# Inversion

An inversion is an improper isometry of Euclidean space.

When used as an operator in the sandwich product, a unitized point $$\mathbf q = q_x\mathbf e_1 + q_y\mathbf e_2 + q_z\mathbf e_3 + \mathbf e_4$$ is a specific kind of flector that performs an inversion through $$\mathbf q$$. The exact inversion calculations for points, lines, and planes are shown in the following table.

Type Inversion
Point

$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$

$$\mathbf q \mathbin{\unicode{x27C7}} \mathbf p \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{q}}} = (2p_wq_x - p_x)\mathbf e_1 + (2p_wq_y - p_y)\mathbf e_2 + (2p_wq_z - p_z)\mathbf e_3 + p_w\mathbf e_4$$
Line

$$\begin{split}\mathbf L =\, &v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} \\ +\, &m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}\end{split}$$

$$\mathbf q \mathbin{\unicode{x27C7}} \mathbf L \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{q}}} = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + (2q_yv_z - 2q_zv_y - m_x)\mathbf e_{23} + (2q_zv_x - 2q_xv_z - m_y)\mathbf e_{31} + (2q_xv_y - 2q_yv_x - m_z)\mathbf e_{12}$$
Plane

$$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$

$$\mathbf q \mathbin{\unicode{x27C7}} \mathbf f \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{q}}} = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} - (2q_xf_x + 2q_yf_y + 2q_zf_z + f_w) \mathbf e_{321}$$