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Combined display of all available logs of Rigid Geometric Algebra. You can narrow down the view by selecting a log type, the username (case-sensitive), or the affected page (also case-sensitive).

• 06:16, 15 July 2023 Eric Lengyel talk contribs created page File:Plane.svg
• 06:16, 15 July 2023 Eric Lengyel talk contribs uploaded File:Plane.svg
• 06:15, 15 July 2023 Eric Lengyel talk contribs created page Grade and antigrade (Created page with "The ''grade'' of a basis element in a geometric algebra is equal to the number of basis vectors present in its factorization. An arbitrary element whose components all have the same grade is also said to have that grade. The ''antigrade'' of a basis element is equal to the number of basis vectors absent from its factorization. The grade of an element $$\mathbf x$$ is denoted by $$\operatorname{gr}(\mathbf x)$$, and the antigrade is denoted by $$\operatorname{ag}(\mathb...")
• 06:14, 15 July 2023 Eric Lengyel talk contribs created page Reverses (Created page with "''Reverses'' are unary operations in geometric algebra that are analogs of conjugate or transpose operations. For any element $$\mathbf x$$ that is the wedge product of $$k$$ vectors, the ''reverse'' of $$\mathbf x$$, which we denote by $$\mathbf{\tilde x}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{234}$$ is $$\mathbf e_4 \wedge \mathbf e_3 \wedge \mathbf e_2$$, which we would write as $$-\math...")
• 06:14, 15 July 2023 Eric Lengyel talk contribs created page File:Reverses.svg
• 06:14, 15 July 2023 Eric Lengyel talk contribs uploaded File:Reverses.svg
• 06:12, 15 July 2023 Eric Lengyel talk contribs created page File:Plane line.svg
• 06:12, 15 July 2023 Eric Lengyel talk contribs uploaded File:Plane line.svg
• 06:12, 15 July 2023 Eric Lengyel talk contribs created page File:Line point.svg
• 06:12, 15 July 2023 Eric Lengyel talk contribs uploaded File:Line point.svg
• 06:12, 15 July 2023 Eric Lengyel talk contribs created page File:Plane point.svg
• 06:12, 15 July 2023 Eric Lengyel talk contribs uploaded File:Plane point.svg
• 06:11, 15 July 2023 Eric Lengyel talk contribs created page File:Line plane.svg
• 06:11, 15 July 2023 Eric Lengyel talk contribs uploaded File:Line plane.svg
• 06:11, 15 July 2023 Eric Lengyel talk contribs created page File:Point line.svg
• 06:11, 15 July 2023 Eric Lengyel talk contribs uploaded File:Point line.svg
• 06:11, 15 July 2023 Eric Lengyel talk contribs created page File:Point plane.svg
• 06:11, 15 July 2023 Eric Lengyel talk contribs uploaded File:Point plane.svg
• 06:11, 15 July 2023 Eric Lengyel talk contribs created page Projections (Created page with "Projections and antiprojections of one geometric object onto another can be accomplished using interior products as described below. The formulas on this page are general and do not require the geometric objects to be unitized. Most of them become simpler if unitization can be assumed. == Projection == The geometric projection of an object $$\mathbf x$$ onto an object $$\mathbf y$$ is given by the general formula $$(\mathbf y_\unicode{x25CB} \mathbin{\unicode{...")
• 06:10, 15 July 2023 Eric Lengyel talk contribs created page Euclidean distance (Created page with "The Euclidean distance between geometric objects can be measured by using commutators to calculate homogeneous magnitudes. The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be unitized. Most of them become simpler if unitization can be assumed. The points, lines, and p...")
• 06:10, 15 July 2023 Eric Lengyel talk contribs created page File:Distance line line.svg
• 06:10, 15 July 2023 Eric Lengyel talk contribs uploaded File:Distance line line.svg
• 06:10, 15 July 2023 Eric Lengyel talk contribs created page File:Distance point plane.svg
• 06:10, 15 July 2023 Eric Lengyel talk contribs uploaded File:Distance point plane.svg
• 06:09, 15 July 2023 Eric Lengyel talk contribs created page File:Distance point line.svg
• 06:09, 15 July 2023 Eric Lengyel talk contribs uploaded File:Distance point line.svg
• 06:09, 15 July 2023 Eric Lengyel talk contribs created page File:Distance point point.svg
• 06:09, 15 July 2023 Eric Lengyel talk contribs uploaded File:Distance point point.svg
• 06:06, 15 July 2023 Eric Lengyel talk contribs created page Flector (Created page with "400px|thumb|right|'''Figure 1.''' A flector represents an improper Euclidean isometry, which can always be regarded as a rotation about a line $$\boldsymbol l$$ and a reflection across a plane perpendicular to the same line. A ''flector'' is an operator that performs an improper isometry in Euclidean space. Such isometries encompass all possible combinations of an odd number of reflections, inversions, transflections, and rotorefle...")
• 06:06, 15 July 2023 Eric Lengyel talk contribs created page File:Improper isom.svg
• 06:06, 15 July 2023 Eric Lengyel talk contribs uploaded File:Improper isom.svg
• 06:04, 15 July 2023 Eric Lengyel talk contribs created page File:Complements.svg
• 06:04, 15 July 2023 Eric Lengyel talk contribs uploaded File:Complements.svg
• 06:04, 15 July 2023 Eric Lengyel talk contribs created page Complements (Created page with "''Complements'' are unary operations in geometric algebra that perform a specific type of dualization. Every basis element $$\mathbf x$$ has a ''right complement'', which we denote by $$\overline{\mathbf x}$$, that satisfies the equation :$$\mathbf x \wedge \overline{\mathbf x} = {\large\unicode{x1D7D9}}$$ . There is also a ''left complement'', which we denote by $$\underline{\mathbf x}$$, that satisfies the equation :$$\underline{\mathbf x} \wedge \mathbf x = {\larg...")
