my}, Q_{mz})$$.<br /> <br /> {| class=&quot;wikitable&quot;<br /> ! Type || Transformation<br /> |-<br /> | style=&quot;padding: 12px;&quot; | [[Point]]<br /> <br /> $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf a = \mathbf v \times \mathbf p_{xyz} + p_w\mathbf m$$<br /> <br /> $$\mathbf p'_{xyz} = \mathbf p_{xyz} + 2(Q_{vw}\mathbf a + \mathbf v \times \mathbf a - Q_{mw}p_w\mathbf v)$$<br /> <br /> $$p'_w = p_w$$<br /> |-<br /> | style=&quot;padding: 12px;&quot; | [[Line]]<br /> <br /> $$\begin{split}\boldsymbol l =\, &amp;l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} \\ +\, &amp;l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}\end{split}$$<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf a = \mathbf v \times \boldsymbol l_{\mathbf v}$$<br /> <br /> $$\mathbf b = \mathbf v \times \boldsymbol l_{\mathbf m}$$<br /> <br /> $$\mathbf c = \mathbf m \times \boldsymbol l_{\mathbf v}$$<br /> <br /> $$\boldsymbol l'_{\mathbf v} = \boldsymbol l_{\mathbf v} + 2(Q_{vw}\mathbf a + \mathbf v \times \mathbf a)$$<br /> <br /> $$\boldsymbol l'_{\mathbf m} = \boldsymbol l_{\mathbf m} + 2(Q_{mw}\mathbf a + Q_{vw}(\mathbf b + \mathbf c) + \mathbf v \times (\mathbf b + \mathbf c) + \mathbf m \times \mathbf a)$$<br /> |-<br /> | style=&quot;padding: 12px;&quot; | [[Plane]]<br /> <br /> $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf a = \mathbf v \times \mathbf g_{xyz}$$<br /> <br /> $$\mathbf g'_{xyz} = \mathbf g_{xyz} + 2(Q_{vw}\mathbf a + \mathbf v \times \mathbf a)$$<br /> <br /> $$g'_w = g_w + 2[(\mathbf m \times \mathbf g_{xyz} + Q_{mw}\mathbf g_{xyz}) \cdot \mathbf v - Q_{vw}(\mathbf m \cdot \mathbf g_{xyz})]$$<br /> |}<br /> <br /> == In the Book ==<br /> <br /> * General motion operators (motors) are discussed in Section 3.6.<br /> <br /> == See Also ==<br /> <br /> * [[Flector]]<br /> * [[Translation]]<br /> * [[Rotation]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Metrics&diff=493 2026-03-22T23:33:18Z

<p>Eric Lengyel: </p> <hr /> <div>The ''metric'' used in the 4D rigid geometric algebra over 3D Euclidean space is the $$4 \times 4$$ matrix $$\mathfrak g$$ given by<br /> <br /> :$$\mathfrak g = \begin{bmatrix}<br /> 1 &amp; 0 &amp; 0 &amp; 0 \\<br /> 0 &amp; 1 &amp; 0 &amp; 0 \\<br /> 0 &amp; 0 &amp; 1 &amp; 0 \\<br /> 0 &amp; 0 &amp; 0 &amp; 0 \end{bmatrix}$$ .<br /> <br /> The ''metric exomorphism matrix'' $$\mathbf G$$, often just called the &quot;metric&quot; itself, corresponding to the metric $$\mathfrak g$$ is the $$16 \times 16$$ matrix shown below.<br /> <br /> [[Image:metric-rga-3d.svg|420px]]<br /> <br /> The ''metric antiexomorphism matrix'' $$\mathbb G$$, often called the &quot;antimetric&quot;, corresponding to the metric $$\mathfrak g$$ is the $$16 \times 16$$ matrix shown below.<br /> <br /> [[Image:antimetric-rga-3d.svg|420px]]<br /> <br /> The product of the metric exomorphism matrix $$\mathbf G$$ and metric antiexomorphism matrix $$\mathbb G$$ for any metric $$\mathfrak g$$ is always equal to the $$16 \times 16$$ identity matrix times the determinant of $$\mathfrak g$$. That is, $$\mathbf G \mathbb G = \det(\mathfrak g) \mathbf I$$.<br /> <br /> The metric and antimetric determine [[bulk and weight]], [[duals]], [[dot products]], and [[geometric products]].<br /> <br /> == In the Book ==<br /> <br /> * The metric and antimetric are introduced in Sections 2.8.1 and 2.8.2.<br /> <br /> == See Also ==<br /> <br /> * [[Bulk and weight]]<br /> * [[Duals]]<br /> * [[Dot products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Geometric_products&diff=492 2026-02-06T22:21:52Z

<p>Eric Lengyel: </p> <hr /> <div>The ''geometric product'' is a method of multiplication in geometric algebra derived from the [[exterior product]] and the [[metric]]. There are two products with symmetric properties called the geometric product and geometric antiproduct.<br /> <br /> == Geometric Product ==<br /> <br /> The geometric product between two elements $$\mathbf a$$ and $$\mathbf b$$ has often been written by simply juxtaposing its operands as $$\mathbf{ab}$$ without the use of any infix operator. However, this clearly becomes impractical when both a product and antiproduct are present in the same context, which is now known to be necessary for a proper understanding of the algebra. To remedy the situation, we write the geometric product between elements $$\mathbf a$$ and $$\mathbf b$$ as $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ and read it as &quot;$$\mathbf a$$ wedge-dot $$\mathbf b$$&quot;.<br /> <br /> The geometric product is characterized by a metric that defines the products of the basis vectors with themselves. The subscript in $$\mathcal G_{3,0,1}$$ means that three basis vectors square to +'''1''', zero basis vectors square to &amp;minus;'''1''', and one basis vector squares to 0. The geometric product between two different basis vectors is given by the [[wedge product]]. We can write these rules as follows.<br /> <br /> :$$\mathbf e_1 \mathbin{\unicode{x27D1}} \mathbf e_1 = \mathbf 1$$<br /> <br /> :$$\mathbf e_2 \mathbin{\unicode{x27D1}} \mathbf e_2 = \mathbf 1$$<br /> <br /> :$$\mathbf e_3 \mathbin{\unicode{x27D1}} \mathbf e_3 = \mathbf 1$$<br /> <br /> :$$\mathbf e_4 \mathbin{\unicode{x27D1}} \mathbf e_4 = 0$$<br /> <br /> :$$\mathbf e_i \mathbin{\unicode{x27D1}} \mathbf e_j = \mathbf e_i \wedge \mathbf e_j$$, for $$i \neq j$$.<br /> <br /> The geometric product is equal to the sum of the [[transwedge products]] of orders 0 through $$n$$, where $$n$$ is the dimension of the algebra, in this case 4.<br /> <br /> When $$\mathbf a$$ is a vector, the geometric product $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ can be decomposed as<br /> <br /> :$$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf b \vee \mathbf a^{\unicode[&quot;segoe ui symbol&quot;]{x2605}}$$,<br /> <br /> and the geometric product $$\mathbf b \mathbin{\unicode{x27D1}} \mathbf a$$ can be decomposed as<br /> <br /> :$$\mathbf b \mathbin{\unicode{x27D1}} \mathbf a = \mathbf b \wedge \mathbf a + \mathbf a_{\unicode[&quot;segoe ui symbol&quot;]{x2605}} \vee \mathbf b$$.<br /> <br /> If $$\mathbf a$$ and $$\mathbf b$$ are both vectors, then both of these decompositions reduce to<br /> <br /> :$$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \cdot \mathbf b$$.<br /> <br /> The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.<br /> <br /> Cells highlighted in green correspond to the contribution from the [[wedge product]].<br /> <br /> <br /> [[Image:GeometricProduct.svg|720px]]<br /> <br /> == Geometric Antiproduct ==<br /> <br /> The geometric antiproduct is dual to the geometric product. The geometric antiproduct between elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x27C7}} \mathbf b$$ and is read as &quot;$$\mathbf a$$ antiwedge-dot $$\mathbf b$$&quot;.<br /> <br /> The same metric that defines products of basis vectors under the geometric product also applies to the geometric antiproduct, except that now it defines products of basis [[antivectors]]. Three basis antivectors square to $$+{\large\unicode{x1D7D9}}$$, zero basis antivectors square to $$-{\large\unicode{x1D7D9}}$$, and one basis antivector squares to 0. The geometric antiproduct between two different basis antivectors is given by the [[antiwedge product]]. We can write these rules as follows.<br /> <br /> :$$\overline{\mathbf e_1} \mathbin{\unicode{x27C7}} \overline{\mathbf e_1} = {\large\unicode{x1D7D9}}$$<br /> <br /> :$$\overline{\mathbf e_2} \mathbin{\unicode{x27C7}} \overline{\mathbf e_2} = {\large\unicode{x1D7D9}}$$<br /> <br /> :$$\overline{\mathbf e_3} \mathbin{\unicode{x27C7}} \overline{\mathbf e_3} = {\large\unicode{x1D7D9}}$$<br /> <br /> :$$\overline{\mathbf e_4} \mathbin{\unicode{x27C7}} \overline{\mathbf e_4} = 0$$<br /> <br /> :$$\overline{\mathbf e_i} \mathbin{\unicode{x27C7}} \overline{\mathbf e_j} = \overline{\mathbf e_i} \vee \overline{\mathbf e_j}$$, for $$i \neq j$$.<br /> <br /> The following Cayley table shows the geometric antiproducts between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric antiproduct is the [[antiscalar]] basis element $${\large\unicode{x1D7D9}}$$.<br /> <br /> Cells highlighted in green correspond to the contribution from the [[antiwedge product]].<br /> <br /> <br /> [[Image:GeometricAntiproduct.svg|720px]]<br /> <br /> == De Morgan Laws ==<br /> <br /> The relationship between the product and antiproduct is based on an exchange of full and empty dimensions. We can then express each product in terms of the other through an analog of De Morgan's laws as follows.<br /> <br /> :$$\overline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \overline{\mathbf b}$$<br /> <br /> :$$\overline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \overline{\mathbf b}$$<br /> <br /> :$$\underline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \underline{\mathbf b}$$<br /> <br /> :$$\underline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \underline{\mathbf b}$$<br /> <br /> == In the Book ==<br /> <br /> * The geometric product and antiproduct are introduced in Section 3.1.<br /> <br /> == See Also ==<br /> <br /> * [[Transwedge products]]<br /> * [[Exterior products]]<br /> * [[Dot products]]<br /> * [[Complements]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Errata&diff=491 2026-02-03T08:06:55Z

<p>Eric Lengyel: </p> <hr /> <div>__NOTOC__<br /> This is the corrigenda and errata page for the ''Projective Geometric Algebra Illuminated''. To find out which printing you have, look on the copyright page. If you believe you have found an error that should be added to this list, please email the author (lengyel@terathon.com).<br /> <br /> == Sixth and Seventh Printings Only ==<br /> <br /> * '''Page 29.''' In Equation (1.88), the minus sign after $$\mathbf a_0$$ should be a plus sign.<br /> <br /> == Sixth Printing Only ==<br /> <br /> * '''Page 20.''' In Equation (1.44), the last line should read $$ki = -ik = j$$.<br /> <br /> == Fifth Printing and Earlier ==<br /> <br /> * '''Page 142.''' In Table 3.8, the inversion formula for a line should have a minus sign in the $$\mathbf e_{12}$$ term similar to the other terms.<br /> <br /> * '''Page 174.''' In Equation (3.219), $$Q_{vw}$$ should be the coefficient of the scalar term $$\mathbf 1$$, and $$Q_{mw}$$ should be the coefficient of the antiscalar term $${\large\unicode{x1D7D9}}$$.<br /> <br /> == Fourth Printing and Earlier ==<br /> <br /> * '''Page 12.''' Right before Equation (1.25), the expression for the magnitude of the projection of $$\mathbf u$$ onto the line should be $$|\mathbf u \cdot \mathbf v| / \Vert\mathbf v\Vert$$.<br /> <br /> == Third Printing and Earlier ==<br /> <br /> * '''Page 70.''' In the line following Equation (2.71), the product of the bulks should use the antiwedge product and appear as $$\mathbf a_\unicode[&quot;segoe ui symbol&quot;]{x25CF} \vee \mathbf b_\unicode[&quot;segoe ui symbol&quot;]{x25CF}$$.<br /> <br /> == Second Printing and Earlier ==<br /> <br /> * '''Page 248.''' In Equation (5.27), the operator $$\mathbf D$$ should be dualized as $$\mathbf D^\unicode[&quot;segoe ui symbol&quot;]{x2605}$$. It should also be dualized in the first factor of the sandwich product in the sentence that follows this equation.<br /> <br /> == First Printing ==<br /> <br /> * '''Page 84.''' In Table 2.16, both bulk duals of $$\mathbf e_4$$ should be 0. The table should look like the one on the wiki page for [[duals]].<br /> <br /> * '''Page 156.''' The number of multiply-adds required for transforming a line with a motor did not include the work needed to calculate $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$. Each cross product adds 6 multiply-adds, so the correct total for transforming a line is 54.<br /> <br /> * '''Page 156.''' Similarly, the number of multiply-adds required for transforming a plane with a motor did not include the work needed to calculate $$\mathbf a$$. The correct total for transforming a plane is 35.<br /> <br /> == See Also ==<br /> <br /> * [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]<br /> * [[Contents]]<br /> * [[Exercises]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Duals&diff=490 2026-01-25T19:57:00Z

<p>Eric Lengyel: </p> <hr /> <div>Every object in projective geometric algebra has two duals derived from the metric tensor, called the ''metric dual'' and ''metric antidual''.<br /> <br /> == Dual ==<br /> <br /> The (right) ''metric dual'' or just &quot;dual&quot; of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode[&quot;segoe ui symbol&quot;]{x2605}$$ and defined as<br /> <br /> :$$\mathbf u^\unicode[&quot;segoe ui symbol&quot;]{x2605} = \overline{\mathbf{Gu}}$$ ,<br /> <br /> where $$\mathbf G$$ is the $$16 \times 16$$ [[metric exomorphism matrix]]. In projective geometric algebra, this dual is also called the ''bulk dual'' because it is the [[complement]] of the bulk components, as expressed by<br /> <br /> :$$\mathbf u^\unicode[&quot;segoe ui symbol&quot;]{x2605} = \overline{\mathbf u_\unicode[&quot;segoe ui symbol&quot;]{x25CF}}$$ .<br /> <br /> The bulk dual satisfies the following identity based on the [[geometric product]]:<br /> <br /> :$$\mathbf u^\unicode[&quot;segoe ui symbol&quot;]{x2605} = \tilde{\mathbf u} \mathbin{\unicode{x27D1}} {\large\unicode{x1D7D9}}$$ .<br /> <br /> The right bulk dual produces results equivalent to the Hodge star operator and thus satisfies the identity<br /> <br /> :$$\mathbf a \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605} = (\mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b){\large\unicode{x1D7D9}}$$<br /> <br /> when $$\mathbf a$$ and $$\mathbf b$$ have the same grade.<br /> <br /> == Antidual ==<br /> <br /> The (right) ''metric antidual'' or just &quot;antidual&quot; of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode[&quot;segoe ui symbol&quot;]{x2606}$$ and defined as<br /> <br /> :$$\mathbf u^\unicode[&quot;segoe ui symbol&quot;]{x2606} = \overline{\mathbb G \mathbf u}$$ ,<br /> <br /> where $$\mathbb G$$ is the $$16 \times 16$$ [[metric antiexomorphism matrix]]. In projective geometric algebra, this dual is also called the ''weight dual'' because it is the [[complement]] of the weight components, as expressed by<br /> <br /> :$$\mathbf u^\unicode[&quot;segoe ui symbol&quot;]{x2606} = \overline{\mathbf u_\unicode[&quot;segoe ui symbol&quot;]{x25CB}}$$ .<br /> <br /> The weight dual satisfies the following identity based on the [[geometric antiproduct]]:<br /> <br /> :$$\mathbf u^\unicode[&quot;segoe ui symbol&quot;]{x2606} = \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} \mathbin{\unicode{x27C7}} \mathbf 1$$ .<br /> <br /> == Duals of Basis Elements ==<br /> <br /> The following table lists the bulk and weight duals for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$. In addition to the right duals, there are also left bulk and weight duals denoted by $$\mathbf u_\unicode[&quot;segoe ui symbol&quot;]{x2605}$$ and $$\mathbf u_\unicode[&quot;segoe ui symbol&quot;]{x2606}$$ for which the right complement operation is replaced by the left complement operation.<br /> <br /> [[Image:Duals.svg|720px]]<br /> <br /> Taking either dual twice causes the sign to change according to the formula<br /> <br /> :$$\mathbf u^{\unicode[&quot;segoe ui symbol&quot;]{x2605}\unicode[&quot;segoe ui symbol&quot;]{x2605}} = \mathbf u^{\unicode[&quot;segoe ui symbol&quot;]{x2606}\unicode[&quot;segoe ui symbol&quot;]{x2606}} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}\,\det(\mathfrak g)\mathbf u$$ .<br /> <br /> Since the determinant of the [[metric]] $$\mathfrak g$$ is zero in the projective algebra $$\mathcal G_{3,0,1}$$, applying either dual twice always produces zero here.<br /> <br /> == Duals of Geometries ==<br /> <br /> The bulk duals and weight duals of geometries in the 4D rigid geometric algebra are listed in the following table.<br /> <br /> {| class=&quot;wikitable&quot;<br /> ! Type !! Bulk Dual !! Weight Dual<br /> |-<br /> | style=&quot;padding: 12px;&quot; | [[Point]]<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf p^\unicode[&quot;segoe ui symbol&quot;]{x2605} = p_x \mathbf e_{423} + p_y \mathbf e_{431} + p_z \mathbf e_{412}$$<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf p^\unicode[&quot;segoe ui symbol&quot;]{x2606} = p_w \mathbf e_{321}$$<br /> |-<br /> | style=&quot;padding: 12px;&quot; | [[Line]]<br /> | style=&quot;padding: 12px;&quot; | $$\boldsymbol l^\unicode[&quot;segoe ui symbol&quot;]{x2605} = -l_{mx} \mathbf e_{41} - l_{my} \mathbf e_{42} - l_{mz} \mathbf e_{43}$$<br /> | style=&quot;padding: 12px;&quot; | $$\boldsymbol l^\unicode[&quot;segoe ui symbol&quot;]{x2606} = -l_{vx} \mathbf e_{23} - l_{vy} \mathbf e_{31} - l_{vz} \mathbf e_{12}$$<br /> |-<br /> | style=&quot;padding: 12px;&quot; | [[Plane]]<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf g^\unicode[&quot;segoe ui symbol&quot;]{x2605} = -g_w \mathbf e_4$$<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf g^\unicode[&quot;segoe ui symbol&quot;]{x2606} = -g_x \mathbf e_1 - g_y \mathbf e_2 - g_z \mathbf e_3$$<br /> |}<br /> <br /> == In the Book ==<br /> <br /> * Duals are introduced in Section 2.12.<br /> <br /> == See Also ==<br /> <br /> * [[Complements]]<br /> * [[Bulk and weight]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Interior_products&diff=489 2026-01-01T21:57:03Z

