Transflection

A transflection is an improper isometry of Euclidean space consisting of a reflection through a plane and a translation parallel to the same plane. All combinations of a reflection and a translation, even if the original translation vector is not parallel to the original reflection plane, can be formulated as a transflection with respect to some plane.

The specific kind of flector


 * $$\mathbf F = F_{px} \mathbf e_{1} + F_{py} \mathbf e_{2} + F_{pz} \mathbf e_{3} + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$ ,

in which $$F_{pw} = 0$$, performs a reflection through the plane $$\mathbf g = F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$ and a translation by twice the displacement vector given by the cross product $$(F_{gx}, F_{gy}, F_{gz}) \times (F_{px}, F_{py}, F_{pz})$$.

By the geometric property, we must have $$F_{px} F_{gx} + F_{py} F_{gy} + F_{pz} F_{gz} = 0$$, so the vector $$(F_{px}, F_{py}, F_{pz})$$ and the displacement vector are both parallel to the plane $$\mathbf g$$.

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