Rotation

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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 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 $$\boldsymbol l$$.

Calculation

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

Type Transformation
Point

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

$$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \mathbf p \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2R_{vy}^2 - 2R_{vz}^2)p_x + 2(R_{vx}R_{vy} - R_{vz}R_{vw})p_y + 2(R_{vz}R_{vx} + R_{vy}R_{vw})p_z + 2(R_{vy}R_{mz} - R_{vz}R_{my} + R_{vw}R_{mx})p_w\right]\mathbf e_1 \\ +\, &\left[(1 - 2R_{vz}^2 - 2R_{vx}^2)p_y + 2(R_{vy}R_{vz} - R_{vx}R_{vw})p_z + 2(R_{vx}R_{vy} + R_{vz}R_{vw})p_x + 2(R_{vz}R_{mx} - R_{vx}R_{mz} + R_{vw}R_{my})p_w\right]\mathbf e_2 \\ +\, &\left[(1 - 2R_{vx}^2 - 2R_{vy}^2)p_z + 2(R_{vz}R_{vx} - R_{vy}R_{vw})p_x + 2(R_{vy}R_{vz} + R_{vx}R_{vw})p_y + 2(R_{vx}R_{my} - R_{vy}R_{mx} + R_{vw}R_{mz})p_w\right]\mathbf e_3 \\ +\, &p_w\mathbf e_4\end{split}$$
Line

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

$$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2R_{vy}^2 - 2R_{vz}^2)l_{vx} + 2(R_{vx}R_{vy} - R_{vz}R_{vw})l_{vy} + 2(R_{vz}R_{vx} + R_{vy}R_{vw})l_{vz}\right]\mathbf e_{41} \\ +\, &\left[(1 - 2R_{vz}^2 - 2R_{vx}^2)l_{vy} + 2(R_{vy}R_{vz} - R_{vx}R_{vw})l_{vz} + 2(R_{vx}R_{vy} + R_{vz}R_{vw})l_{vx}\right]\mathbf e_{42} \\ +\, &\left[(1 - 2R_{vx}^2 - 2R_{vy}^2)l_{vz} + 2(R_{vz}R_{vx} - R_{vy}R_{vw})l_{vx} + 2(R_{vy}R_{vz} + R_{vx}R_{vw})l_{vy}\right]\mathbf e_{43} \\ +\, &\left[-4(R_{vy}R_{my} + R_{vz}R_{mz})l_{vx} + 2(R_{vy}R_{mx} + R_{vx}R_{my} - R_{vw}R_{mz})l_{vy} + 2(R_{vz}R_{mx} + R_{vx}R_{mz} + R_{vw}R_{my})l_{vz} + (1 - 2R_{vy}^2 - 2R_{vz}^2)l_{mx} + 2(R_{vx}R_{vy} - R_{vz}R_{vw})l_{my} + 2(R_{vz}R_{vx} + R_{vy}R_{vw})l_{mz}\right]\mathbf e_{23} \\ +\, &\left[-4(R_{vz}R_{mz} + R_{vx}R_{mx})l_{vy} + 2(R_{vz}R_{my} + R_{vy}R_{mz} - R_{vw}R_{mx})l_{vz} + 2(R_{vx}R_{my} + R_{vy}R_{mx} + R_{vw}R_{mz})l_{vx} + (1 - 2R_{vz}^2 - 2R_{vx}^2)l_{my} + 2(R_{vy}R_{vz} - R_{vx}R_{vw})l_{mz} + 2(R_{vx}R_{vy} + R_{vz}R_{vw})l_{mx}\right]\mathbf e_{31} \\ +\, &\left[-4(R_{vx}R_{mx} + R_{vy}R_{my})l_{vz} + 2(R_{vx}R_{mz} + R_{vz}R_{mx} - R_{vw}R_{my})l_{vx} + 2(R_{vy}R_{mz} + R_{vz}R_{my} + R_{vw}R_{mx})l_{vy} + (1 - 2R_{vx}^2 - 2R_{vy}^2)l_{mz} + 2(R_{vz}R_{vx} - R_{vy}R_{vw})l_{mx} + 2(R_{vy}R_{vz} + R_{vx}R_{vw})l_{my}\right]\mathbf e_{12}\end{split}$$
Plane

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

$$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \mathbf g \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2R_{vy}^2 - 2R_{vz}^2)g_x + 2(R_{vx}R_{vy} - R_{vz}R_{vw})g_y + 2(R_{vz}R_{vx} + R_{vy}R_{vw})g_z\right]\mathbf e_{423} \\ +\, &\left[(1 - 2R_{vz}^2 - 2R_{vx}^2)g_y + 2(R_{vy}R_{vz} - R_{vx}R_{vw})g_z + 2(R_{vx}R_{vy} + R_{vz}R_{vw})g_x\right]\mathbf e_{431} \\ +\, &\left[(1 - 2R_{vx}^2 - 2R_{vy}^2)g_z + 2(R_{vz}R_{vx} - R_{vy}R_{vw})g_x + 2(R_{vy}R_{vz} + R_{vx}R_{vw})g_y\right]\mathbf e_{412} \\ +\, &\left[2(R_{vy}R_{mz} - R_{vz}R_{my} - R_{vw}R_{mx})g_x + 2(R_{vz}R_{mx} - R_{vx}R_{mz} - R_{vw}R_{my})g_y + 2(R_{vx}R_{my} - R_{vy}R_{mx} - R_{vw}R_{mz})g_z + g_w\right]\mathbf e_{321}\end{split}$$

See Also