Inversion
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.
| Type | Inversion |
|---|---|
| Point
$$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$ |
$$\mathbf F \mathbin{\unicode{x27C7}} \mathbf q \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} = (2q_w F_{px} - q_x)\mathbf e_1 + (2q_w F_{py} - q_y)\mathbf e_2 + (2q_w F_{pz} - q_z)\mathbf e_3 + q_w\mathbf e_4$$ |
| Line
$$\begin{split}\boldsymbol l =\, &l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} \\ +\, &l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}\end{split}$$ |
$$\mathbf F \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + (2F_{py} l_{vz} - 2F_{pz} l_{vy} - l_{mx})\mathbf e_{23} + (2F_{pz} l_{vx} - 2F_{px} l_{vz} - l_{my})\mathbf e_{31} + (2F_{px} l_{vy} - 2F_{py} l_{vx} - l_{mz})\mathbf e_{12}$$ |
| Plane
$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$ |
$$\mathbf F \mathbin{\unicode{x27C7}} \mathbf g \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} - (2F_{px} g_x + 2F_{py} g_y + 2F_{pz} g_z + g_w) \mathbf e_{321}$$ |