Reflection
A reflection is an improper isometry of Euclidean space.
When used as an operator in the sandwich antiproduct, a unitized plane $$\mathbf F = F_{gx}\mathbf e_{423} + F_{gy}\mathbf e_{431} + F_{gz}\mathbf e_{412} + F_{gw}\mathbf e_{321}$$ is a specific kind of flector that performs a reflection through $$\mathbf F$$.
Calculation
The exact reflection calculations for points, lines, and planes are shown in the following table.
| Type | Reflection |
|---|---|
| Point
$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ |
$$\begin{split}\mathbf F \mathbin{\unicode{x27C7}} \mathbf p \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} =\, &((2F_{gy}^2 + 2F_{gz}^2 - 1)p_x \,&-\, 2F_{gx} F_{gy} p_y \,&-\, 2F_{gz} F_{gx} p_z \,&-\, 2F_{gx} F_{gw} p_w)&\mathbf e_1 \\ +\, &((2F_{gz}^2 + 2F_{gx}^2 - 1)p_y \,&-\, 2F_{gy} F_{gz} p_z \,&-\, 2F_{gx} F_{gy} p_x \,&-\, 2F_{gy} F_{gw} p_w)&\mathbf e_2 \\ +\, &((2F_{gx}^2 + 2F_{gy}^2 - 1)p_z \,&-\, 2F_{gz} F_{gx} p_x \,&-\, 2F_{gy} F_{gz} p_y \,&-\, 2F_{gz} F_{gw} p_w)&\mathbf e_3 \\ +\, &p_w\mathbf e_4\end{split}$$ |
| Line
$$\begin{split}\boldsymbol l =\, &l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} \\ +\, &l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}\end{split}$$ |
$$\begin{split}\mathbf F \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} =\, &((1 - 2F_{gy}^2 - 2F_{gz}^2)l_{vx} \,&-\, 2F_{gx} F_{gy} l_{vy} \,&+\, 2F_{gz} F_{gx} l_{vz})&\mathbf e_{41} \\ +\, &((1 - 2F_{gz}^2 - 2F_{gx}^2)l_{vy} \,&-\, 2F_{gy} F_{gz} l_{vz} \,&+\, 2F_{gx} F_{gy} l_{vx})&\mathbf e_{42} \\ +\, &((1 - 2F_{gx}^2 - 2F_{gy}^2)l_{vz} \,&-\, 2F_{gz} F_{gx} l_{vx} \,&+\, 2F_{gy} F_{gz} l_{vy})&\mathbf e_{43} \\ +\, &((2F_{gy}^2 + 2F_{gz}^2 - 1)l_{mx} \,&-\, 2F_{gx} F_{gy} l_{my} \,&-\, 2F_{gz} F_{gx} l_{mz} \,&+\, 2F_{gw} F_{gy} l_{vz} \,&-\, 2F_{gw} F_{gz} l_{vy})&\mathbf e_{23} \\ +\, &((2F_{gz}^2 + 2F_{gx}^2 - 1)l_{my} \,&-\, 2F_{gy} F_{gz} l_{mz} \,&-\, 2F_{gx} F_{gy} l_{mx} \,&+\, 2F_{gw} F_{gz} l_{vx} \,&-\, 2F_{gw} F_{gx} l_{vz})&\mathbf e_{31} \\ +\, &((2F_{gx}^2 + 2F_{gy}^2 - 1)l_{mz} \,&-\, 2F_{gz} F_{gx} l_{mx} \,&-\, 2F_{gy} F_{gz} l_{my} \,&+\, 2F_{gw} F_{gx} l_{vy} \,&-\, 2F_{gw} F_{gy} l_{vx})&\mathbf e_{12}\end{split}$$ |
| Plane
$$\mathbf h = h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412} + h_w \mathbf e_{321}$$ |
$$\begin{split}\mathbf F \mathbin{\unicode{x27C7}} \mathbf h \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} =\, &((1 - 2F_{gy}^2 - 2F_{gz}^2)h_x \,&+\, 2F_{gx} F_{gy} h_y + 2F_{gz} F_{gx} h_z)&\mathbf e_{423} \\ +\, &((1 - 2F_{gz}^2 - 2F_{gx}^2)h_y \,&+\, 2F_{gy} F_{gz} h_z + 2F_{gx} F_{gy} h_x)&\mathbf e_{431} \\ +\, &((1 - 2F_{gx}^2 - 2F_{gy}^2)h_z \,&+\, 2F_{gz} F_{gx} h_x + 2F_{gy} F_{gz} h_y)&\mathbf e_{412} \\ +\, &\rlap{(2F_{gx} F_{gw} h_x + 2F_{gy} F_{gw} h_y + 2F_{gz} F_{gw} h_z - h_w)\mathbf e_{321}}\end{split}$$ |