<p>Eric Lengyel: /* Expansions */</p> <hr /> <div>The ''interior products'' are special products in geometric algebra that are useful for a number of operations including [[projections]]. They are defined as the [[wedge product]] or [[antiwedge product]] between one object and either the [[bulk dual]] or [[weight dual]] of another object.<br /> <br /> == Contractions ==<br /> <br /> The ''bulk contraction'' between elements $$\mathbf a$$ and $$\mathbf b$$ is defined as<br /> <br /> :$$\mathrm{bulk\ contraction}(\mathbf a, \mathbf b) = \mathbf a \vee \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605}$$ .<br /> <br /> The ''weight contraction'' between elements $$\mathbf a$$ and $$\mathbf b$$ is defined as<br /> <br /> :$$\mathrm{weight\ contraction}(\mathbf a, \mathbf b) = \mathbf a \vee \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2606}$$ .<br /> <br /> The contractions have the effect of subtracting grades so that<br /> <br /> :$$\operatorname{gr}(\mathbf a \vee \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605}) = \operatorname{gr}(\mathbf a \vee \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2606}) = \operatorname{gr}(\mathbf a) - \operatorname{gr}(\mathbf b)$$ .<br /> <br /> In the case that the grades of $$\mathbf a$$ and $$\mathbf b$$ are the same, the bulk and weight contractions are equal to dot products written as<br /> <br /> :$$\mathbf a \vee \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605} = \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$<br /> <br /> and<br /> <br /> :$$\mathbf a \vee \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2606} = (\mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2218}} \mathbf b) \vee \mathbf 1$$ .<br /> <br /> The bulk contraction satisfies the identity<br /> <br /> :$$\mathbf a \vee \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605} = \langle\mathbf{\tilde b} \mathbin{\unicode{x27D1}} \mathbf a\rangle_{\operatorname{gr}(\mathbf a) - \operatorname{gr}(\mathbf b)}$$ .<br /> <br /> The following Cayley table shows the bulk and weight contraction products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk contraction are highlighted in green, and the values of the weight contraction are highlighted in purple. The darker green cells along the diagonal correspond to the nonzero entries of the metric $$\mathbf G$$.<br /> <br /> <br /> [[Image:Constractions.svg|720px]]<br /> <br /> == Expansions ==<br /> <br /> The ''bulk expansion'' between elements $$\mathbf a$$ and $$\mathbf b$$ is defined as<br /> <br /> :$$\mathrm{bulk\ expansion}(\mathbf a, \mathbf b) = \mathbf a \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605}$$ .<br /> <br /> The ''weight expansion'' between elements $$\mathbf a$$ and $$\mathbf b$$ is defined as<br /> <br /> :$$\mathrm{weight\ expansion}(\mathbf a, \mathbf b) = \mathbf a \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2606}$$ .<br /> <br /> The expansions have the effect of subtracting antigrades so that<br /> <br /> :$$\operatorname{ag}(\mathbf a \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605}) = \operatorname{ag}(\mathbf a \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2606}) = \operatorname{ag}(\mathbf a) - \operatorname{ag}(\mathbf b)$$ .<br /> <br /> In the case that the grades (and thus antigrades) of $$\mathbf a$$ and $$\mathbf b$$ are the same, the bulk and weight expansions are equal to dot products written as<br /> <br /> :$$\mathbf a \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2606} = \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2218}} \mathbf b$$<br /> <br /> and<br /> <br /> :$$\mathbf a \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605} = (\mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b) \wedge \mathbf {\large\unicode{x1D7D9}}$$ .<br /> <br /> The weight expansion satisfies the identity<br /> <br /> :$$\mathbf a \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2606} = \langle\!\langle\smash{\mathbf{\underset{\Large\unicode{x7E}}{b}}} \mathbin{\unicode{x27C7}} \mathbf a\rangle\!\rangle_{\operatorname{ag}(\mathbf a) - \operatorname{ag}(\mathbf b)}$$ .<br /> <br /> The following Cayley table shows the bulk and weight expansion products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The values of the bulk expansion are highlighted in green, and the values of the weight expansion are highlighted in purple. The darker purple cells along the diagonal correspond to the nonzero entries of the antimetric $$\mathbb G$$.<br /> <br /> <br /> [[Image:Expansions.svg|720px]]<br /> <br /> == In the Book ==<br /> <br /> * Interior products, contractions, and expansions are discussed in Section 2.13.<br /> <br /> == See Also ==<br /> <br /> * [[Projections]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=488 2025-12-17T09:12:56Z

<p>Eric Lengyel: </p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf c^{\unicode{x2605}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, which is an interior product. If $$\mathbf a$$ and $$\mathbf b$$ have the same grade, then this interior product reduces to the inner product $$\mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$.<br /> <br /> If $$k &gt; \operatorname{gr}(\mathbf a)$$ or $$k &gt; \operatorname{gr}(\mathbf b)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf c_{\unicode{x2605}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products, negating for orders 2 and 3 modulo 4, yields the [[geometric product]]. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> When the [[metric]] is diagonal, the [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> <br /> For vectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$.<br /> <br /> For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b - \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf{\tilde b}$$,<br /> <br /> where the [[dot product]] is again attributable to the right contraction $$\mathbf{\tilde b} \vee \mathbf a^{\unicode{x2605}}$$, but this time with a reversal to account for the sign change. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf c^{\unicode{x2606}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf c_{\unicode{x2606}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], and we are now summing over all basis elements of [[antigrade]] $$k$$.<br /> <br /> The sum of all possible transwedge antiproducts, again negating for orders 2 and 3 modulo 4, yields the [[geometric antiproduct]]. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, 3, and 4.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Exercises&diff=487 2025-12-11T22:47:54Z

<p>Eric Lengyel: </p> <hr /> <div>These are exercises accompanying the book [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''Projective Geometric Algebra Illuminated'']. More will be posted over time.<br /> <br /> == Exercises for Chapter 2 ==<br /> <br /> '''1.''' Show that Equation (2.35) properly constructs a line containing two points $$\mathbf p$$ and $$\mathbf q$$ with non-unit weights by considering $$\mathbf p / p_w \wedge \mathbf q / q_w$$ and then scaling by $$p_wq_w$$.<br /> <br /> '''2.''' Let $$\mathbf u$$ be a basis element of the 4D projective algebra. Prove that if $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ and $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$, then it must also be true that $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ and $$\underline{\mathbf u} \vee \mathbf u = \mathbf 1$$. That is, show that right and left complements under the wedge product are also the right and left complements under the antiwedge product.<br /> <br /> '''3.''' Suppose that the 4D trivectors $$\mathbf g$$ and $$\mathbf h$$ represent parallel planes in 3D space. Show that the magnitude of the moment of $$\mathbf g \vee \mathbf h$$ is the distance between the planes multiplied by both their weights.<br /> <br /> '''4.''' Let $$\mathbf m$$ be a $$4 \times 4$$ matrix that performs a rotation about the $$z$$ axis in homogeneous coordinates. Calculate the $$16 \times 16$$ exomorphism matrix $$\mathbf M$$ corresponding to $$\mathbf m$$.<br /> <br /> '''5.''' Suppose that $$\mathbf G$$ is a metric exomorphism. Use the fact that $$\mathbf G$$ is an exomorphism to prove that the associated antimetric $$\mathbb G$$ must satisfy $$\mathbb G(\mathbf a \vee \mathbf b) = \mathbb G\mathbf a \vee \mathbb G\mathbf b$$ for any $$\mathbf a$$ and $$\mathbf b$$.<br /> <br /> '''6.''' Suppose that the metric tensor $$\mathfrak g$$ is invertible. Show that the wedge and antiwedge products satisfy the relationship $$\mathbf a \vee \mathbf b = (\mathbf a^\unicode[&quot;segoe ui symbol&quot;]{x2605} \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605})^\unicode[&quot;segoe ui symbol&quot;]{x2606}$$.<br /> <br /> '''7.''' Suppose that $$\mathbf a$$ and $$\mathbf b$$ are basis elements of an $$n$$-dimensional exterior algebra and $$\operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b) = n$$. Show that $$(\mathbf a \wedge \mathbf b)^\unicode[&quot;segoe ui symbol&quot;]{x2605} = \mathbf a^\unicode[&quot;segoe ui symbol&quot;]{x2605} \mathbin{\unicode{x25CF}} \mathbf b$$.<br /> <br /> '''8.''' Show that the geometric norm is idempotent. That is, show that $$\Vert \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}} \Vert = \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}}$$.<br /> <br /> '''9.''' Derive the relationship between left and right interior products shown in Equation (2.110).<br /> <br /> '''10.''' Derive Equation (2.159), which is the expansion analog of Equation (2.129).<br /> <br /> '''11.''' Derive a formula for $$\mathbf u^{\unicode[&quot;segoe ui symbol&quot;]{x2605}\unicode[&quot;segoe ui symbol&quot;]{x2605}}$$, the double bulk dual of $$\mathbf u$$, that uses only $$\operatorname{gr}(\mathbf u)$$, $$\operatorname{ag}(\mathbf u)$$, and the determinant of the metric tensor $$\mathfrak g$$.<br /> <br /> '''12.''' Assuming that the antivector basis elements are written in the same order as their vector complements, prove that the $$(n - 1)$$-th compound matrix $$C_{n - 1}(\mathbf m)$$ of a matrix $$\mathbf m$$ is always equal to the adjugate transpose of $$\mathbf m$$.<br /> <br /> '''13.''' Develop the two-dimensional projective exterior algebra having the basis elements $$\mathbf 1$$, $$\mathbf e_1$$, $$\mathbf e_2$$, and $${\large\unicode{x1D7D9}} = \mathbf e_{12}$$. Give the metric exomorphism for this algebra, give the homogeneous representation of a one-dimensional point and identify its bulk and weight, give the bulk dual and weight dual of a point, and give the geometric norm of a point.<br /> <br /> == Exercises for Chapter 3 ==<br /> <br /> '''1.''' Determine in what numbers of dimensions the reverse and antireverse operations are equivalent, each always producing the same results as the other.<br /> <br /> '''2.''' Derive a translation operator, not necessarily unitized, that moves a given 3D point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ to the origin.<br /> <br /> '''3.''' Show that the geometric constraint is closed under the geometric product. That is, if $$\mathbf a$$ and $$\mathbf b$$ both satisfy the geometric constraint, show that $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ also satisfies the geometric constraint.<br /> <br /> '''4.''' Derive a formula that gives the square root of the complement motor $$\mathbf Q$$ appearing in Equation (3.221).<br /> <br /> '''5.''' Formulate motors and flectors in the two-dimensional projective exterior algebra that models one-dimensional space. What geometric transformations does each type of operator perform?<br /> <br /> == Exercises for Chapter 4 ==<br /> <br /> '''1.''' In the 5D conformal geometric algebra modeling 3D space, show that $$\mathbf u^{\unicode[&quot;segoe ui symbol&quot;]{x2605}\unicode[&quot;segoe ui symbol&quot;]{x2605}} = -\mathbf u$$.<br /> <br /> '''2.''' Calculate the wedge product of two round points having the same center but different radii, and interpret the result.<br /> <br /> '''3.''' Let $$\mathbf s_1$$ and $$\mathbf s_2$$ be two intersecting unitized spheres that have real radii. Show that the dot product $$\mathbf s_1 \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf s_2$$ is equal to the product of the radii multiplied by the cosine of the exterior angle wherever the surfaces of the two spheres meet.<br /> <br /> '''4.''' Consider the 3D conformal geometric algebra modeling 1D space, where $$\mathbf e_2$$ represents the origin and $$\mathbf e_3$$ represents the point at infinity. Determine formulas for the join of two round points $$\mathbf a$$ and $$\mathbf b$$, the meet of two dipoles $$\mathbf d$$ and $$\mathbf f$$, the meet of a dipole $$\mathbf d$$ and a flat point $$\mathbf p$$, the weight expansion of a round point $$\mathbf a$$ onto a dipole $$\mathbf d$$, and the weight expansion of a round point $$\mathbf a$$ onto a flat point $$\mathbf p$$.<br /> <br /> == See Also ==<br /> <br /> * [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]<br /> * [[Contents]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Exercises&diff=486 2025-12-05T23:18:53Z

<p>Eric Lengyel: </p> <hr /> <div>These are exercises accompanying the book [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''Projective Geometric Algebra Illuminated'']. More will be posted over time.<br /> <br /> == Exercises for Chapter 2 ==<br /> <br /> '''1.''' Show that Equation (2.35) properly constructs a line containing two points $$\mathbf p$$ and $$\mathbf q$$ with non-unit weights by considering $$\mathbf p / p_w \wedge \mathbf q / q_w$$ and then scaling by $$p_wq_w$$.<br /> <br /> '''2.''' Let $$\mathbf u$$ be a basis element of the 4D projective algebra. Prove that if $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ and $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$, then it must also be true that $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ and $$\underline{\mathbf u} \vee \mathbf u = \mathbf 1$$. That is, show that right and left complements under the wedge product are also the right and left complements under the antiwedge product.<br /> <br /> '''3.''' Suppose that the 4D trivectors $$\mathbf g$$ and $$\mathbf h$$ represent parallel planes in 3D space. Show that the magnitude of the moment of $$\mathbf g \vee \mathbf h$$ is the distance between the planes multiplied by both their weights.<br /> <br /> '''4.''' Let $$\mathbf m$$ be a $$4 \times 4$$ matrix that performs a rotation about the $$z$$ axis in homogeneous coordinates. Calculate the $$16 \times 16$$ exomorphism matrix $$\mathbf M$$ corresponding to $$\mathbf m$$.<br /> <br /> '''5.''' Suppose that $$\mathbf G$$ is a metric exomorphism. Use the fact that $$\mathbf G$$ is an exomorphism to prove that the associated antimetric $$\mathbb G$$ must satisfy $$\mathbb G(\mathbf a \vee \mathbf b) = \mathbb G\mathbf a \vee \mathbb G\mathbf b$$ for any $$\mathbf a$$ and $$\mathbf b$$.<br /> <br /> '''6.''' Suppose that the metric tensor $$\mathfrak g$$ is invertible. Show that the wedge and antiwedge products satisfy the relationship $$\mathbf a \vee \mathbf b = (\mathbf a^\unicode[&quot;segoe ui symbol&quot;]{x2605} \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605})^\unicode[&quot;segoe ui symbol&quot;]{x2606}$$.<br /> <br /> '''7.''' Suppose that $$\mathbf a$$ and $$\mathbf b$$ are basis elements of an $$n$$-dimensional exterior algebra and $$\operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b) = n$$. Show that $$(\mathbf a \wedge \mathbf b)^\unicode[&quot;segoe ui symbol&quot;]{x2605} = \mathbf a^\unicode[&quot;segoe ui symbol&quot;]{x2605} \mathbin{\unicode{x25CF}} \mathbf b$$.<br /> <br /> '''8.''' Show that the geometric norm is idempotent. That is, show that $$\Vert \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}} \Vert = \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}}$$.<br /> <br /> '''9.''' Derive the relationship between left and right interior products shown in Equation (2.110).<br /> <br /> '''10.''' Derive Equation (2.159), which is the expansion analog of Equation (2.129).<br /> <br /> '''11.''' Derive a formula for $$\mathbf u^{\unicode[&quot;segoe ui symbol&quot;]{x2605}\unicode[&quot;segoe ui symbol&quot;]{x2605}}$$, the double bulk dual of $$\mathbf u$$, that uses only $$\operatorname{gr}(\mathbf u)$$, $$\operatorname{ag}(\mathbf u)$$, and the determinant of the metric tensor $$\mathfrak g$$.<br /> <br /> '''12.''' Assuming that the antivector basis elements are written in the same order as their vector complements, prove that the $$(n - 1)$$-th compound matrix $$C_{n - 1}(\mathbf m)$$ of a matrix $$\mathbf m$$ is always equal to the adjugate transpose of $$\mathbf m$$.<br /> <br /> '''13.''' Develop the two-dimensional projective exterior algebra having the basis elements $$\mathbf 1$$, $$\mathbf e_1$$, $$\mathbf e_2$$, and $${\large\unicode{x1D7D9}} = \mathbf e_{12}$$. Give the metric exomorphism for this algebra, give the homogeneous representation of a one-dimensional point and identify its bulk and weight, give the bulk dual and weight dual of a point, and give the geometric norm of a point.<br /> <br /> == Exercises for Chapter 3 ==<br /> <br /> '''1.''' Determine in what numbers of dimensions the reverse and antireverse operations are equivalent, each always producing the same results as the other.<br /> <br /> '''2.''' Derive a translation operator, not necessarily unitized, that moves a given 3D point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ to the origin.<br /> <br /> '''3.''' Show that the geometric constraint is closed under the geometric product. That is, if $$\mathbf a$$ and $$\mathbf b$$ both satisfy the geometric constraint, show that $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ also satisfies the geometric constraint.<br /> <br /> '''4.''' Derive a formula that gives the square root of the complement motor $$\mathbf Q$$ appearing in Equation (3.221).<br /> <br /> '''5.''' Formulate motors and flectors in the two-dimensional projective exterior algebra that models one-dimensional space. What geometric transformations does each type of operator perform?<br /> <br /> == Exercises for Chapter 4 ==<br /> <br /> '''1.''' In the 5D conformal geometric algebra modeling 3D space, show that $$\mathbf u^{\unicode[&quot;segoe ui symbol&quot;]{x2605}\unicode[&quot;segoe ui symbol&quot;]{x2605}} = -\mathbf u$$.<br /> <br /> '''2.''' Calculate the wedge product of two round points having the same center but different radii, and interpret the result.<br /> <br /> '''3.''' Consider the 3D conformal geometric algebra modeling 1D space, where $$\mathbf e_2$$ represents the origin and $$\mathbf e_3$$ represents the point at infinity. Determine formulas for the join of two round points $$\mathbf a$$ and $$\mathbf b$$, the meet of two dipoles $$\mathbf d$$ and $$\mathbf f$$, the meet of a dipole $$\mathbf d$$ and a flat point $$\mathbf p$$, the weight expansion of a round point $$\mathbf a$$ onto a dipole $$\mathbf d$$, and the weight expansion of a round point $$\mathbf a$$ onto a flat point $$\mathbf p$$.<br /> <br /> == See Also ==<br /> <br /> * [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]<br /> * [[Contents]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Exercises&diff=485 2025-12-05T07:04:57Z

