# Transformation groups

In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, every Euclidean isometry of 3D space can be represented by a motor $$\mathbf Q$$ of the form

$$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$

or by a flector $$\mathbf G$$ of the form

$$\mathbf G = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$ .

Under the geometric antiproduct $$\unicode{x27C7}$$, arbitrary products of these operators form the Euclidean group E(3) with $${\large\unicode{x1D7D9}}$$ as the identity, and they covariantly transform any object $$\mathbf x$$ in the algebra through the sandwich products $$\mathbf x' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$ and $$\mathbf x' = -\mathbf G \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{G}}}$$.

Symmetrically, every dual Euclidean isometry of 3D space can be represented by an antimotor $$\mathbf Q$$ of the form

$$\mathbf Q = r_x \mathbf e_{23} + r_y \mathbf e_{31} + r_z \mathbf e_{12} - r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{41} + u_y \mathbf e_{42} + u_z \mathbf e_{43} - u_w$$

or by an antiflector $$\mathbf G$$ of the form

$$\mathbf G = s_x \mathbf e_{234} + s_y \mathbf e_{314} + s_z \mathbf e_{124} + s_w \mathbf e_{321} - h_x \mathbf e_1 - h_y \mathbf e_2 - h_z \mathbf e_3 - h_w \mathbf e_4$$ .

Under the geometric product $$\unicode{x27D1}$$, arbitrary products of these operators form the dual Euclidean group DE(3) with $$\mathbf 1$$ as the identity, and they covariantly transform any object $$\mathbf x$$ in the algebra through the sandwich products $$\mathbf x' = \mathbf Q \mathbin{\unicode{x27D1}} \mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde Q}$$ and $$\mathbf x' = -\mathbf G \mathbin{\unicode{x27D1}} \mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde G}$$.