Eric Lengyel: Created page with "An ''inversion'' is an improper isometry of Euclidean space. When used as an operator in the sandwich antiproduct, a unitized point $$\mathbf F = F_{px}\mathbf e_1 + F_{py}\mathbf e_2 + F_{pz}\mathbf e_3 + \mathbf e_4$$ is a specific kind of flector that performs an inversion through $$\mathbf F$$. == Calculation == The exact inversion calculations for points, lines, and planes are shown in the following table. {| class="wikitable" ! Type || Inversion |-..."

2023-07-15T05:55:49Z

Created page with "An ''inversion'' is an improper isometry of Euclidean space. When used as an operator in the sandwich antiproduct, a unitized point $$\mathbf F = F_{px}\mathbf e_1 + F_{py}\mathbf e_2 + F_{pz}\mathbf e_3 + \mathbf e_4$$ is a specific kind of flector that performs an inversion through $$\mathbf F$$. == Calculation == The exact inversion calculations for points, lines, and planes are shown in the following table. {| class="wikitable" ! Type || Inversion |-..."

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An ''inversion'' is an improper isometry of Euclidean space.

When used as an operator in the sandwich antiproduct, a [[unitized]] [[point]] $$\mathbf F = F_{px}\mathbf e_1 + F_{py}\mathbf e_2 + F_{pz}\mathbf e_3 + \mathbf e_4$$ is a specific kind of [[flector]] that performs an inversion through $$\mathbf F$$.

== Calculation ==

The exact inversion calculations for points, lines, and planes are shown in the following table.

{| class="wikitable"
! Type || Inversion
|-
| style="padding: 12px;" | [[Point]]

$$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$
| style="padding: 12px;" | $$\mathbf F \mathbin{\unicode{x27C7}} \mathbf q \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} = (2q_w F_{px} - q_x)\mathbf e_1 + (2q_w F_{py} - q_y)\mathbf e_2 + (2q_w F_{pz} - q_z)\mathbf e_3 + q_w\mathbf e_4$$
|-
| style="padding: 12px;" | [[Line]]

$$\begin{split}\boldsymbol l =\, &l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} \\ +\, &l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}\end{split}$$
| style="padding: 12px;" | $$\mathbf F \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + (2F_{py} l_{vz} - 2F_{pz} l_{vy} - l_{mx})\mathbf e_{23} + (2F_{pz} l_{vx} - 2F_{px} l_{vz} - l_{my})\mathbf e_{31} + (2F_{px} l_{vy} - 2F_{py} l_{vx} - l_{mz})\mathbf e_{12}$$
|-
| style="padding: 12px;" | [[Plane]]

$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
| style="padding: 12px;" | $$\mathbf F \mathbin{\unicode{x27C7}} \mathbf g \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} - (2F_{px} g_x + 2F_{py} g_y + 2F_{pz} g_z + g_w) \mathbf e_{321}$$
|}

== See Also ==

* [[Translation]]
* [[Rotation]]
* [[Reflection]]
* [[Transflection]]

Inversion - Revision history