<p>Eric Lengyel: </p> <hr /> <div>These are exercises accompanying the book [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''Projective Geometric Algebra Illuminated''].<br /> <br /> == Exercises for Chapter 2 ==<br /> <br /> '''1.''' Show that Equation (2.35) properly constructs a line containing two points $$\mathbf p$$ and $$\mathbf q$$ with non-unit weights by considering $$\mathbf p / p_w \wedge \mathbf q / q_w$$ and then scaling by $$p_wq_w$$.<br /> <br /> '''2.''' Let $$\mathbf u$$ be a basis element of the 4D projective algebra. Prove that if $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ and $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$, then it must also be true that $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ and $$\underline{\mathbf u} \vee \mathbf u = \mathbf 1$$. That is, show that right and left complements under the wedge product are also the right and left complements under the antiwedge product.<br /> <br /> '''3.''' Suppose that the 4D trivectors $$\mathbf g$$ and $$\mathbf h$$ represent parallel planes in 3D space. Show that the magnitude of the moment of $$\mathbf g \vee \mathbf h$$ is the distance between the planes multiplied by both their weights.<br /> <br /> '''4.''' Let $$\mathbf m$$ be a $$4 \times 4$$ matrix that performs a rotation about the $$z$$ axis in homogeneous coordinates. Calculate the $$16 \times 16$$ exomorphism matrix $$\mathbf M$$ corresponding to $$\mathbf m$$.<br /> <br /> '''5.''' Suppose that $$\mathbf G$$ is a metric exomorphism. Use the fact that $$\mathbf G$$ is an exomorphism to prove that the associated antimetric $$\mathbb G$$ must satisfy $$\mathbb G(\mathbf a \vee \mathbf b) = \mathbb G\mathbf a \vee \mathbb G\mathbf b$$ for any $$\mathbf a$$ and $$\mathbf b$$.<br /> <br /> '''6.''' Suppose that the metric tensor $$\mathfrak g$$ is invertible. Show that the wedge and antiwedge products satisfy the relationship $$\mathbf a \vee \mathbf b = (\mathbf a^\unicode[&quot;segoe ui symbol&quot;]{x2605} \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605})^\unicode[&quot;segoe ui symbol&quot;]{x2606}$$.<br /> <br /> '''7.''' Suppose that $$\mathbf a$$ and $$\mathbf b$$ are basis elements of an $$n$$-dimensional exterior algebra and $$\operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b) = n$$. Show that $$(\mathbf a \wedge \mathbf b)^\unicode[&quot;segoe ui symbol&quot;]{x2605} = \mathbf a^\unicode[&quot;segoe ui symbol&quot;]{x2605} \mathbin{\unicode{x25CF}} \mathbf b$$.<br /> <br /> '''8.''' Show that the geometric norm is idempotent. That is, show that $$\Vert \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}} \Vert = \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}}$$.<br /> <br /> '''9.''' Derive the relationship between left and right interior products shown in Equation (2.110).<br /> <br /> '''10.''' Derive Equation (2.159), which is the expansion analog of Equation (2.129).<br /> <br /> '''11.''' Derive a formula for $$\mathbf u^{\unicode[&quot;segoe ui symbol&quot;]{x2605}\unicode[&quot;segoe ui symbol&quot;]{x2605}}$$, the double bulk dual of $$\mathbf u$$, that uses only $$\operatorname{gr}(\mathbf u)$$, $$\operatorname{ag}(\mathbf u)$$, and the determinant of the metric tensor $$\mathfrak g$$.<br /> <br /> '''12.''' Assuming that the antivector basis elements are written in the same order as their vector complements, prove that the $$(n - 1)$$-th compound matrix $$C_{n - 1}(\mathbf m)$$ of a matrix $$\mathbf m$$ is always equal to the adjugate transpose of $$\mathbf m$$.<br /> <br /> '''13.''' Develop the two-dimensional projective exterior algebra having the basis elements $$\mathbf 1$$, $$\mathbf e_1$$, $$\mathbf e_2$$, and $${\large\unicode{x1D7D9}} = \mathbf e_{12}$$. Give the metric exomorphism for this algebra, give the homogeneous representation of a one-dimensional point and identify its bulk and weight, give the bulk dual and weight dual of a point, and give the geometric norm of a point.<br /> <br /> == Exercises for Chapter 3 ==<br /> <br /> '''1.''' Determine in what numbers of dimensions the reverse and antireverse operations are equivalent, each always producing the same results as the other.<br /> <br /> '''2.''' Derive a translation operator, not necessarily unitized, that moves a given 3D point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ to the origin.<br /> <br /> '''3.''' Show that the geometric constraint is closed under the geometric product. That is, if $$\mathbf a$$ and $$\mathbf b$$ both satisfy the geometric constraint, show that $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ also satisfies the geometric constraint.<br /> <br /> '''4.''' Derive a formula that gives the square root of the complement motor $$\mathbf Q$$ appearing in Equation (3.221).<br /> <br /> '''5.''' Formulate motors and flectors in the two-dimensional projective exterior algebra that models one-dimensional space. What geometric transformations does each type of operator perform?<br /> <br /> == See Also ==<br /> <br /> * [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]<br /> * [[Contents]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Exercises&diff=484 2025-12-05T06:53:24Z

<p>Eric Lengyel: </p> <hr /> <div>These are exercises accompanying the book [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''Projective Geometric Algebra Illuminated''].<br /> <br /> == Exercises for Chapter 2 ==<br /> <br /> '''1.''' Show that Equation (2.35) properly constructs a line containing two points $$\mathbf p$$ and $$\mathbf q$$ with non-unit weights by considering $$\mathbf p / p_w \wedge \mathbf q / q_w$$ and then scaling by $$p_wq_w$$.<br /> <br /> '''2.''' Let $$\mathbf u$$ be a basis element of the 4D projective algebra. Prove that if $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ and $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$, then it must also be true that $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ and $$\underline{\mathbf u} \vee \mathbf u = \mathbf 1$$. That is, show that right and left complements under the wedge product are also the right and left complements under the antiwedge product.<br /> <br /> '''3.''' Suppose that the 4D trivectors $$\mathbf g$$ and $$\mathbf h$$ represent parallel planes in 3D space. Show that the magnitude of the moment of $$\mathbf g \vee \mathbf h$$ is the distance between the planes multiplied by both their weights.<br /> <br /> '''4.''' Let $$\mathbf m$$ be a $$4 \times 4$$ matrix that performs a rotation about the $$z$$ axis in homogeneous coordinates. Calculate the $$16 \times 16$$ exomorphism matrix $$\mathbf M$$ corresponding to $$\mathbf m$$.<br /> <br /> '''5.''' Suppose that $$\mathbf G$$ is a metric exomorphism. Use the fact that $$\mathbf G$$ is an exomorphism to prove that the associated antimetric $$\mathbb G$$ must satisfy $$\mathbb G(\mathbf a \vee \mathbf b) = \mathbb G\mathbf a \vee \mathbb G\mathbf b$$ for any $$\mathbf a$$ and $$\mathbf b$$.<br /> <br /> '''6.''' Suppose that the metric tensor $$\mathfrak g$$ is invertible. Show that the wedge and antiwedge products satisfy the relationship $$\mathbf a \vee \mathbf b = (\mathbf a^\unicode[&quot;segoe ui symbol&quot;]{x2605} \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605})^\unicode[&quot;segoe ui symbol&quot;]{x2606}$$.<br /> <br /> '''7.''' Suppose that $$\mathbf a$$ and $$\mathbf b$$ are basis elements of an $$n$$-dimensional exterior algebra and $$\operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b) = n$$. Show that $$(\mathbf a \wedge \mathbf b)^\unicode[&quot;segoe ui symbol&quot;]{x2605} = \mathbf a^\unicode[&quot;segoe ui symbol&quot;]{x2605} \mathbin{\unicode{x25CF}} \mathbf b$$.<br /> <br /> '''8.''' Show that the geometric norm is idempotent. That is, show that $$\Vert \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}} \Vert = \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}}$$.<br /> <br /> '''9.''' Derive the relationship between left and right interior products shown in Equation (2.110).<br /> <br /> '''10.''' Derive Equation (2.159), which is the expansion analog of Equation (2.129).<br /> <br /> '''11.''' Derive a formula for $$\mathbf u^{\unicode[&quot;segoe ui symbol&quot;]{x2605}\unicode[&quot;segoe ui symbol&quot;]{x2605}}$$, the double bulk dual of $$\mathbf u$$, that uses only $$\operatorname{gr}(\mathbf u)$$, $$\operatorname{ag}(\mathbf u)$$, and the determinant of the metric tensor $$\mathfrak g$$.<br /> <br /> '''12.''' Assuming that the antivector basis elements are written in the same order as their vector complements, prove that the $$(n - 1)$$-th compound matrix $$C_{n - 1}(\mathbf m)$$ of a matrix $$\mathbf m$$ is always equal to the adjugate transpose of $$\mathbf m$$.<br /> <br /> '''13.''' Develop the two-dimensional projective exterior algebra having the basis elements $$\mathbf 1$$, $$\mathbf e_1$$, $$\mathbf e_2$$, and $${\large\unicode{x1D7D9}} = \mathbf e_{12}$$. Give the metric exomorphism for this algebra, give the homogeneous representation of a one-dimensional point and identify its bulk and weight, give the bulk dual and weight dual of a point, and give the geometric norm of a point.<br /> <br /> == Exercises for Chapter 3 ==<br /> <br /> '''1.''' Determine in what numbers of dimensions the reverse and antireverse operations are equivalent, each always producing the same results as the other.<br /> <br /> '''2.''' Derive a translation operator, not necessarily unitized, that moves a given 3D point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ to the origin.<br /> <br /> '''3.''' Show that the geometric constraint is closed under the geometric product. That is, if $$\mathbf a$$ and $$\mathbf b$$ both satisfy the geometric constraint, show that $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ also satisfies the geometric constraint.<br /> <br /> '''4.''' Derive a formula that gives the square root of the complement motor $$\mathbf Q$$ appearing in Equation (3.221).<br /> <br /> == See Also ==<br /> <br /> * [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]<br /> * [[Contents]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Exercises&diff=483 2025-12-05T06:41:41Z

<p>Eric Lengyel: </p> <hr /> <div>These are exercises accompanying the book [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''Projective Geometric Algebra Illuminated''].<br /> <br /> == Exercises for Chapter 2 ==<br /> <br /> '''1.''' Show that Equation (2.35) properly constructs a line containing two points $$\mathbf p$$ and $$\mathbf q$$ with non-unit weights by considering $$\mathbf p / p_w \wedge \mathbf q / q_w$$ and then scaling by $$p_wq_w$$.<br /> <br /> '''2.''' Let $$\mathbf u$$ be a basis element of the 4D projective algebra. Prove that if $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ and $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$, then it must also be true that $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ and $$\underline{\mathbf u} \vee \mathbf u = \mathbf 1$$. That is, show that right and left complements under the wedge product are also the right and left complements under the antiwedge product.<br /> <br /> '''3.''' Suppose that the 4D trivectors $$\mathbf g$$ and $$\mathbf h$$ represent parallel planes in 3D space. Show that the magnitude of the moment of $$\mathbf g \vee \mathbf h$$ is the distance between the planes multiplied by both their weights.<br /> <br /> '''4.''' Let $$\mathbf m$$ be a $$4 \times 4$$ matrix that performs a rotation about the $$z$$ axis in homogeneous coordinates. Calculate the $$16 \times 16$$ exomorphism matrix $$\mathbf M$$ corresponding to $$\mathbf m$$.<br /> <br /> '''5.''' Suppose that $$\mathbf G$$ is a metric exomorphism. Use the fact that $$\mathbf G$$ is an exomorphism to prove that the associated antimetric $$\mathbb G$$ must satisfy $$\mathbb G(\mathbf a \vee \mathbf b) = \mathbb G\mathbf a \vee \mathbb G\mathbf b$$ for any $$\mathbf a$$ and $$\mathbf b$$.<br /> <br /> '''6.''' Suppose that the metric tensor $$\mathfrak g$$ is invertible. Show that the wedge and antiwedge products satisfy the relationship $$\mathbf a \vee \mathbf b = (\mathbf a^\unicode[&quot;segoe ui symbol&quot;]{x2605} \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605})^\unicode[&quot;segoe ui symbol&quot;]{x2606}$$.<br /> <br /> '''7.''' Suppose that $$\mathbf a$$ and $$\mathbf b$$ are basis elements of an $$n$$-dimensional exterior algebra and $$\operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b) = n$$. Show that $$(\mathbf a \wedge \mathbf b)^\unicode[&quot;segoe ui symbol&quot;]{x2605} = \mathbf a^\unicode[&quot;segoe ui symbol&quot;]{x2605} \mathbin{\unicode{x25CF}} \mathbf b$$.<br /> <br /> '''8.''' Show that the geometric norm is idempotent. That is, show that $$\Vert \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}} \Vert = \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}}$$.<br /> <br /> '''9.''' Derive the relationship between left and right interior products shown in Equation (2.110).<br /> <br /> '''10.''' Derive Equation (2.159), which is the expansion analog of Equation (2.129).<br /> <br /> '''11.''' Derive a formula for $$\mathbf u^{\unicode[&quot;segoe ui symbol&quot;]{x2605}\unicode[&quot;segoe ui symbol&quot;]{x2605}}$$, the double bulk dual of $$\mathbf u$$, that uses only $$\operatorname{gr}(\mathbf u)$$, $$\operatorname{ag}(\mathbf u)$$, and the determinant of the metric tensor $$\mathfrak g$$.<br /> <br /> '''12.''' Assuming that the antivector basis elements are written in the same order as their vector complements, prove that the $$(n - 1)$$-th compound matrix $$C_{n - 1}(\mathbf m)$$ of a matrix $$\mathbf m$$ is always equal to the adjugate transpose of $$\mathbf m$$.<br /> <br /> == Exercises for Chapter 3 ==<br /> <br /> '''1.''' Determine in what numbers of dimensions the reverse and antireverse operations are equivalent, each always producing the same results as the other.<br /> <br /> '''2.''' Derive a translation operator, not necessarily unitized, that moves a given 3D point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ to the origin.<br /> <br /> '''3.''' Show that the geometric constraint is closed under the geometric product. That is, if $$\mathbf a$$ and $$\mathbf b$$ both satisfy the geometric constraint, show that $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ also satisfies the geometric constraint.<br /> <br /> '''4.''' Derive a formula that gives the square root of the complement motor $$\mathbf Q$$ appearing in Equation (3.221).<br /> <br /> == See Also ==<br /> <br /> * [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]<br /> * [[Contents]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Errata&diff=482 2025-11-09T07:17:51Z

