Dual rotation

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

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


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

performs a dual 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 dual 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 dual 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 dual 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 $$R_{vx}R_{mx} + R_{vy}R_{my} + R_{vz}R_{mz} = 0$$.