From Rigid Geometric Algebra
(Redirected from Inversions)
Jump to navigation Jump to search

An inversion is an improper isometry of Euclidean space.

When used as an operator in the sandwich antiproduct, 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

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

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

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

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

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

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

See Also