<p>Eric Lengyel: </p> <hr /> <div>__NOTOC__<br /> This is the corrigenda and errata page for the ''Projective Geometric Algebra Illuminated''. To find out which printing you have, look on the copyright page. If you believe you have found an error that should be added to this list, please email the author (lengyel@terathon.com).<br /> <br /> == Fifth Printing and Earlier ==<br /> <br /> * '''Page 142.''' In Table 3.8, the inversion formula for a line should have a minus sign in the $$\mathbf e_{12}$$ term similar to the other terms.<br /> <br /> * '''Page 174.''' In Equation (3.219), $$Q_{vw}$$ should be the coefficient of the scalar term $$\mathbf 1$$, and $$Q_{mw}$$ should be the coefficient of the antiscalar term $${\large\unicode{x1D7D9}}$$.<br /> <br /> == Fourth Printing and Earlier ==<br /> <br /> * '''Page 12.''' Right before Equation (1.25), the expression for the magnitude of the projection of $$\mathbf u$$ onto the line should be $$|\mathbf u \cdot \mathbf v| / \Vert\mathbf v\Vert$$.<br /> <br /> == Third Printing and Earlier ==<br /> <br /> * '''Page 70.''' In the line following Equation (2.71), the product of the bulks should use the antiwedge product and appear as $$\mathbf a_\unicode[&quot;segoe ui symbol&quot;]{x25CF} \vee \mathbf b_\unicode[&quot;segoe ui symbol&quot;]{x25CF}$$.<br /> <br /> == Second Printing and Earlier ==<br /> <br /> * '''Page 248.''' In Equation (5.27), the operator $$\mathbf D$$ should be dualized as $$\mathbf D^\unicode[&quot;segoe ui symbol&quot;]{x2605}$$. It should also be dualized in the first factor of the sandwich product in the sentence that follows this equation.<br /> <br /> == First Printing ==<br /> <br /> * '''Page 84.''' In Table 2.16, both bulk duals of $$\mathbf e_4$$ should be 0. The table should look like the one on the wiki page for [[duals]].<br /> <br /> * '''Page 156.''' The number of multiply-adds required for transforming a line with a motor did not include the work needed to calculate $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$. Each cross product adds 6 multiply-adds, so the correct total for transforming a line is 54.<br /> <br /> * '''Page 156.''' Similarly, the number of multiply-adds required for transforming a plane with a motor did not include the work needed to calculate $$\mathbf a$$. The correct total for transforming a plane is 35.<br /> <br /> == See Also ==<br /> <br /> * [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]<br /> * [[Contents]]<br /> * [[Exercises]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transformation_groups&diff=481 2025-11-09T07:12:22Z

<p>Eric Lengyel: </p> <hr /> <div>In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, every Euclidean isometry of 3D space can be represented by a [[motor]] $$\mathbf Q$$ of the form<br /> <br /> :$$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$<br /> <br /> or by a [[flector]] $$\mathbf F$$ of the form<br /> <br /> :$$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$ .<br /> <br /> Under the [[geometric antiproduct]] $$\unicode{x27C7}$$, arbitrary products of these operators form the Euclidean group $$\mathrm E(3)$$ with $${\large\unicode{x1D7D9}}$$ as the identity, and they covariantly transform any object $$\mathbf x$$ in the algebra through the sandwich products $$\mathbf x' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$ and $$\mathbf x' = -\mathbf F \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}}$$.<br /> <br /> Symmetrically, every complement Euclidean isometry of 3D space can be represented by a [[complement motor]] $$\mathbf Q$$ of the form<br /> <br /> :$$\mathbf Q = Q_{vx} \mathbf e_{23} + Q_{vy} \mathbf e_{31} + Q_{vz} \mathbf e_{12} - Q_{vw} \mathbf 1 + Q_{mx} \mathbf e_{41} + Q_{my} \mathbf e_{42} + Q_{mz} \mathbf e_{43} - Q_{mw} {\large\unicode{x1d7d9}}$$<br /> <br /> or by a [[complement flector]] $$\mathbf F$$ of the form<br /> <br /> :$$\mathbf F = F_{px} \mathbf e_{423} + F_{py} \mathbf e_{431} + F_{pz} \mathbf e_{412} + F_{pw} \mathbf e_{321} - F_{gx} \mathbf e_1 - F_{gy} \mathbf e_2 - F_{gz} \mathbf e_3 - F_{gw} \mathbf e_4$$ .<br /> <br /> Under the [[geometric product]] $$\unicode{x27D1}$$, arbitrary products of these operators form the complement Euclidean group $$\overline{\mathrm E}(3)$$ with $$\mathbf 1$$ as the identity, and they covariantly transform any object $$\mathbf x$$ in the algebra through the sandwich products $$\mathbf x' = \mathbf Q \mathbin{\unicode{x27D1}} \mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde Q}$$ and $$\mathbf x' = -\mathbf F \mathbin{\unicode{x27D1}} \mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde F}$$.<br /> <br /> The [[geometric product]] corresponds to transform composition in the group $$\overline{\mathrm E}(3)$$, and the [[geometric antiproduct]] corresponds to transform composition in the group $$\mathrm E(3)$$. [[Reflections]] across planes are represented by [[antivectors]] (having [[antigrade]] one), and they meet at lower-dimensional invariants under the geometric antiproduct. Symmetrically, [[complement reflections]] across points are represented by [[vectors]] (having [[grade]] one), and they join at higher-dimensional invariants under the geometric product. A sandwich product $$\mathbf Q \mathbin{\unicode{x27D1}} \mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde Q}$$ transforms the space of $$\mathbf x$$ with an element of $$\overline{\mathrm E}(3)$$, and it transforms the antispace of $$\mathbf x$$ with the complementary element of $$\mathrm E(3)$$. Symmetrically, a sandwich product $$\mathbf Q \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$ transforms the space of $$\mathbf x$$ with an element of $$\mathrm E(3)$$, and it transforms the antispace of $$\mathbf x$$ with the complementary element of $$\overline{\mathrm E}(3)$$.<br /> <br /> The groups $$\mathrm E(n)$$ and $$\overline{\mathrm E}(n)$$ are isomorphic, and they each contain the orthogonal group $$\mathrm O(n)$$ as a common subgroup. The [[complement]] operation provides a two-way mapping between transforms associated with members of $$\mathrm E(n)$$ and $$\overline{\mathrm E}(n)$$. The groups $$\mathrm E(n)$$ and $$\overline{\mathrm E}(n)$$ have a number of subgroups, and the hierarchical relationships among them are shown in the figure below. In particular, the Euclidean group $$\mathrm E(n)$$ contains the special Euclidean subgroup $$\mathrm{SE}(n)$$ consisting of all combinations of ordinary [[rotations]] and [[translations]], which are covered by the elements of $$\mathcal G_{n,0,1}$$ having even antigrade. Correspondingly, the complement Euclidean group $$\overline{\mathrm E}(n)$$ contains the complement special Euclidean subgroup $$\mathrm S\overline{\mathrm E}(n)$$ consisting of all combinations of [[complement rotations]] and [[complement translations]], which are covered by the elements of $$\mathcal G_{n,0,1}$$ having even grade. The subgroups $$\mathrm{SE}(n)$$ and $$\mathrm S\overline{\mathrm E}(n)$$ further contain translation subgroups $$\mathrm T(n)$$ and $$\overline{\mathrm T}(n)$$, respectively.<br /> <br /> [[Image:Groups.svg|800px]]<br /> <br /> Transforms about invariants containing the origin are the same in both $$\mathrm E(n)$$ and $$\overline{\mathrm E}(n)$$, and they constitute the common subgroup $$\mathrm O(n)$$. Every member of $$\mathrm O(n)$$ has a representation that transforms elements with the geometric product and a complementary representation that transforms elements with the geometric antiproduct. For example, conventional [[quaternions]] $$\mathbf q$$ have two representations, one that transforms any object $$\mathbf x$$ through the sandwich product $$\mathbf x' = \mathbf q \mathbin{\unicode{x27D1}} \mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde q}$$ and another that transforms any object $$\mathbf x$$ through the sandwich product $$\mathbf x' = \mathbf q \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{q}}}$$.<br /> <br /> In terms of matrix multiplication, a general element of the group $$\mathrm E(n)$$ transforms a point by multiplying on the left by an $$(n + 1) \times (n + 1)$$ matrix of the form<br /> <br /> :$$\begin{bmatrix} \mathbf M_{n \times n} &amp; \boldsymbol \tau_{n \times 1} \\ \mathbf 0_{1 \times n} &amp; 1 \end{bmatrix}$$ ,<br /> <br /> where the $$n \times n$$ submatrix $$\mathbf M$$ is orthogonal. A general element of the corresponding group $$\overline{\mathrm E}(n)$$ transforms points with matrices of the form<br /> <br /> :$$\begin{bmatrix} \mathbf M_{n \times n} &amp; \mathbf 0_{n \times 1} \\ \boldsymbol \tau_{1 \times n} &amp; 1 \end{bmatrix}$$ .<br /> <br /> In the special subgroups $$\mathrm{SE}(n)$$ and $$\mathrm S\overline{\mathrm E}(n)$$, the submatrix $$\mathbf M$$ has a determinant of +1. In the translation subgroups $$\mathrm T(n)$$ and $$\overline{\mathrm T}(n)$$, $$\mathbf M$$ is the identity matrix. Finally, when $$\boldsymbol \tau = \mathbf 0$$, the two matrices above have the same form and belong to $$\mathrm O(n)$$.<br /> <br /> The isomorphic mapping between $$\mathrm E(n)$$ and $$\overline{\mathrm E}(n)$$ is given by the inverse transpose operation on the matrix representatives. That is, if $$\mathbf M$$ is an $$(n + 1) \times (n + 1)$$ matrix representing an element of $$\mathrm E(n)$$, then the corresponding element of $$\overline{\mathrm E}(n)$$ is given by $$(\mathbf M^{-1})^{\text T}$$. Of course, this operation is an involution, and the mapping works both ways.<br /> <br /> == In the Book ==<br /> <br /> * Transformation groups are discussed in Section 3.9.2.<br /> <br /> == See Also ==<br /> <br /> * [[Motor]]<br /> * [[Flector]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Errata&diff=480 2025-10-24T03:30:11Z

<p>Eric Lengyel: </p> <hr /> <div>__NOTOC__<br /> This is the corrigenda and errata page for the ''Projective Geometric Algebra Illuminated''. To find out which printing you have, look on the copyright page. If you believe you have found an error that should be added to this list, please email the author (lengyel@terathon.com).<br /> <br /> == Fourth Printing and Earlier ==<br /> <br /> * '''Page 12.''' Right before Equation (1.25), the expression for the magnitude of the projection of $$\mathbf u$$ onto the line should be $$|\mathbf u \cdot \mathbf v| / \Vert\mathbf v\Vert$$.<br /> <br /> == Third Printing and Earlier ==<br /> <br /> * '''Page 70.''' In the line following Equation (2.71), the product of the bulks should use the antiwedge product and appear as $$\mathbf a_\unicode[&quot;segoe ui symbol&quot;]{x25CF} \vee \mathbf b_\unicode[&quot;segoe ui symbol&quot;]{x25CF}$$.<br /> <br /> == Second Printing and Earlier ==<br /> <br /> * '''Page 248.''' In Equation (5.27), the operator $$\mathbf D$$ should be dualized as $$\mathbf D^\unicode[&quot;segoe ui symbol&quot;]{x2605}$$. It should also be dualized in the first factor of the sandwich product in the sentence that follows this equation.<br /> <br /> == First Printing ==<br /> <br /> * '''Page 84.''' In Table 2.16, both bulk duals of $$\mathbf e_4$$ should be 0. The table should look like the one on the wiki page for [[duals]].<br /> <br /> * '''Page 156.''' The number of multiply-adds required for transforming a line with a motor did not include the work needed to calculate $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$. Each cross product adds 6 multiply-adds, so the correct total for transforming a line is 54.<br /> <br /> * '''Page 156.''' Similarly, the number of multiply-adds required for transforming a plane with a motor did not include the work needed to calculate $$\mathbf a$$. The correct total for transforming a plane is 35.<br /> <br /> == See Also ==<br /> <br /> * [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]<br /> * [[Contents]]<br /> * [[Exercises]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Errata&diff=479 2025-06-28T07:47:15Z

<p>Eric Lengyel: /* See Also */</p> <hr /> <div>__NOTOC__<br /> This is the errata page for the ''Projective Geometric Algebra Illuminated''. To find out which printing you have, look on the copyright page. If you believe you have found an error that should be added to this list, please email the author (lengyel@terathon.com).<br /> <br /> == Fourth Printing and Earlier ==<br /> <br /> * '''Page 12.''' Right before Equation (1.25), the expression for the magnitude of the projection of $$\mathbf u$$ onto the line should be $$|\mathbf u \cdot \mathbf v| / \Vert\mathbf v\Vert$$.<br /> <br /> == Third Printing and Earlier ==<br /> <br /> * '''Page 70.''' In the line following Equation (2.71), the product of the bulks should use the antiwedge product and appear as $$\mathbf a_\unicode[&quot;segoe ui symbol&quot;]{x25CF} \vee \mathbf b_\unicode[&quot;segoe ui symbol&quot;]{x25CF}$$.<br /> <br /> == Second Printing and Earlier ==<br /> <br /> * '''Page 248.''' In Equation (5.27), the operator $$\mathbf D$$ should be dualized as $$\mathbf D^\unicode[&quot;segoe ui symbol&quot;]{x2605}$$. It should also be dualized in the first factor of the sandwich product in the sentence that follows this equation.<br /> <br /> == First Printing ==<br /> <br /> * '''Page 84.''' In Table 2.16, both bulk duals of $$\mathbf e_4$$ should be 0. The table should look like the one on the wiki page for [[duals]].<br /> <br /> * '''Page 156.''' The number of multiply-adds required for transforming a line with a motor did not include the work needed to calculate $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$. Each cross product adds 6 multiply-adds, so the correct total for transforming a line is 54.<br /> <br /> * '''Page 156.''' Similarly, the number of multiply-adds required for transforming a plane with a motor did not include the work needed to calculate $$\mathbf a$$. The correct total for transforming a plane is 35.<br /> <br /> == See Also ==<br /> <br /> * [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]<br /> * [[Contents]]<br /> * [[Exercises]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Exercises&diff=478 2025-06-28T07:46:58Z

<p>Eric Lengyel: /* See Also */</p> <hr /> <div>These are exercises accompanying the book [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''Projective Geometric Algebra Illuminated''].<br /> <br /> == Exercises for Chapter 2 ==<br /> <br /> '''1.''' Show that Equation (2.35) properly constructs a line containing two points $$\mathbf p$$ and $$\mathbf q$$ with non-unit weights by considering $$\mathbf p / p_w \wedge \mathbf q / q_w$$ and then scaling by $$p_wq_w$$.<br /> <br /> '''2.''' Let $$\mathbf u$$ be a basis element of the 4D projective algebra. Prove that if $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ and $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$, then it must also be true that $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ and $$\underline{\mathbf u} \vee \mathbf u = \mathbf 1$$. That is, show that right and left complements under the wedge product are also the right and left complements under the antiwedge product.<br /> <br /> '''3.''' Suppose that the 4D trivectors $$\mathbf g$$ and $$\mathbf h$$ represent parallel planes in 3D space. Show that the magnitude of the moment of $$\mathbf g \vee \mathbf h$$ is the distance between the planes multiplied by both their weights.<br /> <br /> '''4.''' Let $$\mathbf m$$ be a $$4 \times 4$$ matrix that performs a rotation about the $$z$$ axis in homogeneous coordinates. Calculate the $$16 \times 16$$ exomorphism matrix $$\mathbf M$$ corresponding to $$\mathbf m$$.<br /> <br /> '''5.''' Suppose that $$\mathbf G$$ is a metric exomorphism. Use the fact that $$\mathbf G$$ is an exomorphism to prove that the associated antimetric $$\mathbb G$$ must satisfy $$\mathbb G(\mathbf a \vee \mathbf b) = \mathbb G\mathbf a \vee \mathbb G\mathbf b$$ for any $$\mathbf a$$ and $$\mathbf b$$.<br /> <br /> '''6.''' Suppose that the metric tensor $$\mathfrak g$$ is invertible. Show that the wedge and antiwedge products satisfy the relationship $$\mathbf a \vee \mathbf b = (\mathbf a^\unicode[&quot;segoe ui symbol&quot;]{x2605} \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605})^\unicode[&quot;segoe ui symbol&quot;]{x2606}$$.<br /> <br /> '''7.''' Suppose that $$\mathbf a$$ and $$\mathbf b$$ are basis elements of an $$n$$-dimensional exterior algebra and $$\operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b) = n$$. Show that $$(\mathbf a \wedge \mathbf b)^\unicode[&quot;segoe ui symbol&quot;]{x2605} = \mathbf a^\unicode[&quot;segoe ui symbol&quot;]{x2605} \mathbin{\unicode{x25CF}} \mathbf b$$.<br /> <br /> '''8.''' Show that the geometric norm is idempotent. That is, show that $$\Vert \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}} \Vert = \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}}$$.<br /> <br /> '''9.''' Derive the relationship between left and right interior products shown in Equation (2.110).<br /> <br /> '''10.''' Derive Equation (2.159), which is the expansion analog of Equation (2.129).<br /> <br /> '''11.''' Derive a formula for $$\mathbf u^{\unicode[&quot;segoe ui symbol&quot;]{x2605}\unicode[&quot;segoe ui symbol&quot;]{x2605}}$$, the double bulk dual of $$\mathbf u$$, that uses only $$\operatorname{gr}(\mathbf u)$$, $$\operatorname{ag}(\mathbf u)$$, and the determinant of the metric tensor $$\mathfrak g$$.<br /> <br /> '''12.''' Assuming that the antivector basis elements are written in the same order as their vector complements, prove that the $$(n - 1)$$-th compound matrix $$C_{n - 1}(\mathbf m)$$ of a matrix $$\mathbf m$$ is always equal to the adjugate transpose of $$\mathbf m$$.<br /> <br /> == See Also ==<br /> <br /> * [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]<br /> * [[Contents]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Contents&diff=477 2025-06-28T07:46:45Z

