# Translation

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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 $$\mathbf t$$. The exact translation calculations for points, lines, and planes are shown in the following table.

Type Translation
Point

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

$$\mathbf T \mathbin{\unicode{x27C7}} \mathbf p \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = (p_x + 2t_x)\mathbf e_1 + (p_y + 2t_y)\mathbf e_2 + (p_z + 2t_z)\mathbf e_3 + p_w\mathbf e_4$$
Line

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

$$\mathbf T \mathbin{\unicode{x27C7}} \mathbf L \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + (m_x + 2t_yv_z - 2t_zv_y)\mathbf e_{23} + (m_y + 2t_zv_x - 2t_xv_z)\mathbf e_{31} + (m_z + 2t_xv_y - 2t_yv_x)\mathbf e_{12}$$
Plane

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

$$\mathbf T \mathbin{\unicode{x27C7}} \mathbf f \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + (f_w - 2t_xf_x - 2t_yf_y - 2t_zf_z) \mathbf e_{321}$$

### 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

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