Translation

A translation is a proper isometry of Euclidean space.

The specific kind of motor


 * $$\mathbf T = {t_x \mathbf e_{23} + t_y \mathbf e_{31} + t_z \mathbf e_{12} + \large\unicode{x1d7d9}}$$

performs a translation by twice the displacement vector $$\mathbf t = (t_x, t_y, t_z)$$ when used as an operator in the sandwich antiproduct. This can be interpreted as a rotation about the line at infinity perpendicular to the direction $$\mathbf t$$.

Translation to Origin
A point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ is translated to the origin by the operator


 * $$\mathbf T = {-\dfrac{p_{x\vphantom{y}}}{2p_w} \mathbf e_{23} - \dfrac{p_y}{2p_w} \mathbf e_{31} - \dfrac{p_{z\vphantom{y}}}{2p_w} \mathbf e_{12} + \large\unicode{x1d7d9}}$$.

Exponential Form
A direction vector $$\mathbf v = v_x \mathbf e_1 + v_y \mathbf e_2 + v_z \mathbf e_3$$ is converted to a translation operator $$\mathbf T$$ through the exponential


 * $$\mathbf T = \overline{\exp_\unicode{x27D1}\left(\dfrac{1}{2}\mathbf v \wedge \mathbf e_4\right)} = \exp_\unicode{x27C7}{\left(\dfrac{1}{2}\overline{\mathbf v} \vee \mathbf e_{321}\right)}$$.

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