<p>Eric Lengyel: /* See Also */</p> <hr /> <div>This is the table of contents for ''Projective Geometric Algebra Illuminated''.<br /> <br /> * Preface<br /> * 1 Conventional Mathematics<br /> ** 1.1 The Cross Product<br /> ** 1.2 Homogeneous Coordinates<br /> ** 1.3 Lines and Planes<br /> *** 1.3.1 Parametric Forms<br /> *** 1.3.2 Implicit Forms<br /> *** 1.3.3 Distance Between a Point and a Line<br /> *** 1.3.4 Intersection of a Line and a Plane<br /> *** 1.3.5 Intersection of Multiple Planes<br /> *** 1.3.6 Reflection Across a Plane<br /> *** 1.3.7 Homogeneous Formulas<br /> *** 1.3.8 Plane Transformation<br /> *** 1.3.9 Line Transformation<br /> ** 1.4 Quaternions<br /> *** 1.4.1 Quaternion Fundamentals<br /> *** 1.4.2 Rotations With Quaternions<br /> *** 1.4.3 Interpolating Quaternions<br /> *** 1.4.4 Dual Quaternions<br /> **Historical Remarks<br /> * 2 Flat Projective Geometry<br /> ** 2.1 Algebraic Structure<br /> *** 2.1.1 The Wedge Product<br /> *** 2.1.2 Bivectors<br /> *** 2.1.3 Trivectors<br /> *** 2.1.4 Basis Elements<br /> ** 2.2 Complements<br /> ** 2.3 Antiproducts<br /> ** 2.4 3D Flat Geometry<br /> *** 2.4.1 Points<br /> *** 2.4.2 Lines<br /> *** 2.4.3 Planes<br /> ** 2.5 Join and Meet<br /> ** 2.6 Duality<br /> ** 2.7 Exomorphisms<br /> ** 2.8 Metric Transformations<br /> *** 2.8.1 The Metric<br /> *** 2.8.2 The Antimetric<br /> *** 2.8.3 Bulk and Weight<br /> *** 2.8.4 Attitude<br /> ** 2.9 Inner Products<br /> ** 2.10 Norms<br /> *** 2.10.1 Bulk and Weight Norms<br /> *** 2.10.2 Unitization<br /> *** 2.10.3 The Geometric Norm<br /> ** 2.11 Euclidean Distances<br /> ** 2.12 Duals<br /> ** 2.13 Interior Products<br /> *** 2.13.1 Contractions<br /> *** 2.13.2 Projection and Rejection<br /> *** 2.13.3 Euclidean Angles<br /> *** 2.13.4 Parametric Forms<br /> *** 2.13.5 Expansions<br /> *** 2.13.6 Geometric Projection<br /> ** 2.14 2D Flat Geometry<br /> ** 2.15 Dependencies<br /> ** Historical Remarks<br /> * 3 Rigid Transformations<br /> ** 3.1 The Geometric Product<br /> ** 3.2 Dual Numbers<br /> ** 3.3 Reflection and Rotation<br /> ** 3.4 Reversion<br /> *** 3.4.1 Reverse and Antireverse<br /> *** 3.4.2 Dual Identities<br /> *** 3.4.3 Geometric Constraint<br /> ** 3.5 Euclidean Isometries<br /> *** 3.5.1 Reflection<br /> *** 3.5.2 Rotation<br /> *** 3.5.3 Translation<br /> *** 3.5.4 Inversion<br /> *** 3.5.5 Transflection<br /> ** 3.6 Motors<br /> *** 3.6.1 Motion Operator<br /> *** 3.6.2 Parameterization<br /> *** 3.6.3 Line to Line Motion<br /> *** 3.6.4 Matrix Conversion<br /> *** 3.6.5 Implementation<br /> ** 3.7 Flectors<br /> *** 3.7.1 Reflection Operator<br /> *** 3.7.2 Matrix Conversion<br /> *** 3.7.3 Implementation<br /> ** 3.8 2D Rigid Transformations<br /> ** 3.9 Operator Duality<br /> *** 3.9.1 Complement Isometries<br /> *** 3.9.2 Transformation Groups<br /> *** 3.9.3 Quaternions Revisited<br /> ** Historical Remarks<br /> * 4 Round Projective Geometry<br /> ** 4.1 Construction<br /> ** 4.2 3D Round Geometry<br /> *** 4.2.1 Representations<br /> *** 4.2.2 Duals<br /> *** 4.2.3 Carriers<br /> *** 4.2.4 Centers<br /> *** 4.2.5 Containers<br /> *** 4.2.6 Partners<br /> *** 4.2.7 Attitude<br /> ** 4.3 Norms<br /> ** 4.4 Alignment<br /> ** 4.5 Dot Products<br /> *** 4.5.1 Round Points<br /> *** 4.5.2 Spheres<br /> *** 4.5.3 Partners<br /> *** 4.5.4 Conjugates<br /> ** 4.6 Containment<br /> ** 4.7 Join and Meet<br /> ** 4.8 Expansions<br /> ** 4.9 2D Round Geometry<br /> ** 4.10 Degrees of Freedom<br /> * 5 Conformal Transformations<br /> ** 5.1 Generalized Operators<br /> *** 5.1.1 Rigid Transformations<br /> *** 5.1.2 Sphere Inversion<br /> *** 5.1.3 Circle Rotation<br /> ** 5.2 Dilation<br /> ** 5.3 Duals and Complements<br /> ** 5.4 2D Conformal Transformations<br /> * A Multiplication Tables<br /> * B Geometric Properties<br /> * C Notation Reference<br /> * Bibliography<br /> * Index<br /> <br /> == See Also ==<br /> <br /> * [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]<br /> * [[Exercises]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Exercises&diff=476 2025-06-28T07:46:22Z

<p>Eric Lengyel: </p> <hr /> <div>These are exercises accompanying the book [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''Projective Geometric Algebra Illuminated''].<br /> <br /> == Exercises for Chapter 2 ==<br /> <br /> '''1.''' Show that Equation (2.35) properly constructs a line containing two points $$\mathbf p$$ and $$\mathbf q$$ with non-unit weights by considering $$\mathbf p / p_w \wedge \mathbf q / q_w$$ and then scaling by $$p_wq_w$$.<br /> <br /> '''2.''' Let $$\mathbf u$$ be a basis element of the 4D projective algebra. Prove that if $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ and $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$, then it must also be true that $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ and $$\underline{\mathbf u} \vee \mathbf u = \mathbf 1$$. That is, show that right and left complements under the wedge product are also the right and left complements under the antiwedge product.<br /> <br /> '''3.''' Suppose that the 4D trivectors $$\mathbf g$$ and $$\mathbf h$$ represent parallel planes in 3D space. Show that the magnitude of the moment of $$\mathbf g \vee \mathbf h$$ is the distance between the planes multiplied by both their weights.<br /> <br /> '''4.''' Let $$\mathbf m$$ be a $$4 \times 4$$ matrix that performs a rotation about the $$z$$ axis in homogeneous coordinates. Calculate the $$16 \times 16$$ exomorphism matrix $$\mathbf M$$ corresponding to $$\mathbf m$$.<br /> <br /> '''5.''' Suppose that $$\mathbf G$$ is a metric exomorphism. Use the fact that $$\mathbf G$$ is an exomorphism to prove that the associated antimetric $$\mathbb G$$ must satisfy $$\mathbb G(\mathbf a \vee \mathbf b) = \mathbb G\mathbf a \vee \mathbb G\mathbf b$$ for any $$\mathbf a$$ and $$\mathbf b$$.<br /> <br /> '''6.''' Suppose that the metric tensor $$\mathfrak g$$ is invertible. Show that the wedge and antiwedge products satisfy the relationship $$\mathbf a \vee \mathbf b = (\mathbf a^\unicode[&quot;segoe ui symbol&quot;]{x2605} \wedge \mathbf b^\unicode[&quot;segoe ui symbol&quot;]{x2605})^\unicode[&quot;segoe ui symbol&quot;]{x2606}$$.<br /> <br /> '''7.''' Suppose that $$\mathbf a$$ and $$\mathbf b$$ are basis elements of an $$n$$-dimensional exterior algebra and $$\operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b) = n$$. Show that $$(\mathbf a \wedge \mathbf b)^\unicode[&quot;segoe ui symbol&quot;]{x2605} = \mathbf a^\unicode[&quot;segoe ui symbol&quot;]{x2605} \mathbin{\unicode{x25CF}} \mathbf b$$.<br /> <br /> '''8.''' Show that the geometric norm is idempotent. That is, show that $$\Vert \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}} \Vert = \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}}$$.<br /> <br /> '''9.''' Derive the relationship between left and right interior products shown in Equation (2.110).<br /> <br /> '''10.''' Derive Equation (2.159), which is the expansion analog of Equation (2.129).<br /> <br /> '''11.''' Derive a formula for $$\mathbf u^{\unicode[&quot;segoe ui symbol&quot;]{x2605}\unicode[&quot;segoe ui symbol&quot;]{x2605}}$$, the double bulk dual of $$\mathbf u$$, that uses only $$\operatorname{gr}(\mathbf u)$$, $$\operatorname{ag}(\mathbf u)$$, and the determinant of the metric tensor $$\mathfrak g$$.<br /> <br /> '''12.''' Assuming that the antivector basis elements are written in the same order as their vector complements, prove that the $$(n - 1)$$-th compound matrix $$C_{n - 1}(\mathbf m)$$ of a matrix $$\mathbf m$$ is always equal to the adjugate transpose of $$\mathbf m$$.<br /> <br /> == See Also ==<br /> <br /> * [[Contents]]<br /> * [[Errata]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Errata&diff=475 2025-06-28T07:46:04Z

<p>Eric Lengyel: </p> <hr /> <div>__NOTOC__<br /> This is the errata page for the ''Projective Geometric Algebra Illuminated''. To find out which printing you have, look on the copyright page. If you believe you have found an error that should be added to this list, please email the author (lengyel@terathon.com).<br /> <br /> == Fourth Printing and Earlier ==<br /> <br /> * '''Page 12.''' Right before Equation (1.25), the expression for the magnitude of the projection of $$\mathbf u$$ onto the line should be $$|\mathbf u \cdot \mathbf v| / \Vert\mathbf v\Vert$$.<br /> <br /> == Third Printing and Earlier ==<br /> <br /> * '''Page 70.''' In the line following Equation (2.71), the product of the bulks should use the antiwedge product and appear as $$\mathbf a_\unicode[&quot;segoe ui symbol&quot;]{x25CF} \vee \mathbf b_\unicode[&quot;segoe ui symbol&quot;]{x25CF}$$.<br /> <br /> == Second Printing and Earlier ==<br /> <br /> * '''Page 248.''' In Equation (5.27), the operator $$\mathbf D$$ should be dualized as $$\mathbf D^\unicode[&quot;segoe ui symbol&quot;]{x2605}$$. It should also be dualized in the first factor of the sandwich product in the sentence that follows this equation.<br /> <br /> == First Printing ==<br /> <br /> * '''Page 84.''' In Table 2.16, both bulk duals of $$\mathbf e_4$$ should be 0. The table should look like the one on the wiki page for [[duals]].<br /> <br /> * '''Page 156.''' The number of multiply-adds required for transforming a line with a motor did not include the work needed to calculate $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$. Each cross product adds 6 multiply-adds, so the correct total for transforming a line is 54.<br /> <br /> * '''Page 156.''' Similarly, the number of multiply-adds required for transforming a plane with a motor did not include the work needed to calculate $$\mathbf a$$. The correct total for transforming a plane is 35.<br /> <br /> == See Also ==<br /> <br /> * [[Contents]]<br /> * [[Exercises]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Errata&diff=474 2025-06-28T07:45:47Z

<p>Eric Lengyel: </p> <hr /> <div>__NOTOC__<br /> This is the errata page for the ''Projective Geometric Algebra Illuminated''. To find out which printing you have, look on the copyright page. If you believe you have found an error that should be added to this list, please email the author (lengyel@terathon.com).<br /> <br /> == Fourth Printing and Earlier ==<br /> <br /> * '''Page 12.''' Right before Equation (1.25), the expression for the magnitude of the projection of $$\mathbf u$$ onto the line should be $$|\mathbf u \cdot \mathbf v| / \Vert\mathbf v\Vert$$.<br /> <br /> == Third Printing and Earlier ==<br /> <br /> * '''Page 70.''' In the line following Equation (2.71), the product of the bulks should use the antiwedge product and appear as $$\mathbf a_\unicode[&quot;segoe ui symbol&quot;]{x25CF} \vee \mathbf b_\unicode[&quot;segoe ui symbol&quot;]{x25CF}$$.<br /> <br /> == Second Printing and Earlier ==<br /> <br /> * '''Page 248.''' In Equation (5.27), the operator $$\mathbf D$$ should be dualized as $$\mathbf D^\unicode[&quot;segoe ui symbol&quot;]{x2605}$$. It should also be dualized in the first factor of the sandwich product in the sentence that follows this equation.<br /> <br /> == First Printing ==<br /> <br /> * '''Page 84.''' In Table 2.16, both bulk duals of $$\mathbf e_4$$ should be 0. The table should look like the one on the wiki page for [[duals]].<br /> <br /> * '''Page 156.''' The number of multiply-adds required for transforming a line with a motor did not include the work needed to calculate $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$. Each cross product adds 6 multiply-adds, so the correct total for transforming a line is 54.<br /> <br /> * '''Page 156.''' Similarly, the number of multiply-adds required for transforming a plane with a motor did not include the work needed to calculate $$\mathbf a$$. The correct total for transforming a plane is 35.</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Main_Page&diff=473 2025-06-28T07:45:01Z

