# Difference between pages "Line" and "Rotation"

(Difference between pages)

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 $$u_w = 0$$. The line $$\boldsymbol l$$ and its weight complement $$\underline{\boldsymbol l_\smash{\unicode{x25CB}}}$$ are invariant under this operation. The line $$\underline{\boldsymbol l_\smash{\unicode{x25CB}}}$$ lies in the horizon in directions perpendicular to the direction of $$\boldsymbol l$$.

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

Type Transformation
Point

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

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

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

$$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2r_y^2 - 2r_z^2)l_{vx} + 2(r_xr_y - r_zr_w)l_{vy} + 2(r_zr_x + r_yr_w)l_{vz}\right]\mathbf e_{41} \\ +\, &\left[(1 - 2r_z^2 - 2r_x^2)l_{vy} + 2(r_yr_z - r_xr_w)l_{vz} + 2(r_xr_y + r_zr_w)l_{vx}\right]\mathbf e_{42} \\ +\, &\left[(1 - 2r_x^2 - 2r_y^2)l_{vz} + 2(r_zr_x - r_yr_w)l_{vx} + 2(r_yr_z + r_xr_w)l_{vy}\right]\mathbf e_{43} \\ +\, &\left[-4(r_yu_y + r_zu_z)l_{vx} + 2(r_yu_x + r_xu_y - r_wu_z)l_{vy} + 2(r_zu_x + r_xu_z + r_wu_y)l_{vz} + (1 - 2r_y^2 - 2r_z^2)l_{mx} + 2(r_xr_y - r_zr_w)l_{my} + 2(r_zr_x + r_yr_w)l_{mz}\right]\mathbf e_{23} \\ +\, &\left[-4(r_zu_z + r_xu_x)l_{vy} + 2(r_zu_y + r_yu_z - r_wu_x)l_{vz} + 2(r_xu_y + r_yu_x + r_wu_z)l_{vx} + (1 - 2r_z^2 - 2r_x^2)l_{my} + 2(r_yr_z - r_xr_w)l_{mz} + 2(r_xr_y + r_zr_w)l_{mx}\right]\mathbf e_{31} \\ +\, &\left[-4(r_xu_x + r_yu_y)l_{vz} + 2(r_xu_z + r_zu_x - r_wu_y)l_{vx} + 2(r_yu_z + r_zu_y + r_wu_x)l_{vy} + (1 - 2r_x^2 - 2r_y^2)l_{mz} + 2(r_zr_x - r_yr_w)l_{mx} + 2(r_yr_z + r_xr_w)l_{my}\right]\mathbf e_{12}\end{split}$$
Plane

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

$$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \mathbf g \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2r_y^2 - 2r_z^2)g_x + 2(r_xr_y - r_zr_w)g_y + 2(r_zr_x + r_yr_w)g_z\right]\mathbf e_{423} \\ +\, &\left[(1 - 2r_z^2 - 2r_x^2)g_y + 2(r_yr_z - r_xr_w)g_z + 2(r_xr_y + r_zr_w)g_x\right]\mathbf e_{431} \\ +\, &\left[(1 - 2r_x^2 - 2r_y^2)g_z + 2(r_zr_x - r_yr_w)g_x + 2(r_yr_z + r_xr_w)g_y\right]\mathbf e_{412} \\ +\, &\left[2(r_yu_z - r_zu_y - r_wu_x)g_x + 2(r_zu_x - r_xu_z - r_wu_y)g_y + 2(r_xu_y - r_yu_x - r_wu_z)g_z + g_w\right]\mathbf e_{321}\end{split}$$