# Rotation

A rotation is a proper isometry of Euclidean space.

For a unitized line $$\mathbf L$$, the specific kind of motor

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

performs a rotation by twice the angle $$\phi$$ about the line $$\mathbf L$$. This differs from a general motor only in that it is always the case that $$u_w = 0$$. The exact rotation calculations for points, lines, and planes are shown in the following table.

Type Transformation
Point

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

$$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \mathbf p \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2(r_y^2 + r_z^2))p_x + 2(r_xr_y - r_zr_w)p_y + 2(r_zr_x + 2r_yr_w)p_z + 2(r_yu_z - r_zu_y + r_wu_x)p_w\right]\mathbf e_1 \\ +\, &\left[(1 - 2(r_z^2 + r_x^2))p_y + 2(r_yr_z - r_xr_w)p_z + 2(r_xr_y + 2r_zr_w)p_x + 2(r_zu_x - r_xu_z + r_wu_y)p_w\right]\mathbf e_2 \\ +\, &\left[(1 - 2(r_x^2 + r_y^2))p_z + 2(r_zr_x - r_yr_w)p_x + 2(r_yr_z + 2r_xr_w)p_y + 2(r_xu_y - r_yu_x + r_wu_z)p_w\right]\mathbf e_3 \\ +\, &p_w\mathbf e_4\end{split}$$
Line

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

$$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \mathbf L \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2r_y^2 - 2r_z^2)v_x + 2(r_xr_y - r_zr_w)v_y + 2(r_zr_x + 2r_yr_w)v_z\right]\mathbf e_{41} \\ +\, &\left[(1 - 2r_z^2 - 2r_x^2)v_y + 2(r_yr_z - r_xr_w)v_z + 2(r_xr_y + 2r_zr_w)v_x\right]\mathbf e_{42} \\ +\, &\left[(1 - 2r_x^2 - 2r_y^2)v_z + 2(r_zr_x - r_yr_w)v_x + 2(r_yr_z + 2r_xr_w)v_y\right]\mathbf e_{43} \\ +\, &\left[-4(r_yu_y + r_zu_z)v_x + 2(r_yu_x + r_xu_y - r_wu_z)v_y + 2(r_zu_x + r_xu_z + r_wu_y)v_z + (1 - 2r_y^2 - 2r_z^2)m_x + 2(r_xr_y - r_zr_w)m_y + 2(r_zr_x + 2r_yr_w)m_z\right]\mathbf e_{23} \\ +\, &\left[-4(r_zu_z + r_xu_x)v_y + 2(r_zu_y + r_yu_z - r_wu_x)v_z + 2(r_xu_y + r_yu_x + r_wu_z)v_x + (1 - 2r_z^2 - 2r_x^2)m_y + 2(r_yr_z - r_xr_w)m_z + 2(r_xr_y + 2r_zr_w)m_x\right]\mathbf e_{31} \\ +\, &\left[-4(r_xu_x + r_yu_y)v_z + 2(r_xu_z + r_zu_x - r_wu_y)v_x + 2(r_yu_z + r_zu_y + r_wu_x)v_y + (1 - 2r_x^2 - 2r_y^2)m_z + 2(r_zr_x - r_yr_w)m_x + 2(r_yr_z + 2r_xr_w)m_y\right]\mathbf e_{12}\end{split}$$
Plane

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

$$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \mathbf f \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2r_y^2 - 2r_z^2)f_x + 2(r_xr_y - r_zr_w)f_y + 2(r_zr_x + r_yr_w)f_z\right]\mathbf e_{234} \\ +\, &\left[(1 - 2r_z^2 - 2r_x^2)f_y + 2(r_yr_z - r_xr_w)f_z + 2(r_xr_y + r_zr_w)f_x\right]\mathbf e_{314} \\ +\, &\left[(1 - 2r_x^2 - 2r_y^2)f_z + 2(r_zr_x - r_yr_w)f_x + 2(r_yr_z + r_xr_w)f_y\right]\mathbf e_{124} \\ +\, &\left[2(r_yu_z - r_zu_y - r_wu_x)f_x + 2(r_zu_x - r_xu_z - r_wu_y)f_y + 2(r_xu_y - r_yu_x - r_wu_z)f_z + f_w\right]\mathbf e_{321}\end{split}$$