<p>Eric Lengyel: </p> <hr /> <div>__NOTOC__<br /> == Rigid Geometric Algebra ==<br /> <br /> This wiki is a repository of information about Rigid Geometric Algebra (RGA), and specifically the four-dimensional Clifford algebra $$\mathcal G_{3,0,1}$$. This wiki is associated with the following websites:<br /> <br /> * [http://projectivegeometricalgebra.org Projective Geometric Algebra overview site]<br /> * [http://conformalgeometricalgebra.org/wiki/index.php?title=Main_Page Conformal Geometric Algebra companion site]<br /> <br /> Rigid geometric algebra is a mathematical model that naturally incorporates representations for Euclidean [[points]], [[lines]], and [[planes]] in 3D space as well as operations for performing [[rotations]], [[reflections]], and [[translations]] in a single algebraic structure. It completely subsumes conventional models that include homogeneous coordinates, Plücker coordinates, [[quaternions]], and screw theory (which makes use of dual quaternions). This makes rigid geometric algebra a natural fit for areas of computer science that routinely use these mathematical concepts, especially computer graphics and robotics. [http://conformalgeometricalgebra.org/wiki/index.php?title=Main_Page Conformal Geometric Algebra] (CGA) is a larger algebra that contains the complete RGA and also includes round objects like circles and spheres.<br /> <br /> Rigid geometric algebra is an area of active research, and new information is frequently being added to this wiki.<br /> <br /> '''If you are experiencing problems with the LaTeX on this site, please clear the cookies for rigidgeometricalgebra.org and reload.'''<br /> <br /> == Introduction ==<br /> <br /> [[Image:basis.svg|thumb|right|400px|'''Table 1.''' The 16 basis elements of the 4D rigid geometric algebra.]]<br /> In the four-dimensional rigid geometric algebra, there are 16 graded basis elements. These are listed in Table 1.<br /> <br /> There is a single ''[[scalar]]'' basis element that we denote by $$\mathbf 1$$, in bold, and its multiples correspond to the real numbers, which are values that have no dimensions.<br /> <br /> There are four ''[[vector]]'' basis elements named $$\mathbf e_1$$, $$\mathbf e_2$$, $$\mathbf e_3$$, and $$\mathbf e_4$$ that have one-dimensional extents. A general vector $$\mathbf v = (v_x, v_y, v_z, v_w)$$ has the form<br /> <br /> :$$\mathbf v = v_x \mathbf e_1 + v_y \mathbf e_2 + v_z \mathbf e_3 + v_w \mathbf e_4$$ .<br /> <br /> There are six ''[[bivector]]'' basis elements named $$\mathbf e_{23}$$, $$\mathbf e_{31}$$, $$\mathbf e_{12}$$, $$\mathbf e_{41}$$, $$\mathbf e_{42}$$, and $$\mathbf e_{43}$$ having two-dimensional extents. These correspond to all possible [[wedge products]] between pairs of vector basis elements up to order. We use the multiple subscript notation $$\mathbf e_{ij}$$ as shorthand for the wedge product $$\mathbf e_i \wedge \mathbf e_j$$. Numerical subscripts for the bivector basis elements are always written in the order shown in Table 1, and the bivectors are negated when basis vectors are multiplied in the opposite order. For example, $$\mathbf e_3 \wedge \mathbf e_2 = -\mathbf e_{23}$$.<br /> <br /> There are four ''[[trivector]]'' basis elements named $$\mathbf e_{423}$$, $$\mathbf e_{431}$$, $$\mathbf e_{412}$$, and $$\mathbf e_{321}$$ having three-dimensional extents. These correspond to all possible wedge products of three different vector basis elements. Again, numerical subscripts will always be written exactly as shown in the table, and negation will be applied for any odd permutation of the multiplication order.<br /> <br /> Finally, there is a single ''quadrivector'' basis element $$\mathbf e_1 \wedge \mathbf e_2 \wedge \mathbf e_3 \wedge \mathbf e_4$$ having four-dimensional extents. Because the quadrivector element has only one component, it is often called the ''pseudoscalar'', and it is often denoted by $$\mathbf I_4$$. The subscript 4 corresponds to the number of dimensions, and it is usually dropped when the dimensionality is clear from the context. Because the quadrivector contains all four dimensions, it is also called the ''volume element'' of the algebra, and this is often denoted by $$\mathbf E_4$$. We use the notation<br /> <br /> :$${\large\unicode{x1D7D9}} = \mathbf e_1 \wedge \mathbf e_2 \wedge \mathbf e_3 \wedge \mathbf e_4$$ ,<br /> <br /> with a blackboard bold $${\large\unicode{x1D7D9}}$$, to emphasize that the volume element is in symmetric opposition to the scalar basis element $$\mathbf 1$$ and is equally functional within the algebra. We refer to multiples of the basis element $${\large\unicode{x1D7D9}}$$ as ''[[antiscalars]]''. Scalars and antiscalars are two sides of the same coin, and neither has a place of greater importance. We eschew the term pseudoscalar due to its portrayal of the element $${\large\unicode{x1D7D9}}$$ as different from and perhaps somewhat less significant than the element $$\mathbf 1$$. It is not.<br /> <br /> As shown in the rightmost column in the table, each of the basis elements can be identified by which specific multiplicative combination of the four available dimensions it represents. This is essentially a four-bit code in which black bars correspond to the dimensions that are present or ''full'', and white bars correspond to the dimensions that are absent or ''empty''. The ''[[grade]]'' of a basis element $$\mathbf u$$, denoted by $$\operatorname{gr}(\mathbf u)$$, is the number of black bars it has, which is the same as the number of vector basis elements in its factorization.<br /> <br /> For a thorough understanding of the algebraic structure, it is critically important to recognize that there is a fundamental symmetry at work. We have assigned a dimensionality to each basis element according to the number of full dimensions it has, but it is equally valid to assign a dimensionality according to the number of empty dimensions each one has. Vectors, bivectors, and trivectors have dimensions one, two, and three when we count the black bars. However, from the opposite perspective, vectors, bivectors, and trivectors have dimensions three, two, and one when we count the white bars. Both of these interpretations are simultaneously correct, and together they establish the concept of ''[[duality]]''. [[Duality]] is always present, and it pervades geometric algebra. It can be found not only in the elements of the algebra but in the operations that act on those elements.<br /> <br /> In addition to the grade, we can assign an ''[[antigrade]]'' to each basis element $$\mathbf u$$. Denoted by $$\operatorname{ag}(\mathbf u)$$, the antigrade of $$\mathbf u$$ is the number of vector basis elements missing from its factorization, which is the number of white bars in the table. Of course, it is always the case that<br /> <br /> :$$\operatorname{gr}(\mathbf u) + \operatorname{ag}(\mathbf u) = n$$ ,<br /> <br /> where $$n$$ is the total number of dimensions in the algebra. Whenever we can make a statement about how an operation relates to the grade of its inputs and outputs, we can make the same statement about how the dual operation relates to the antigrade of its inputs and outputs.<br /> <br /> In an $$n$$-dimensional algebra, the elements with grade $$n - 1$$ are called ''[[antivectors]]''. Antivectors have the same number of components as vectors, and the two can be regarded as the dimensional inverses of each other. Vectors have grade one because they have one full dimension, and antivectors have antigrade one because they have one empty dimension.<br /> <br /> == Pages ==<br /> <br /> === The five main types of rigid geometric objects ===<br /> <br /> * [[Point]]<br /> * [[Line]]<br /> * [[Plane]]<br /> * [[Motor]]<br /> * [[Flector]]<br /> <br /> === Various properties and unary operations ===<br /> <br /> * [[Grade and antigrade]]<br /> * [[Complements]]<br /> * [[Metrics]]<br /> * [[Bulk and weight]]<br /> * [[Duals]]<br /> * [[Reverses]]<br /> * [[Attitude]]<br /> * [[Geometric norm]]<br /> * [[Geometric constraint]]<br /> * [[Unitization]]<br /> * [[Duality]]<br /> <br /> === Products and other binary operations ===<br /> <br /> * [[Exterior products]]<br /> * [[Dot products]]<br /> * [[Interior products]]<br /> * [[Geometric products]]<br /> * [[Transwedge products]]<br /> * [[Join and meet]]<br /> * [[Projections]]<br /> * [[Euclidean distance]]<br /> * [[Euclidean angle]]<br /> <br /> === Isometries of 3D space ===<br /> <br /> * [[Transformation groups]]<br /> * [[Translation]]<br /> * [[Rotation]]<br /> * [[Reflection]]<br /> * [[Inversion]]<br /> * [[Transflection]]<br /> <br /> === Projective Geometric Algebra Illuminated ===<br /> <br /> * [[Contents]]<br /> * [[Exercises]]<br /> * [[Errata]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Main_Page&diff=472 2025-06-28T07:44:30Z

<p>Eric Lengyel: /* Pages */</p> <hr /> <div>__NOTOC__<br /> == Rigid Geometric Algebra ==<br /> <br /> This wiki is a repository of information about Rigid Geometric Algebra (RGA), and specifically the four-dimensional Clifford algebra $$\mathcal G_{3,0,1}$$. This wiki is associated with the following websites:<br /> <br /> * [http://projectivegeometricalgebra.org Projective Geometric Algebra overview site]<br /> * [http://conformalgeometricalgebra.org/wiki/index.php?title=Main_Page Conformal Geometric Algebra companion site]<br /> <br /> Rigid geometric algebra is a mathematical model that naturally incorporates representations for Euclidean [[points]], [[lines]], and [[planes]] in 3D space as well as operations for performing [[rotations]], [[reflections]], and [[translations]] in a single algebraic structure. It completely subsumes conventional models that include homogeneous coordinates, Plücker coordinates, [[quaternions]], and screw theory (which makes use of dual quaternions). This makes rigid geometric algebra a natural fit for areas of computer science that routinely use these mathematical concepts, especially computer graphics and robotics. [http://conformalgeometricalgebra.org/wiki/index.php?title=Main_Page Conformal Geometric Algebra] (CGA) is a larger algebra that contains the complete RGA and also includes round objects like circles and spheres.<br /> <br /> Rigid geometric algebra is an area of active research, and new information is frequently being added to this wiki.<br /> <br /> '''If you are experiencing problems with the LaTeX on this site, please clear the cookies for rigidgeometricalgebra.org and reload.'''<br /> <br /> == Introduction ==<br /> <br /> [[Image:basis.svg|thumb|right|400px|'''Table 1.''' The 16 basis elements of the 4D rigid geometric algebra.]]<br /> In the four-dimensional rigid geometric algebra, there are 16 graded basis elements. These are listed in Table 1.<br /> <br /> There is a single ''[[scalar]]'' basis element that we denote by $$\mathbf 1$$, in bold, and its multiples correspond to the real numbers, which are values that have no dimensions.<br /> <br /> There are four ''[[vector]]'' basis elements named $$\mathbf e_1$$, $$\mathbf e_2$$, $$\mathbf e_3$$, and $$\mathbf e_4$$ that have one-dimensional extents. A general vector $$\mathbf v = (v_x, v_y, v_z, v_w)$$ has the form<br /> <br /> :$$\mathbf v = v_x \mathbf e_1 + v_y \mathbf e_2 + v_z \mathbf e_3 + v_w \mathbf e_4$$ .<br /> <br /> There are six ''[[bivector]]'' basis elements named $$\mathbf e_{23}$$, $$\mathbf e_{31}$$, $$\mathbf e_{12}$$, $$\mathbf e_{41}$$, $$\mathbf e_{42}$$, and $$\mathbf e_{43}$$ having two-dimensional extents. These correspond to all possible [[wedge products]] between pairs of vector basis elements up to order. We use the multiple subscript notation $$\mathbf e_{ij}$$ as shorthand for the wedge product $$\mathbf e_i \wedge \mathbf e_j$$. Numerical subscripts for the bivector basis elements are always written in the order shown in Table 1, and the bivectors are negated when basis vectors are multiplied in the opposite order. For example, $$\mathbf e_3 \wedge \mathbf e_2 = -\mathbf e_{23}$$.<br /> <br /> There are four ''[[trivector]]'' basis elements named $$\mathbf e_{423}$$, $$\mathbf e_{431}$$, $$\mathbf e_{412}$$, and $$\mathbf e_{321}$$ having three-dimensional extents. These correspond to all possible wedge products of three different vector basis elements. Again, numerical subscripts will always be written exactly as shown in the table, and negation will be applied for any odd permutation of the multiplication order.<br /> <br /> Finally, there is a single ''quadrivector'' basis element $$\mathbf e_1 \wedge \mathbf e_2 \wedge \mathbf e_3 \wedge \mathbf e_4$$ having four-dimensional extents. Because the quadrivector element has only one component, it is often called the ''pseudoscalar'', and it is often denoted by $$\mathbf I_4$$. The subscript 4 corresponds to the number of dimensions, and it is usually dropped when the dimensionality is clear from the context. Because the quadrivector contains all four dimensions, it is also called the ''volume element'' of the algebra, and this is often denoted by $$\mathbf E_4$$. We use the notation<br /> <br /> :$${\large\unicode{x1D7D9}} = \mathbf e_1 \wedge \mathbf e_2 \wedge \mathbf e_3 \wedge \mathbf e_4$$ ,<br /> <br /> with a blackboard bold $${\large\unicode{x1D7D9}}$$, to emphasize that the volume element is in symmetric opposition to the scalar basis element $$\mathbf 1$$ and is equally functional within the algebra. We refer to multiples of the basis element $${\large\unicode{x1D7D9}}$$ as ''[[antiscalars]]''. Scalars and antiscalars are two sides of the same coin, and neither has a place of greater importance. We eschew the term pseudoscalar due to its portrayal of the element $${\large\unicode{x1D7D9}}$$ as different from and perhaps somewhat less significant than the element $$\mathbf 1$$. It is not.<br /> <br /> As shown in the rightmost column in the table, each of the basis elements can be identified by which specific multiplicative combination of the four available dimensions it represents. This is essentially a four-bit code in which black bars correspond to the dimensions that are present or ''full'', and white bars correspond to the dimensions that are absent or ''empty''. The ''[[grade]]'' of a basis element $$\mathbf u$$, denoted by $$\operatorname{gr}(\mathbf u)$$, is the number of black bars it has, which is the same as the number of vector basis elements in its factorization.<br /> <br /> For a thorough understanding of the algebraic structure, it is critically important to recognize that there is a fundamental symmetry at work. We have assigned a dimensionality to each basis element according to the number of full dimensions it has, but it is equally valid to assign a dimensionality according to the number of empty dimensions each one has. Vectors, bivectors, and trivectors have dimensions one, two, and three when we count the black bars. However, from the opposite perspective, vectors, bivectors, and trivectors have dimensions three, two, and one when we count the white bars. Both of these interpretations are simultaneously correct, and together they establish the concept of ''[[duality]]''. [[Duality]] is always present, and it pervades geometric algebra. It can be found not only in the elements of the algebra but in the operations that act on those elements.<br /> <br /> In addition to the grade, we can assign an ''[[antigrade]]'' to each basis element $$\mathbf u$$. Denoted by $$\operatorname{ag}(\mathbf u)$$, the antigrade of $$\mathbf u$$ is the number of vector basis elements missing from its factorization, which is the number of white bars in the table. Of course, it is always the case that<br /> <br /> :$$\operatorname{gr}(\mathbf u) + \operatorname{ag}(\mathbf u) = n$$ ,<br /> <br /> where $$n$$ is the total number of dimensions in the algebra. Whenever we can make a statement about how an operation relates to the grade of its inputs and outputs, we can make the same statement about how the dual operation relates to the antigrade of its inputs and outputs.<br /> <br /> In an $$n$$-dimensional algebra, the elements with grade $$n - 1$$ are called ''[[antivectors]]''. Antivectors have the same number of components as vectors, and the two can be regarded as the dimensional inverses of each other. Vectors have grade one because they have one full dimension, and antivectors have antigrade one because they have one empty dimension.<br /> <br /> == Pages ==<br /> <br /> === The five main types of rigid geometric objects ===<br /> <br /> * [[Point]]<br /> * [[Line]]<br /> * [[Plane]]<br /> * [[Motor]]<br /> * [[Flector]]<br /> <br /> === Various properties and unary operations ===<br /> <br /> * [[Grade and antigrade]]<br /> * [[Complements]]<br /> * [[Metrics]]<br /> * [[Bulk and weight]]<br /> * [[Duals]]<br /> * [[Reverses]]<br /> * [[Attitude]]<br /> * [[Geometric norm]]<br /> * [[Geometric constraint]]<br /> * [[Unitization]]<br /> * [[Duality]]<br /> <br /> === Products and other binary operations ===<br /> <br /> * [[Exterior products]]<br /> * [[Dot products]]<br /> * [[Interior products]]<br /> * [[Geometric products]]<br /> * [[Transwedge products]]<br /> * [[Join and meet]]<br /> * [[Projections]]<br /> * [[Euclidean distance]]<br /> * [[Euclidean angle]]<br /> <br /> === Isometries of 3D space ===<br /> <br /> * [[Transformation groups]]<br /> * [[Translation]]<br /> * [[Rotation]]<br /> * [[Reflection]]<br /> * [[Inversion]]<br /> * [[Transflection]]<br /> <br /> === Projective Geometric Algebra Illuminated ==<br /> <br /> * [[Contents]]<br /> * [[Exercises]]<br /> * [[Errata]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Contents&diff=471 2025-06-28T07:21:28Z

<p>Eric Lengyel: Created page with &quot;This is the table of contents for ''Projective Geometric Algebra Illuminated''. * Preface * 1 Conventional Mathematics ** 1.1 The Cross Product ** 1.2 Homogeneous Coordinates ** 1.3 Lines and Planes *** 1.3.1 Parametric Forms *** 1.3.2 Implicit Forms *** 1.3.3 Distance Between a Point and a Line *** 1.3.4 Intersection of a Line and a Plane *** 1.3.5 Intersection of Multiple Planes *** 1.3.6 Reflection Across a Plane *** 1.3.7 Homogeneous Formulas *** 1.3.8 P...&quot;</p> <hr /> <div>This is the table of contents for ''Projective Geometric Algebra Illuminated''.<br /> <br /> * Preface<br /> * 1 Conventional Mathematics<br /> ** 1.1 The Cross Product<br /> ** 1.2 Homogeneous Coordinates<br /> ** 1.3 Lines and Planes<br /> *** 1.3.1 Parametric Forms<br /> *** 1.3.2 Implicit Forms<br /> *** 1.3.3 Distance Between a Point and a Line<br /> *** 1.3.4 Intersection of a Line and a Plane<br /> *** 1.3.5 Intersection of Multiple Planes<br /> *** 1.3.6 Reflection Across a Plane<br /> *** 1.3.7 Homogeneous Formulas<br /> *** 1.3.8 Plane Transformation<br /> *** 1.3.9 Line Transformation<br /> ** 1.4 Quaternions<br /> *** 1.4.1 Quaternion Fundamentals<br /> *** 1.4.2 Rotations With Quaternions<br /> *** 1.4.3 Interpolating Quaternions<br /> *** 1.4.4 Dual Quaternions<br /> **Historical Remarks<br /> * 2 Flat Projective Geometry<br /> ** 2.1 Algebraic Structure<br /> *** 2.1.1 The Wedge Product<br /> *** 2.1.2 Bivectors<br /> *** 2.1.3 Trivectors<br /> *** 2.1.4 Basis Elements<br /> ** 2.2 Complements<br /> ** 2.3 Antiproducts<br /> ** 2.4 3D Flat Geometry<br /> *** 2.4.1 Points<br /> *** 2.4.2 Lines<br /> *** 2.4.3 Planes<br /> ** 2.5 Join and Meet<br /> ** 2.6 Duality<br /> ** 2.7 Exomorphisms<br /> ** 2.8 Metric Transformations<br /> *** 2.8.1 The Metric<br /> *** 2.8.2 The Antimetric<br /> *** 2.8.3 Bulk and Weight<br /> *** 2.8.4 Attitude<br /> ** 2.9 Inner Products<br /> ** 2.10 Norms<br /> *** 2.10.1 Bulk and Weight Norms<br /> *** 2.10.2 Unitization<br /> *** 2.10.3 The Geometric Norm<br /> ** 2.11 Euclidean Distances<br /> ** 2.12 Duals<br /> ** 2.13 Interior Products<br /> *** 2.13.1 Contractions<br /> *** 2.13.2 Projection and Rejection<br /> *** 2.13.3 Euclidean Angles<br /> *** 2.13.4 Parametric Forms<br /> *** 2.13.5 Expansions<br /> *** 2.13.6 Geometric Projection<br /> ** 2.14 2D Flat Geometry<br /> ** 2.15 Dependencies<br /> ** Historical Remarks<br /> * 3 Rigid Transformations<br /> ** 3.1 The Geometric Product<br /> ** 3.2 Dual Numbers<br /> ** 3.3 Reflection and Rotation<br /> ** 3.4 Reversion<br /> *** 3.4.1 Reverse and Antireverse<br /> *** 3.4.2 Dual Identities<br /> *** 3.4.3 Geometric Constraint<br /> ** 3.5 Euclidean Isometries<br /> *** 3.5.1 Reflection<br /> *** 3.5.2 Rotation<br /> *** 3.5.3 Translation<br /> *** 3.5.4 Inversion<br /> *** 3.5.5 Transflection<br /> ** 3.6 Motors<br /> *** 3.6.1 Motion Operator<br /> *** 3.6.2 Parameterization<br /> *** 3.6.3 Line to Line Motion<br /> *** 3.6.4 Matrix Conversion<br /> *** 3.6.5 Implementation<br /> ** 3.7 Flectors<br /> *** 3.7.1 Reflection Operator<br /> *** 3.7.2 Matrix Conversion<br /> *** 3.7.3 Implementation<br /> ** 3.8 2D Rigid Transformations<br /> ** 3.9 Operator Duality<br /> *** 3.9.1 Complement Isometries<br /> *** 3.9.2 Transformation Groups<br /> *** 3.9.3 Quaternions Revisited<br /> ** Historical Remarks<br /> * 4 Round Projective Geometry<br /> ** 4.1 Construction<br /> ** 4.2 3D Round Geometry<br /> *** 4.2.1 Representations<br /> *** 4.2.2 Duals<br /> *** 4.2.3 Carriers<br /> *** 4.2.4 Centers<br /> *** 4.2.5 Containers<br /> *** 4.2.6 Partners<br /> *** 4.2.7 Attitude<br /> ** 4.3 Norms<br /> ** 4.4 Alignment<br /> ** 4.5 Dot Products<br /> *** 4.5.1 Round Points<br /> *** 4.5.2 Spheres<br /> *** 4.5.3 Partners<br /> *** 4.5.4 Conjugates<br /> ** 4.6 Containment<br /> ** 4.7 Join and Meet<br /> ** 4.8 Expansions<br /> ** 4.9 2D Round Geometry<br /> ** 4.10 Degrees of Freedom<br /> * 5 Conformal Transformations<br /> ** 5.1 Generalized Operators<br /> *** 5.1.1 Rigid Transformations<br /> *** 5.1.2 Sphere Inversion<br /> *** 5.1.3 Circle Rotation<br /> ** 5.2 Dilation<br /> ** 5.3 Duals and Complements<br /> ** 5.4 2D Conformal Transformations<br /> * A Multiplication Tables<br /> * B Geometric Properties<br /> * C Notation Reference<br /> * Bibliography<br /> * Index<br /> <br /> == See Also ==<br /> <br /> * [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''PGA Illuminated'' on Amazon.com]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=470 2025-05-26T20:54:30Z

