Complement rotation

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A reciprocal rotation is a proper isometry of reciprocal Euclidean space.

For a bulk normalized line $$\boldsymbol l$$, the specific kind of reciprocal motor

$$\mathbf R = \boldsymbol l\sin\phi + \mathbf 1\cos\phi$$ ,

performs a reciprocal rotation of an object $$\mathbf x$$ by twice the angle $$\phi$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27D1}} \mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde R}$$. The line $$\boldsymbol l$$ and its bulk complement $$\underline{\boldsymbol l_\smash{\unicode{x25CF}}}$$ are invariant under this operation. The line $$\underline{\boldsymbol l_\smash{\unicode{x25CF}}}$$ passes through the origin and runs perpendicular to the line's moment bivector.

Under a reciprocal rotation, a point $$\mathbf p$$ follows an orbit of constant eccentricity as the angle $$\phi$$ ranges from 0 to $$\pi$$. The line $$\boldsymbol l$$ is the directrix for the orbit, and the intersection of $$\underline{\boldsymbol l_\smash{\unicode{x25CF}}}$$ with the plane $$\boldsymbol l \wedge \mathbf p$$ is the focus. The eccentricity is given by the distance from $$\mathbf p$$ to the focus divided by the distance from $$\mathbf p$$ to the directrix.

Example

The left image below shows the flow field in the x-y plane for the rotation $$\mathbf R = (\mathbf e_{43} - \frac{1}{2} \mathbf e_{31})\sin\phi + {\large\unicode{x1d7d9}}\cos\phi$$. The axis of rotation runs along the z direction through the yellow point. The right image shows the flow field in the x-y plane for the reciprocal rotation $$\mathbf R = (\frac{1}{2} \mathbf e_{42} - \mathbf e_{12})\sin\phi + \mathbf 1\cos\phi$$. Points follow orbits of constant eccentricity with respect to a focus at the origin and a directrix given by the yellow line.

Calculation

The exact reciprocal rotation calculations for points, lines, and planes transformed by the operator $$\mathbf R = R_{vx}\mathbf e_{41} + R_{vy}\mathbf e_{42} + R_{vz}\mathbf e_{43} + R_{mx}\mathbf e_{23} + R_{my}\mathbf e_{31} + R_{mz}\mathbf e_{12} + R_{mw}\mathbf 1$$ are shown in the following table. Here, it is assumed that $$\mathbf R$$ is bulk normalized so that $$R_{mx}^2 + R_{my}^2 + R_{mz}^2 + R_{mw}^2 = 1$$ and that $$\mathbf R$$ properly satisfies the geometric property so that $$R_{vx}R_{mx} + R_{vy}R_{my} + R_{vz}R_{mz} = 0$$.

Type Transformation
Point

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

$$\begin{split}\mathbf R \mathbin{\unicode{x27D1}} \mathbf p \mathbin{\unicode{x27D1}} \mathbf{\tilde R} =\, &\left[(1 - 2R_{my}^2 - 2R_{mz}^2)p_x + 2(R_{mx}R_{my} + R_{mz}R_{mw})p_y + 2(R_{mz}R_{mx} - R_{my}R_{mw})p_z\right]\mathbf e_1 \\ +\, &\left[(1 - 2R_{mz}^2 - 2R_{mx}^2)p_y + 2(R_{my}R_{mz} + R_{mx}R_{mw})p_z + 2(R_{mx}R_{my} - R_{mz}R_{mw})p_x\right]\mathbf e_2 \\ +\, &\left[(1 - 2R_{mx}^2 - 2R_{my}^2)p_z + 2(R_{mz}R_{mx} + R_{my}R_{mw})p_x + 2(R_{my}R_{mz} - R_{mx}R_{mw})p_y\right]\mathbf e_3 \\ +\, &\left[2(R_{my}R_{vz} - R_{mz}R_{vy} + R_{mw}R_{vx})p_x + 2(R_{mz}R_{vx} - R_{mx}R_{vz} + R_{mw}R_{vy})p_y + 2(R_{mx}R_{vy} - R_{my}R_{vx} + R_{mw}R_{vz})p_z + p_w\right]\mathbf e_4\end{split}$$
Line

