# 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 $$\mathbf t$$.

### 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.

Type Translation
Point

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

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

$$\begin{split}\boldsymbol l =\, &l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} \\ +\, &l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}\end{split}$$

$$\mathbf T \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + (l_{mx} + 2t_y l_{vz} - 2t_z l_{vy})\mathbf e_{23} + (l_{my} + 2t_z l_{vx} - 2t_x l_{vz})\mathbf e_{31} + (l_{mz} + 2t_x l_{vy} - 2t_y l_{vx})\mathbf e_{12}$$
Plane

$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$

$$\mathbf T \mathbin{\unicode{x27C7}} \mathbf g \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + (g_w - 2t_xg_x - 2t_yg_y - 2t_zg_z) \mathbf e_{321}$$