<p>Eric Lengyel: </p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf c^{\unicode{x2605}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, which is an interior product. If $$\mathbf a$$ and $$\mathbf b$$ have the same grade, then this interior product reduces to the inner product $$\mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$.<br /> <br /> If $$k &gt; \operatorname{gr}(\mathbf a)$$ or $$k &gt; \operatorname{gr}(\mathbf b)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf c_{\unicode{x2605}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products, negating for orders 2 and 3 modulo 4, yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> When the [[metric]] is diagonal, the [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> <br /> For vectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$.<br /> <br /> For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b - \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf{\tilde b}$$,<br /> <br /> where the [[dot product]] is again attributable to the right contraction $$\mathbf{\tilde b} \vee \mathbf a^{\unicode{x2605}}$$, but this time with a reversal to account for the sign change. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf c^{\unicode{x2606}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf c_{\unicode{x2606}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], and we are now summing over all basis elements of [[antigrade]] $$k$$.<br /> <br /> The sum of all possible transwedge antiproducts, again negating for orders 2 and 3 modulo 4, yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, 3, and 4.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=469 2025-05-26T20:51:57Z

<p>Eric Lengyel: </p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf c^{\unicode{x2605}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, which is an interior product. If $$\mathbf a$$ and $$\mathbf b$$ have the same grade, then this interior product reduces to the inner product $$\mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$.<br /> <br /> If $$k &gt; \operatorname{gr}(\mathbf a)$$ or $$k &gt; \operatorname{gr}(\mathbf b)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf c_{\unicode{x2605}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products, negating for orders 2 and 3 modulo 4, yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> When the [[metric]] is diagonal, the [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> <br /> For vectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$.<br /> <br /> For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b - \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$, but this time with a reversal to account for the sign change. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf c^{\unicode{x2606}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf c_{\unicode{x2606}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], and we are now summing over all basis elements of [[antigrade]] $$k$$.<br /> <br /> The sum of all possible transwedge antiproducts, again negating for orders 2 and 3 modulo 4, yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, 3, and 4.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=468 2025-05-21T22:02:46Z

<p>Eric Lengyel: </p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf c^{\unicode{x2605}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, which is an interior product. If $$\mathbf a$$ and $$\mathbf b$$ have the same grade, then this interior product reduces to the inner product $$\mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$.<br /> <br /> If $$k &gt; \operatorname{gr}(\mathbf a)$$ or $$k &gt; \operatorname{gr}(\mathbf b)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf c_{\unicode{x2605}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products, negating for orders 2 and 3 modulo 4, yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> The [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> <br /> For vectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$.<br /> <br /> For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b - \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$, but this time with a reversal to account for the sign change. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf c^{\unicode{x2606}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf c_{\unicode{x2606}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], and we are now summing over all basis elements of [[antigrade]] $$k$$.<br /> <br /> The sum of all possible transwedge antiproducts, again negating for orders 2 and 3 modulo 4, yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, 3, and 4.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=467 2025-05-21T22:01:21Z

<p>Eric Lengyel: </p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf c^{\unicode{x2605}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, which is an interior product. If $$\mathbf a$$ and $$\mathbf b$$ have the same grade, then this interior product reduces to the inner product $$\mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$. If $$k &gt; \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf c_{\unicode{x2605}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products, negating for orders 2 and 3 modulo 4, yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> The [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> <br /> For vectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$.<br /> <br /> For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b - \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$, but this time with a reversal to account for the sign change. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf c^{\unicode{x2606}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf c_{\unicode{x2606}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], and we are now summing over all basis elements of [[antigrade]] $$k$$.<br /> <br /> The sum of all possible transwedge antiproducts, again negating for orders 2 and 3 modulo 4, yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, 3, and 4.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=466 2025-05-21T21:26:10Z

<p>Eric Lengyel: </p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf c^{\unicode{x2605}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, which is an interior product. If $$\mathbf a$$ and $$\mathbf b$$ have the same grade, then this interior product reduces to the inner product $$\mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$. If $$k &gt; \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf c_{\unicode{x2605}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products, negating for orders 2 and 3 modulo 4, yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> The [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> <br /> For vectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$.<br /> <br /> For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b - \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf c^{\unicode{x2606}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf c_{\unicode{x2606}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], and we are now summing over all basis elements of [[antigrade]] $$k$$.<br /> <br /> The sum of all possible transwedge antiproducts, again negating for orders 2 and 3 modulo 4, yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, 3, and 4.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=File:TranswedgeAntiproducts.svg&diff=465 2025-05-21T21:17:22Z

<p>Eric Lengyel: Eric Lengyel uploaded a new version of File:TranswedgeAntiproducts.svg</p> <hr /> <div></div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=File:TranswedgeProducts.svg&diff=464 2025-05-21T21:17:08Z

<p>Eric Lengyel: Eric Lengyel uploaded a new version of File:TranswedgeProducts.svg</p> <hr /> <div></div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=463 2025-05-21T21:10:05Z

<p>Eric Lengyel: </p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf c^{\unicode{x2605}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, which is an interior product. If $$k &gt; \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf c_{\unicode{x2605}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products, negating for orders 2 and 3 modulo 4, yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> The [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> <br /> For vectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$.<br /> <br /> For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b - \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf c^{\unicode{x2606}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf c_{\unicode{x2606}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], and we are now summing over all basis elements of [[antigrade]] $$k$$.<br /> <br /> The sum of all possible transwedge antiproducts, again negating for orders 2 and 3 modulo 4, yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, 3, and 4.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_product&diff=462 2025-05-20T20:29:13Z

<p>Eric Lengyel: Redirected page to Transwedge products</p> <hr /> <div>#REDIRECT [[Transwedge products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Geometric_products&diff=461 2025-05-20T20:28:52Z

<p>Eric Lengyel: </p> <hr /> <div>The ''geometric product'' is a method of multiplication in geometric algebra derived from the [[exterior product]] and the [[metric]]. There are two products with symmetric properties called the geometric product and geometric antiproduct.<br /> <br /> == Geometric Product ==<br /> <br /> The geometric product between two elements $$\mathbf a$$ and $$\mathbf b$$ has often been written by simply juxtaposing its operands as $$\mathbf{ab}$$ without the use of any infix operator. However, this clearly becomes impractical when both a product and antiproduct are present in the same context, which is now known to be necessary for a proper understanding of the algebra. To remedy the situation, we write the geometric product between elements $$\mathbf a$$ and $$\mathbf b$$ as $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ and read it as &quot;$$\mathbf a$$ wedge-dot $$\mathbf b$$&quot;.<br /> <br /> The geometric product is characterized by a metric that defines the products of the basis vectors with themselves. The subscript in $$\mathcal G_{3,0,1}$$ means that three basis vectors square to +'''1''', zero basis vectors square to &amp;minus;'''1''', and one basis vector squares to 0. The geometric product between two different basis vectors is given by the [[wedge product]]. We can write these rules as follows.<br /> <br /> :$$\mathbf e_1 \mathbin{\unicode{x27D1}} \mathbf e_1 = \mathbf 1$$<br /> <br /> :$$\mathbf e_2 \mathbin{\unicode{x27D1}} \mathbf e_2 = \mathbf 1$$<br /> <br /> :$$\mathbf e_3 \mathbin{\unicode{x27D1}} \mathbf e_3 = \mathbf 1$$<br /> <br /> :$$\mathbf e_4 \mathbin{\unicode{x27D1}} \mathbf e_4 = 0$$<br /> <br /> :$$\mathbf e_i \mathbin{\unicode{x27D1}} \mathbf e_j = \mathbf e_i \wedge \mathbf e_j$$, for $$i \neq j$$.<br /> <br /> The geometric product is equal to the sum of the [[transwedge products]] of orders 0 through $$n$$, where $$n$$ is the dimension of the algebra, in this case 4.<br /> <br /> The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.<br /> <br /> Cells highlighted in green correspond to the contribution from the [[wedge product]].<br /> <br /> <br /> [[Image:GeometricProduct.svg|720px]]<br /> <br /> == Geometric Antiproduct ==<br /> <br /> The geometric antiproduct is dual to the geometric product. The geometric antiproduct between elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x27C7}} \mathbf b$$ and is read as &quot;$$\mathbf a$$ antiwedge-dot $$\mathbf b$$&quot;.<br /> <br /> The same metric that defines products of basis vectors under the geometric product also applies to the geometric antiproduct, except that now it defines products of basis [[antivectors]]. Three basis antivectors square to $$+{\large\unicode{x1D7D9}}$$, zero basis antivectors square to $$-{\large\unicode{x1D7D9}}$$, and one basis antivector squares to 0. The geometric antiproduct between two different basis antivectors is given by the [[antiwedge product]]. We can write these rules as follows.<br /> <br /> :$$\overline{\mathbf e_1} \mathbin{\unicode{x27C7}} \overline{\mathbf e_1} = {\large\unicode{x1D7D9}}$$<br /> <br /> :$$\overline{\mathbf e_2} \mathbin{\unicode{x27C7}} \overline{\mathbf e_2} = {\large\unicode{x1D7D9}}$$<br /> <br /> :$$\overline{\mathbf e_3} \mathbin{\unicode{x27C7}} \overline{\mathbf e_3} = {\large\unicode{x1D7D9}}$$<br /> <br /> :$$\overline{\mathbf e_4} \mathbin{\unicode{x27C7}} \overline{\mathbf e_4} = 0$$<br /> <br /> :$$\overline{\mathbf e_i} \mathbin{\unicode{x27C7}} \overline{\mathbf e_j} = \overline{\mathbf e_i} \vee \overline{\mathbf e_j}$$, for $$i \neq j$$.<br /> <br /> The following Cayley table shows the geometric antiproducts between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric antiproduct is the [[antiscalar]] basis element $${\large\unicode{x1D7D9}}$$.<br /> <br /> Cells highlighted in green correspond to the contribution from the [[antiwedge product]].<br /> <br /> <br /> [[Image:GeometricAntiproduct.svg|720px]]<br /> <br /> == De Morgan Laws ==<br /> <br /> The relationship between the product and antiproduct is based on an exchange of full and empty dimensions. We can then express each product in terms of the other through an analog of De Morgan's laws as follows.<br /> <br /> :$$\overline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \overline{\mathbf b}$$<br /> <br /> :$$\overline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \overline{\mathbf b}$$<br /> <br /> :$$\underline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \underline{\mathbf b}$$<br /> <br /> :$$\underline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \underline{\mathbf b}$$<br /> <br /> == In the Book ==<br /> <br /> * The geometric product and antiproduct are introduced in Section 3.1.<br /> <br /> == See Also ==<br /> <br /> * [[Transwedge products]]<br /> * [[Exterior products]]<br /> * [[Dot products]]<br /> * [[Complements]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Geometric_products&diff=460 2025-05-20T19:56:06Z

<p>Eric Lengyel: /* See Also */</p> <hr /> <div>The ''geometric product'' is the fundamental product of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct.<br /> <br /> == Geometric Product ==<br /> <br /> The geometric product between two elements $$\mathbf a$$ and $$\mathbf b$$ has often been written by simply juxtaposing its operands as $$\mathbf{ab}$$ without the use of any infix operator. However, this clearly becomes impractical when both a product and antiproduct are present in the same context, which is now known to be necessary for a proper understanding of the algebra. To remedy the situation, we write the geometric product between elements $$\mathbf a$$ and $$\mathbf b$$ as $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ and read it as &quot;$$\mathbf a$$ wedge-dot $$\mathbf b$$&quot;.<br /> <br /> The geometric product is characterized by a metric that defines the products of the basis vectors with themselves. The subscript in $$\mathcal G_{3,0,1}$$ means that three basis vectors square to +'''1''', zero basis vectors square to &amp;minus;'''1''', and one basis vector squares to 0. The geometric product between two different basis vectors is given by the [[wedge product]]. We can write these rules as follows.<br /> <br /> :$$\mathbf e_1 \mathbin{\unicode{x27D1}} \mathbf e_1 = \mathbf 1$$<br /> <br /> :$$\mathbf e_2 \mathbin{\unicode{x27D1}} \mathbf e_2 = \mathbf 1$$<br /> <br /> :$$\mathbf e_3 \mathbin{\unicode{x27D1}} \mathbf e_3 = \mathbf 1$$<br /> <br /> :$$\mathbf e_4 \mathbin{\unicode{x27D1}} \mathbf e_4 = 0$$<br /> <br /> :$$\mathbf e_i \mathbin{\unicode{x27D1}} \mathbf e_j = \mathbf e_i \wedge \mathbf e_j$$, for $$i \neq j$$.<br /> <br /> The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.<br /> <br /> Cells highlighted in green correspond to the contribution from the [[wedge product]].<br /> <br /> <br /> [[Image:GeometricProduct.svg|720px]]<br /> <br /> == Geometric Antiproduct ==<br /> <br /> The geometric antiproduct is a dual to the geometric product. The geometric antiproduct between elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x27C7}} \mathbf b$$ and is read as &quot;$$\mathbf a$$ antiwedge-dot $$\mathbf b$$&quot;.<br /> <br /> The same metric that defines products of basis vectors under the geometric product also applies to the geometric antiproduct, except that now it defines products of basis [[antivectors]]. Three basis antivectors square to $$+{\large\unicode{x1D7D9}}$$, zero basis antivectors square to $$-{\large\unicode{x1D7D9}}$$, and one basis antivector squares to 0. The geometric antiproduct between two different basis antivectors is given by the [[antiwedge product]]. We can write these rules as follows.<br /> <br /> :$$\overline{\mathbf e_1} \mathbin{\unicode{x27C7}} \overline{\mathbf e_1} = {\large\unicode{x1D7D9}}$$<br /> <br /> :$$\overline{\mathbf e_2} \mathbin{\unicode{x27C7}} \overline{\mathbf e_2} = {\large\unicode{x1D7D9}}$$<br /> <br /> :$$\overline{\mathbf e_3} \mathbin{\unicode{x27C7}} \overline{\mathbf e_3} = {\large\unicode{x1D7D9}}$$<br /> <br /> :$$\overline{\mathbf e_4} \mathbin{\unicode{x27C7}} \overline{\mathbf e_4} = 0$$<br /> <br /> :$$\overline{\mathbf e_i} \mathbin{\unicode{x27C7}} \overline{\mathbf e_j} = \overline{\mathbf e_i} \vee \overline{\mathbf e_j}$$, for $$i \neq j$$.<br /> <br /> The following Cayley table shows the geometric antiproducts between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric antiproduct is the [[antiscalar]] basis element $${\large\unicode{x1D7D9}}$$.<br /> <br /> Cells highlighted in green correspond to the contribution from the [[antiwedge product]].<br /> <br /> <br /> [[Image:GeometricAntiproduct.svg|720px]]<br /> <br /> == De Morgan Laws ==<br /> <br /> The relationship between the product and antiproduct is based on an exchange of full and empty dimensions. We can then express each product in terms of the other through an analog of De Morgan's laws as follows.<br /> <br /> :$$\overline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \overline{\mathbf b}$$<br /> <br /> :$$\overline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \overline{\mathbf b}$$<br /> <br /> :$$\underline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \underline{\mathbf b}$$<br /> <br /> :$$\underline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \underline{\mathbf b}$$<br /> <br /> == In the Book ==<br /> <br /> * The geometric product and antiproduct are introduced in Section 3.1.<br /> <br /> == See Also ==<br /> <br /> * [[Transwedge products]]<br /> * [[Exterior products]]<br /> * [[Dot products]]<br /> * [[Complements]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Exterior_products&diff=459 2025-05-20T19:55:46Z