• 06:03, 15 July 2023 Eric Lengyel talk contribs created page Geometric property (Created page with "An element $$\mathbf x$$ of a geometric algebra possesses the ''geometric property'' if and only if the geometric product between $$\mathbf x$$ and its own reverse is a scalar, which is given by the dot product, and the geometric antiproduct between $$\mathbf x$$ and its own antireverse is an antiscalar, which is given by the antidot product. That is, :$$\mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde x} = \mathbf x \mathbin{\unicode{x25CF}} \mathbf{\...")
• 06:03, 15 July 2023 Eric Lengyel talk contribs created page Unitization (Created page with "''Unitization'' is the process of scaling an element of a projective geometric algebra so that its weight norm becomes the antiscalar $$\large\unicode{x1D7D9}$$. An element that has a weight norm of $$\large\unicode{x1D7D9}$$ is said to be ''unitized''. An element $$\mathbf x$$ is unitized by calculating :$$\mathbf{\hat x} = \dfrac{\mathbf x}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf x}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \smash{\ma...")
• 06:02, 15 July 2023 Eric Lengyel talk contribs created page Geometric norm (Created page with "The ''geometric norm'' is a measure of the magnitude of an element. It has two components called the bulk norm and the weight norm. For points, lines, and planes, the geometric norm is equal to the shortest Euclidean distance between the geometry and the origin. For motors and flectors, the geometric norm is equal to half the distance that the origin is moved by the isometry operator. == Bulk Norm == The ''bulk norm'' of an element $$\mathbf x$$, d...")
• 05:59, 15 July 2023 Eric Lengyel talk contribs created page Dual rotation (Created page with "A ''dual rotation'' is a proper isometry of dual Euclidean space. For a bulk normalized line $$\boldsymbol l$$, the specific kind of dual motor :$$\mathbf R = \boldsymbol l\sin\phi + \mathbf 1\cos\phi$$ , performs a dual rotation of an object $$\mathbf x$$ by twice the angle $$\phi$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27D1}} \mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde R}$$. The line $$\boldsymbol l$$ and its bulk complement...")
• 05:59, 15 July 2023 Eric Lengyel talk contribs created page File:DualRotation.svg
• 05:59, 15 July 2023 Eric Lengyel talk contribs uploaded File:DualRotation.svg
• 05:59, 15 July 2023 Eric Lengyel talk contribs created page File:Rotation.svg
• 05:59, 15 July 2023 Eric Lengyel talk contribs uploaded File:Rotation.svg
• 05:59, 15 July 2023 Eric Lengyel talk contribs created page Dual translation (Created page with "__NOTOC__ A ''dual translation'' is a proper isometry of dual Euclidean space. The specific kind of dual motor :$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + \mathbf 1$$ performs a perspective projection in the direction of $$\mathbf t = (t_x, t_y, t_z)$$ with the focal length given by :$$g = \dfrac{1}{2\Vert \mathbf t \Vert}$$ . == Example == The left image below shows the flow field in the ''x''-''z'' plane for the translation $...")
• 05:58, 15 July 2023 Eric Lengyel talk contribs created page File:DualTranslation.svg
• 05:58, 15 July 2023 Eric Lengyel talk contribs uploaded File:DualTranslation.svg
• 05:58, 15 July 2023 Eric Lengyel talk contribs created page File:Translation.svg
• 05:58, 15 July 2023 Eric Lengyel talk contribs uploaded File:Translation.svg
• 05:57, 15 July 2023 Eric Lengyel talk contribs created page Translation (Created page with "__NOTOC__ A ''translation'' is a proper isometry of Euclidean space. The specific kind of motor :$$\mathbf T = {t_x \mathbf e_{23} + t_y \mathbf e_{31} + t_z \mathbf e_{12} + \large\unicode{x1d7d9}}$$ performs a translation by twice the displacement vector $$\mathbf t = (t_x, t_y, t_z)$$ when used as an operator in the sandwich antiproduct. This can be interpreted as a rotation about the line at infinity perpendicular to the direction $$\mathbf t$$. === Trans...")
• 05:57, 15 July 2023 Eric Lengyel talk contribs created page Rotation (Created page with "A ''rotation'' is a proper isometry of Euclidean space. For a unitized line $$\boldsymbol l$$, the specific kind of motor :$$\mathbf R = \boldsymbol l\sin\phi + {\large\unicode{x1d7d9}}\cos\phi$$ , performs a rotation by twice the angle $$\phi$$ about the line $$\boldsymbol l$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{R}}$$. The operator $$\mathb...")
• 05:56, 15 July 2023 Eric Lengyel talk contribs created page Reflection (Created page with "A ''reflection'' is an improper isometry of Euclidean space. When used as an operator in the sandwich antiproduct, a unitized plane $$\mathbf F = F_{gx}\mathbf e_{423} + F_{gy}\mathbf e_{431} + F_{gz}\mathbf e_{412} + F_{gw}\mathbf e_{321}$$ is a specific kind of flector that performs a reflection through $$\mathbf F$$. == Calculation == The exact reflection calculations for points, lines, and planes are shown in the following table. {| class="wikitable"...")