# Rotation

A rotation is a proper isometry of Euclidean space.

For a unitized line $$\boldsymbol l$$, the specific kind of motor

$$\mathbf R = \boldsymbol l\sin\phi + {\large\unicode{x1d7d9}}\cos\phi$$ ,

performs a rotation by twice the angle $$\phi$$ about the line $$\boldsymbol l$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{R}}$$. The operator $$\mathbf R$$ differs from a general motor only in that it is always the case that $$R_{mw} = 0$$. The line $$\boldsymbol l$$ and its weight dual $$\boldsymbol l^\unicode["segoe ui symbol"]{x2606}$$ are invariant under this operation. The line $$\boldsymbol l^\unicode["segoe ui symbol"]{x2606}$$ lies in the horizon in directions perpendicular to the direction of $$\boldsymbol l$$.

## Calculation

The exact rotation calculations for points, lines, and planes are shown in the following table, where $$\mathbf v = (R_{vx}, R_{vy}, R_{vz})$$ and $$\mathbf m = (R_{mx}, R_{my}, R_{mz})$$.

Type Transformation
Point

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

$$\mathbf a = \mathbf v \times \mathbf p_{xyz} + p_w\mathbf m$$

$$\mathbf p'_{xyz} = \mathbf p_{xyz} + 2(R_{vw}\mathbf a + \mathbf v \times \mathbf a)$$

$$p'_w = p_w$$

Line

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

$$\mathbf a = \mathbf v \times \boldsymbol l_{\mathbf v}$$

$$\mathbf b = \mathbf v \times \boldsymbol l_{\mathbf m}$$

$$\mathbf c = \mathbf m \times \boldsymbol l_{\mathbf v}$$

$$\boldsymbol l'_{\mathbf v} = \boldsymbol l_{\mathbf v} + 2(R_{vw}\mathbf a + \mathbf v \times \mathbf a)$$

$$\boldsymbol l'_{\mathbf m} = \boldsymbol l_{\mathbf m} + 2(R_{vw}(\mathbf b + \mathbf c) + \mathbf v \times (\mathbf b + \mathbf c) + \mathbf m \times \mathbf a)$$

Plane

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

$$\mathbf a = \mathbf v \times \mathbf g_{xyz}$$

$$\mathbf g'_{xyz} = \mathbf g_{xyz} + 2(R_{vw}\mathbf a + \mathbf v \times \mathbf a)$$

$$g'_w = g_w + 2[(\mathbf m \times \mathbf g_{xyz}) \cdot \mathbf v - R_{vw}(\mathbf m \cdot \mathbf g_{xyz})]$$