<p>Eric Lengyel: /* See Also */</p> <hr /> <div>The ''exterior product'' is the fundamental product of Grassmann Algebra, and it forms part of the [[geometric product]] in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct.<br /> <br /> The exterior product between two elements $$\mathbf a$$ and $$\mathbf b$$ generally combines their spatial extents, and the magnitude of the result indicates how close they are to being orthogonal. If the spatial extents of $$\mathbf a$$ and $$\mathbf b$$ are parallel, then their exterior product is zero. (Compare this to the [[dot product]], which is zero whenever $$\mathbf a$$ and $$\mathbf b$$ are orthogonal.)<br /> <br /> == Exterior Product ==<br /> <br /> The exterior product is widely known as the ''wedge product'' because it is written with an upward pointing wedge. The exterior product between elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \wedge \mathbf b$$ and read &quot;$$\mathbf a$$ wedge $$\mathbf b$$&quot;. Grassmann called this the progressive combinatorial product.<br /> <br /> The defining characteristic of the wedge product is that multiplying any vector $$\mathbf v$$ by itself produces zero: $$\mathbf v \wedge \mathbf v = 0$$. This implies that the wedge product is anticommutative for vectors, so we always have<br /> <br /> :$$\mathbf v \wedge \mathbf w = -\mathbf w \wedge \mathbf v$$<br /> <br /> for vectors $$\mathbf v$$ and $$\mathbf w$$. The wedge product is not anticommutative in general, however. For general basis elements $$\mathbf a$$ and $$\mathbf b$$, reversing the order of the operands satisfies the relationship<br /> <br /> :$$\mathbf a \wedge \mathbf b = (-1)^{\operatorname{gr}(\mathbf a)\operatorname{gr}(\mathbf b)} \mathbf b \wedge \mathbf a$$ .<br /> <br /> The wedge product adds the [[grades]] of its operands, so we have<br /> <br /> :$$\operatorname{gr}(\mathbf a \wedge \mathbf b) = \operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b)$$ .<br /> <br /> The following Cayley table shows the exterior products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.<br /> <br /> <br /> [[Image:WedgeProduct.svg|720px]]<br /> <br /> == Exterior Antiproduct ==<br /> <br /> The exterior antiproduct is a dual to the exterior product. It is written with a downward pointing wedge and thus called the ''antiwedge product''. The exterior antiproduct $$\mathbf a \vee \mathbf b$$ is read &quot;$$\mathbf a$$ antiwedge $$\mathbf b$$&quot;. Grassmann called this the regressive combinatorial product.<br /> <br /> Whereas the wedge product combines the full dimensions of its operands, the antiwedge product combines the empty dimensions of its operands. The antiwedge product adds the [[antigrades]] of its operands, so we have<br /> <br /> :$$\operatorname{ag}(\mathbf a \vee \mathbf b) = \operatorname{ag}(\mathbf a) + \operatorname{ag}(\mathbf b)$$ .<br /> <br /> The following Cayley table shows the exterior antiproducts between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.<br /> <br /> <br /> [[Image:AntiwedgeProduct.svg|720px]]<br /> <br /> == De Morgan Laws ==<br /> <br /> The relationship between the product and antiproduct is based on an exchange of full and empty dimensions. We can then express each product in terms of the other through an analog of De Morgan's laws as follows.<br /> <br /> :$$\overline{\mathbf a \wedge \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \vee \overline{\mathbf b}$$<br /> <br /> :$$\overline{\mathbf a \vee \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \wedge \overline{\mathbf b}$$<br /> <br /> :$$\underline{\mathbf a \wedge \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \vee \underline{\mathbf b}$$<br /> <br /> :$$\underline{\mathbf a \vee \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \wedge \underline{\mathbf b}$$<br /> <br /> == General Properties ==<br /> <br /> The following table lists several general properties of the wedge product and antiwedge product.<br /> <br /> {| class=&quot;wikitable&quot;<br /> ! Property !! Description<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf a \wedge \mathbf b = -\mathbf b \wedge \mathbf a$$<br /> | style=&quot;padding: 12px;&quot; | Anticommutativity of the wedge product for vectors $$\mathbf a$$ and $$\mathbf b$$.<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf a \vee \mathbf b = -\mathbf b \vee \mathbf a$$<br /> | style=&quot;padding: 12px;&quot; | Anticommutativity of the antiwedge product for antivectors $$\mathbf a$$ and $$\mathbf b$$.<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$(\mathbf a \wedge \mathbf b) \wedge \mathbf c = \mathbf a \wedge (\mathbf b \wedge \mathbf c)$$<br /> | style=&quot;padding: 12px;&quot; | Associative law for the wedge product for any elements $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$.<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$(\mathbf a \vee \mathbf b) \vee \mathbf c = \mathbf a \vee (\mathbf b \vee \mathbf c)$$<br /> | style=&quot;padding: 12px;&quot; | Associative law for the antiwedge product for any elements $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$.<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf a \wedge (\mathbf b + \mathbf c) = \mathbf a \wedge \mathbf b + \mathbf a \wedge \mathbf c$$<br /> | style=&quot;padding: 12px;&quot; | Distributive law for the wedge product for any elements $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$.<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$\mathbf a \vee (\mathbf b + \mathbf c) = \mathbf a \vee \mathbf b + \mathbf a \vee \mathbf c$$<br /> | style=&quot;padding: 12px;&quot; | Distributive law for the antiwedge product for any elements $$\mathbf a$$, $$\mathbf b$$, and $$\mathbf c$$.<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$(t\mathbf a) \wedge \mathbf b = \mathbf a \wedge (t\mathbf b) = t(\mathbf a \wedge \mathbf b)$$<br /> | style=&quot;padding: 12px;&quot; | Scalar factorization of the wedge product.<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$(t\mathbf a) \vee \mathbf b = \mathbf a \vee (t\mathbf b) = t(\mathbf a \vee \mathbf b)$$<br /> | style=&quot;padding: 12px;&quot; | Scalar factorization of the antiwedge product.<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$s \wedge \mathbf a = \mathbf a \wedge s = s\mathbf a$$<br /> | style=&quot;padding: 12px;&quot; | Wedge product of a [[scalar]] $$s$$ and any basis element $$\mathbf a$$.<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$s \vee \mathbf a = \mathbf a \vee s = s\mathbf a$$<br /> | style=&quot;padding: 12px;&quot; | Antiwedge product of an [[antiscalar]] $$s$$ and any basis element $$\mathbf a$$.<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$s \wedge t = st$$<br /> | style=&quot;padding: 12px;&quot; | Wedge product of [[scalars]] $$s$$ and $$t$$.<br /> |-<br /> | style=&quot;padding: 12px;&quot; | $$s \vee t = st$$<br /> | style=&quot;padding: 12px;&quot; | Antiwedge product of [[antiscalars]] $$s$$ and $$t$$.<br /> |}<br /> <br /> == In the Book ==<br /> <br /> * The exterior (wedge) product is introduced in Section 2.1.1.<br /> * The exterior antiproduct is discussed in Section 2.3.<br /> <br /> == See Also ==<br /> <br /> * [[Transwedge products]]<br /> * [[Geometric products]]<br /> * [[Dot products]]<br /> * [[Complements]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=458 2025-05-20T00:50:16Z

<p>Eric Lengyel: /* Transwedge Antiproduct */</p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\tilde c^{\unicode{x2605}}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf{\tilde a^{\unicode{x2605}}}$$, which is an interior product. If $$k &gt; \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\tilde c_{\unicode{x2605}}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> The [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> <br /> For vectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, and we have dropped the reverse operation because $$\mathbf a$$ is a vector.<br /> <br /> For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}^{\unicode{x2606}}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}_{\unicode{x2606}}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], the [[reverse]] has become the [[antireverse]], and we are now summing over all basis elements of [[antigrade]] $$k$$.<br /> <br /> The sum of all possible transwedge antiproducts yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, 3, and 4.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=457 2025-05-20T00:49:53Z

<p>Eric Lengyel: </p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\tilde c^{\unicode{x2605}}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf{\tilde a^{\unicode{x2605}}}$$, which is an interior product. If $$k &gt; \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\tilde c_{\unicode{x2605}}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> The [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> <br /> For vectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, and we have dropped the reverse operation because $$\mathbf a$$ is a vector.<br /> <br /> For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}^{\unicode{x2606}}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}_{\unicode{x2606}}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], the [[reverse]] has become the [[antireverse]], and we are now summing over all basis elements of [[antigrade]] $$k$$.<br /> <br /> The sum of all possible transwedge antiproducts yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, and 3.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=File:TranswedgeAntiproducts.svg&diff=456 2025-05-20T00:48:00Z

<p>Eric Lengyel: Eric Lengyel uploaded a new version of File:TranswedgeAntiproducts.svg</p> <hr /> <div></div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=File:TranswedgeProducts.svg&diff=455 2025-05-20T00:47:40Z

<p>Eric Lengyel: Eric Lengyel uploaded a new version of File:TranswedgeProducts.svg</p> <hr /> <div></div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=454 2025-05-20T00:06:19Z

<p>Eric Lengyel: </p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\tilde c^{\unicode{x2605}}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf{\tilde a^{\unicode{x2605}}}$$, which is an interior product. If $$k &gt; \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\tilde c_{\unicode{x2605}}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> The [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, and 3. (The transwedge product of order 4 is zero in this algebra due to the degenerate metric.)<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> <br /> For vectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, and we have dropped the reverse operation because $$\mathbf a$$ is a vector.<br /> <br /> For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}^{\unicode{x2606}}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}_{\unicode{x2606}}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], the [[reverse]] has become the [[antireverse]], and we are now summing over all basis elements of [[antigrade]] $$k$$.<br /> <br /> The sum of all possible transwedge antiproducts yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, and 3.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=453 2025-05-19T22:06:15Z

<p>Eric Lengyel: </p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\tilde c^{\unicode{x2605}}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf{\tilde a^{\unicode{x2605}}}$$, which is an interior product. If $$k &gt; \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\tilde c_{\unicode{x2605}}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> The [[geometric product]] of each pair of the 16 basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ is given by exactly one of the transwedge products. These are shown in the following table, which color codes the transwedge products of order 0, 1, 2, and 3. (The transwedge product of order 4 is zero in this algebra due to the degenerate metric.)<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> <br /> For vectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, and we have dropped the reverse operation because $$\mathbf a$$ is a vector.<br /> <br /> For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}^{\unicode{x2606}}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}_{\unicode{x2606}}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], the [[reverse]] has become the [[antireverse]], and we are now summing over all basis elements of [[antigrade]] $$k$$.<br /> <br /> The sum of all possible transwedge antiproducts yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, and 3.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=452 2025-05-19T22:02:54Z

<p>Eric Lengyel: </p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\tilde c^{\unicode{x2605}}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf{\tilde a^{\unicode{x2605}}}$$, which is an interior product. If $$k &gt; \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\tilde c_{\unicode{x2605}}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> The [[geometric product]] of each pair of the 16 basis elements in the 3D rigid algebra is given by exactly one of the transwedge products. These are shown in the following table, which color codes the transwedge products of order 0, 1, 2, and 3. (The transwedge product of order 4 is zero in this algebra due to the degenerate metric.)<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> <br /> For vectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, and we have dropped the reverse operation because $$\mathbf a$$ is a vector.<br /> <br /> For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}^{\unicode{x2606}}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}_{\unicode{x2606}}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], the [[reverse]] has become the [[antireverse]], and we are now summing over all basis elements of [[antigrade]] $$k$$.<br /> <br /> The sum of all possible transwedge antiproducts yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, and 3.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=451 2025-05-17T21:15:33Z

<p>Eric Lengyel: /* Transwedge Antiproduct */</p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\tilde c^{\unicode{x2605}}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf{\tilde a^{\unicode{x2605}}}$$, which is an interior product. If $$k &gt; \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\tilde c_{\unicode{x2605}}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> The [[geometric product]] of each pair of the 16 basis elements in the 3D rigid algebra is given by exactly one of the transwedge products. These are shown in the following table, which color codes the transwedge products of order 0, 1, 2, and 3. The transwedge product of order 4 is always zero in this algebra due to the degenerate metric.<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> <br /> For vectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, and we have dropped the reverse operation because $$\mathbf a$$ is a vector.<br /> <br /> For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}^{\unicode{x2606}}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}_{\unicode{x2606}}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], the [[reverse]] has become the [[antireverse]], and we are now summing over all basis elements of [[antigrade]] $$k$$.<br /> <br /> The sum of all possible transwedge antiproducts yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, and 3.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=450 2025-05-17T05:54:44Z

<p>Eric Lengyel: </p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\tilde c^{\unicode{x2605}}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf{\tilde a^{\unicode{x2605}}}$$, which is an interior product. If $$k &gt; \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\tilde c_{\unicode{x2605}}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> The [[geometric product]] of each pair of the 16 basis elements in the 3D rigid algebra is given by exactly one of the transwedge products. These are shown in the following table, which color codes the transwedge products of order 0, 1, 2, and 3. The transwedge product of order 4 is always zero in this algebra due to the degenerate metric.<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> <br /> For vectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, and we have dropped the reverse operation because $$\mathbf a$$ is a vector.<br /> <br /> For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode[&quot;segoe ui symbol&quot;]{x2022}} \mathbf b$$,<br /> <br /> where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}^{\unicode{x2606}}})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], and the [[reverse]] has become the [[antireverse]].<br /> <br /> The sum of all possible transwedge antiproducts yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, and 3.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=449 2025-05-17T05:37:31Z

<p>Eric Lengyel: </p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\tilde c^{\unicode{x2605}}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf{\tilde a^{\unicode{x2605}}}$$, which is an interior product. If $$k &gt; \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\tilde c_{\unicode{x2605}}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> The [[geometric product]] of each pair of the 16 basis elements in the 3D rigid algebra is given by exactly one of the transwedge products. These are shown in the following table, which color codes the transwedge products of order 0, 1, 2, and 3. The transwedge product of order 4 is always zero in this algebra due to the degenerate metric.<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}^{\unicode{x2606}}})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], and the [[reverse]] has become the [[antireverse]].<br /> <br /> The sum of all possible transwedge antiproducts yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, and 3.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Expansion&diff=448 2025-05-17T05:35:50Z

<p>Eric Lengyel: Redirected page to Interior products</p> <hr /> <div>#REDIRECT [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Contraction&diff=447 2025-05-17T05:35:39Z

<p>Eric Lengyel: Redirected page to Interior products</p> <hr /> <div>#REDIRECT [[Interior products]]</div>

Eric Lengyel https://rigidgeometricalgebra.org/wiki/index.php?title=Transwedge_products&diff=446 2025-05-17T05:35:13Z

<p>Eric Lengyel: </p> <hr /> <div>The ''transwedge product'' is a generalization of the [[exterior product]] and [[interior product]] that also includes a transitional sequence of liminal products between exterior and interior.<br /> <br /> == Transwedge Product ==<br /> <br /> The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\tilde c^{\unicode{x2605}}})}$$.<br /> <br /> The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.<br /> <br /> When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf{\tilde a^{\unicode{x2605}}}$$, which is an interior product. If $$k &gt; \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.<br /> <br /> An equivalent definition for the transwedge product is given by<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\tilde c_{\unicode{x2605}}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,<br /> <br /> where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.<br /> <br /> For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it&amp;rsquo;s nonzero.<br /> <br /> The sum of all possible transwedge products yields the geometric product. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,<br /> <br /> where $$n$$ is the dimension of the algebra.<br /> <br /> The [[geometric product]] of each pair of the 16 basis elements in the 3D rigid algebra is given by exactly one of the transwedge products. These are shown in the following table, which color codes the transwedge products of order 0, 1, 2, and 3. The transwedge product of order 4 is always zero in this algebra due to the degenerate metric.<br /> <br /> <br /> [[Image:TranswedgeProducts.svg|720px]]<br /> <br /> == Transwedge Antiproduct ==<br /> <br /> The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}^{\unicode{x2606}}})}$$,<br /> <br /> where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], and the [[reverse]] has become the [[antireverse]].<br /> <br /> The sum of all possible transwedge antiproducts yields the geometric antiproduct. That is,<br /> <br /> :$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,<br /> <br /> The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, and 3.<br /> <br /> <br /> [[Image:TranswedgeAntiproducts.svg|720px]]<br /> <br /> <br /> == See Also ==<br /> <br /> * [[Exterior products]]<br /> * [[Interior products]]</div>

Eric Lengyel