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

$$\begin{split}\mathbf R \mathbin{\unicode{x27D1}} \mathbf L \mathbin{\unicode{x27D1}} \mathbf{\tilde R} =\, &\left[-4(R_{my}R_{vy} + R_{mz}R_{vz})l_{mx} + 2(R_{my}R_{vx} + R_{mx}R_{vy} + R_{mw}R_{vz})l_{my} + 2(R_{mz}R_{vx} + R_{mx}R_{vz} - R_{mw}R_{vy})l_{mz} + (1 - 2R_{my}^2 - 2R_{mz}^2)l_{vx} + 2(R_{mx}R_{my} + R_{mz}R_{mw})l_{vy} + 2(R_{mz}R_{mx} - R_{my}R_{mw})l_{vz}\right]\mathbf e_{41} \\ +\, &\left[-4(R_{mz}R_{vz} + R_{mx}R_{vx})l_{my} + 2(R_{mz}R_{vy} + R_{my}R_{vz} + R_{mw}R_{vx})l_{mz} + 2(R_{mx}R_{vy} + R_{my}R_{vx} - R_{mw}R_{vz})l_{mx} + (1 - 2R_{mz}^2 - 2R_{mx}^2)l_{vy} + 2(R_{my}R_{mz} + R_{mx}R_{mw})l_{vz} + 2(R_{mx}R_{my} - R_{mz}R_{mw})l_{vx}\right]\mathbf e_{42} \\ +\, &\left[-4(R_{mx}R_{vx} + R_{my}R_{vy})l_{mz} + 2(R_{mx}R_{vz} + R_{mz}R_{vx} + R_{mw}R_{vy})l_{mx} + 2(R_{my}R_{vz} + R_{mz}R_{vy} - R_{mw}R_{vx})l_{my} + (1 - 2R_{mx}^2 - 2R_{my}^2)l_{vz} + 2(R_{mz}R_{mx} + R_{my}R_{mw})l_{vx} + 2(R_{my}R_{mz} - R_{mx}R_{mw})l_{vy}\right]\mathbf e_{43} \\ +\, &\left[(1 - 2R_{my}^2 - 2R_{mz}^2)l_{mx} + 2(R_{mx}R_{my} + R_{mz}R_{mw})l_{my} + 2(R_{mz}R_{mx} - R_{my}R_{mw})l_{mz}\right]\mathbf e_{23} \\ +\, &\left[(1 - 2R_{mz}^2 - 2R_{mx}^2)l_{my} + 2(R_{my}R_{mz} + R_{mx}R_{mw})l_{mz} + 2(R_{mx}R_{my} - R_{mz}R_{mw})l_{mx}\right]\mathbf e_{31} \\ +\, &\left[(1 - 2R_{mx}^2 - 2R_{my}^2)l_{mz} + 2(R_{mz}R_{mx} + R_{my}R_{mw})l_{mx} + 2(R_{my}R_{mz} - R_{mx}R_{mw})l_{my}\right]\mathbf e_{12}\end{split}$$
Plane

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

$$\begin{split}\mathbf R \mathbin{\unicode{x27D1}} \mathbf g \mathbin{\unicode{x27D1}} \mathbf{\tilde R} =\, &\left[(1 - 2R_{my}^2 - 2R_{mz}^2)g_x + 2(R_{mx}R_{my} + R_{mz}R_{mw})g_y + 2(R_{mz}R_{mx} - R_{my}R_{mw})g_z + 2(R_{my}R_{vz} - R_{mz}R_{vy} - R_{mw}R_{vx})g_w\right]\mathbf e_{423} \\ +\, &\left[(1 - 2R_{mz}^2 - 2R_{mx}^2)g_y + 2(R_{my}R_{mz} + R_{mx}R_{mw})g_z + 2(R_{mx}R_{my} - R_{mz}R_{mw})g_x + 2(R_{mz}R_{vx} - R_{mx}R_{vz} - R_{mw}R_{vy})g_w\right]\mathbf e_{431} \\ +\, &\left[(1 - 2R_{mx}^2 - 2R_{my}^2)g_z + 2(R_{mz}R_{mx} + R_{my}R_{mw})g_x + 2(R_{my}R_{mz} - R_{mx}R_{mw})g_y + 2(R_{mx}R_{vy} - R_{my}R_{vx} - R_{mw}R_{vz})g_w\right]\mathbf e_{412} \\ +\, &g_w\mathbf e_{321}\end{split}$$