# 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.

Type Transformation
Point

$$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$

$$\begin{split}\mathbf F \mathbin{\unicode{x27C7}} \mathbf q \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} =\, &\left[(2F_{gy}^2 + 2F_{gz}^2 - 1)q_x - 2F_{gx} F_{gy} q_y - 2F_{gz} F_{gx} q_z + 2(F_{gy} F_{pz} - F_{gz} F_{py} - F_{gx} F_{gw})q_w\right]\mathbf e_1 \\ +\, &\left[(2F_{gz}^2 + 2F_{gx}^2 - 1)q_y - 2F_{gy} F_{gz} q_z - 2F_{gx} F_{gy} q_x + 2(F_{gz} F_{px} - F_{gx} F_{pz} - F_{gy} F_{gw})q_w\right]\mathbf e_2 \\ +\, &\left[(2F_{gx}^2 + 2F_{gy}^2 - 1)q_z - 2F_{gz} F_{gx} q_x - 2F_{gy} F_{gz} q_y + 2(F_{gx} F_{py} - F_{gy} F_{px} - F_{gz} F_{gw})q_w\right]\mathbf e_3 \\ +\, &q_w\mathbf e_4\end{split}$$
Line

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

$$\begin{split}\mathbf F \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} =\, &\left[(1 - 2F_{gy}^2 - 2F_{gz}^2)l_{vx} + 2F_{gx} F_{gy} l_{vy} + 2 F_{gz} F_{gx} l_{vz}\right]\mathbf e_{41} \\ +\, &\left[(1 - 2F_{gz}^2 - 2F_{gx}^2)l_{vy} + 2F_{gy} F_{gz} l_{vz} + 2F_{gx} F_{gy} l_{vx}\right]\mathbf e_{42} \\ +\, &\left[(1 - 2F_{gx}^2 - 2F_{gy}^2)l_{vz} + 2F_{gz} F_{gx} l_{vx} + 2F_{gy} F_{gz} l_{vy}\right]\mathbf e_{43} \\ +\, &\left[-4(F_{gy} F_{py} + F_{gz} P_{pz})l_{vx} + 2(F_{gx} F_{py} + F_{gy} F_{px})l_{vy} + 2(F_{gx} F_{pz} + F_{gz} F_{px} + F_{gy} F_{gw})l_{vz} + (2F_{gy}^2 + 2F_{gz}^2 - 1)l_{mx} - 2F_{gx} F_{gy} l_{my} - 2F_{gz} F_{gx} l_{mz}\right]\mathbf e_{23} \\ +\, &\left[-4(F_{gz} F_{pz} + F_{gx} F_{px})l_{vy} + 2(F_{gy} F_{pz} + F_{gz} F_{py})l_{vz} + 2(F_{gy} F_{px} + F_{gx} F_{py} + F_{gz} F_{gw})l_{vx} + (2F_{gz}^2 + 2F_{gx}^2 - 1)l_{my} - 2F_{gy} F_{gz} l_{mz} - 2F_{gx} F_{gy} l_{mx}\right]\mathbf e_{31} \\ +\, &\left[-4(F_{gx} F_{px} + F_{gy} F_{py})l_{vz} + 2(F_{gz} F_{px} + F_{gx} F_{pz})l_{vx} + 2(F_{gz} F_{py} + F_{gy} F_{pz} + F_{gx} F_{gw})l_{vy} + (2F_{gx}^2 + 2F_{gy}^2 - 1)l_{mz} - 2F_{gz} F_{gx} l_{mx} - 2F_{gy} F_{gz} l_{my}\right]\mathbf e_{12}\end{split}$$
Plane

$$\mathbf h = h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412} + h_w \mathbf e_{321}$$

$$\begin{split}\mathbf F \mathbin{\unicode{x27C7}} \mathbf h \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} =\, &\left[(1 - 2F_{gy}^2 - 2F_{gz}^2)h_x + 2F_{gx} F_{gy} h_y + 2F_{gz} F_{gx} h_z\right]\mathbf e_{423} \\ +\, &\left[(1 - 2F_{gz}^2 - 2F_{gx}^2)h_y + 2F_{gy} F_{gz} h_z + 2F_{gx} F_{gy} h_x\right]\mathbf e_{431} \\ +\, &\left[(1 - 2F_{gx}^2 - 2F_{gy}^2)h_z + 2F_{gz} F_{gx} h_x + 2F_{gy} F_{gz} h_y\right]\mathbf e_{412} \\ +\, &\left[2(F_{gy} F_{pz} - F_{gz} F_{py} + F_{gx} F_{gw})h_x + 2(F_{gz} F_{px} - F_{gx} F_{pz} + F_{gy} F_{gw})h_y + 2(F_{gx} F_{py} - F_{gy} F_{px} + F_{gz} F_{gw})h_z - h_w\right]\mathbf e_{321}\end{split}$$