Dual translation

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

The specific kind of dual motor


 * $$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + \mathbf 1$$

performs a perspective projection in the direction of $$\mathbf t = (t_x, t_y, t_z)$$ with the focal length given by


 * $$g = \dfrac{1}{2\Vert \mathbf t \Vert}$$.

Example
The left image below shows the flow field in the x-z plane for the translation $$\mathbf T = -\frac{1}{2} \mathbf e_{12} + {\large\unicode{x1d7d9}}$$. The right image shows the flow field in the x-z plane for the dual translation $$\mathbf T = \frac{1}{2} \mathbf e_{43} + \mathbf 1$$. The yellow line is fixed as a whole, but points on it move to other locations on the line. All points with $$z = 0$$, represented by the blue plane, are fixed. The white plane at $$z = -1$$ represents the division between regions where the signs of projected $$z$$ coordinates are positive and negative.



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

Dual Translation to Horizon
A plane $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ is dual translated to the horizon by the operator


 * $$\mathbf T = \dfrac{f_{x\vphantom{y}}}{2f_w} \mathbf e_{41} + \dfrac{f_y}{2f_w} \mathbf e_{42} + \dfrac{f_{z\vphantom{y}}}{2f_w} \mathbf e_{32} + \mathbf 1$$.