Transflection and Reflection: Difference between pages

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(Created page with "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} \math...")
 
(Created page with "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. {| class="wikitable"...")
 
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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.
A ''reflection'' is an improper isometry of Euclidean space.


The specific kind of [[flector]]
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 $$\mathbf F$$.
 
:$$\mathbf F = F_{px} \mathbf e_{1} + F_{py} \mathbf e_{2} + F_{pz} \mathbf e_{3} + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$ ,
 
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 ==
== Calculation ==


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


{| class="wikitable"
{| class="wikitable"
! Type || Transformation
! Type || Reflection
|-
|-
| style="padding: 12px;" | [[Point]]
| style="padding: 12px;" | [[Point]]


$$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$
$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
| style="padding: 12px;" | $$\begin{split}\mathbf F \mathbin{\unicode{x27C7}} \mathbf q \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} =\, &\left[(2F_{gy}^2 + 2F_{gz}^2 - 1)q_x - 2F_{gx} F_{gy} q_y - 2F_{gz} F_{gx} q_z + 2(F_{gy} F_{pz} - F_{gz} F_{py} - F_{gx} F_{gw})q_w\right]\mathbf e_1 \\ +\, &\left[(2F_{gz}^2 + 2F_{gx}^2 - 1)q_y - 2F_{gy} F_{gz} q_z - 2F_{gx} F_{gy} q_x + 2(F_{gz} F_{px} - F_{gx} F_{pz} - F_{gy} F_{gw})q_w\right]\mathbf e_2 \\ +\, &\left[(2F_{gx}^2 + 2F_{gy}^2 - 1)q_z - 2F_{gz} F_{gx} q_x - 2F_{gy} F_{gz} q_y + 2(F_{gx} F_{py} - F_{gy} F_{px} - F_{gz} F_{gw})q_w\right]\mathbf e_3 \\ +\, &q_w\mathbf e_4\end{split}$$
| style="padding: 12px;" | $$\begin{split}\mathbf F \mathbin{\unicode{x27C7}} \mathbf p \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} =\, &((2F_{gy}^2 + 2F_{gz}^2 - 1)p_x \,&-\, 2F_{gx} F_{gy} p_y \,&-\, 2F_{gz} F_{gx} p_z \,&-\, 2F_{gx} F_{gw} p_w)&\mathbf e_1 \\ +\, &((2F_{gz}^2 + 2F_{gx}^2 - 1)p_y \,&-\, 2F_{gy} F_{gz} p_z \,&-\, 2F_{gx} F_{gy} p_x \,&-\, 2F_{gy} F_{gw} p_w)&\mathbf e_2 \\ +\, &((2F_{gx}^2 + 2F_{gy}^2 - 1)p_z \,&-\, 2F_{gz} F_{gx} p_x \,&-\, 2F_{gy} F_{gz} p_y \,&-\, 2F_{gz} F_{gw} p_w)&\mathbf e_3 \\ +\, &p_w\mathbf e_4\end{split}$$
|-
|-
| style="padding: 12px;" | [[Line]]
| style="padding: 12px;" | [[Line]]


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


$$\mathbf h = h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412} + h_w \mathbf e_{321}$$
$$\mathbf h = h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412} + h_w \mathbf e_{321}$$
| style="padding: 12px;" | $$\begin{split}\mathbf F \mathbin{\unicode{x27C7}} \mathbf h \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} =\, &\left[(1 - 2F_{gy}^2 - 2F_{gz}^2)h_x + 2F_{gx} F_{gy} h_y + 2F_{gz} F_{gx} h_z\right]\mathbf e_{423} \\ +\, &\left[(1 - 2F_{gz}^2 - 2F_{gx}^2)h_y + 2F_{gy} F_{gz} h_z + 2F_{gx} F_{gy} h_x\right]\mathbf e_{431} \\ +\, &\left[(1 - 2F_{gx}^2 - 2F_{gy}^2)h_z + 2F_{gz} F_{gx} h_x + 2F_{gy} F_{gz} h_y\right]\mathbf e_{412} \\ +\, &\left[2(F_{gy} F_{pz} - F_{gz} F_{py} + F_{gx} F_{gw})h_x + 2(F_{gz} F_{px} - F_{gx} F_{pz} + F_{gy} F_{gw})h_y + 2(F_{gx} F_{py} - F_{gy} F_{px} + F_{gz} F_{gw})h_z - h_w\right]\mathbf e_{321}\end{split}$$
| style="padding: 12px;" | $$\begin{split}\mathbf F \mathbin{\unicode{x27C7}} \mathbf h \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{F}}} =\, &((1 - 2F_{gy}^2 - 2F_{gz}^2)h_x \,&+\, 2F_{gx} F_{gy} h_y + 2F_{gz} F_{gx} h_z)&\mathbf e_{423} \\ +\, &((1 - 2F_{gz}^2 - 2F_{gx}^2)h_y \,&+\, 2F_{gy} F_{gz} h_z + 2F_{gx} F_{gy} h_x)&\mathbf e_{431} \\ +\, &((1 - 2F_{gx}^2 - 2F_{gy}^2)h_z \,&+\, 2F_{gz} F_{gx} h_x + 2F_{gy} F_{gz} h_y)&\mathbf e_{412} \\ +\, &\rlap{(2F_{gx} F_{gw} h_x + 2F_{gy} F_{gw} h_y + 2F_{gz} F_{gw} h_z - h_w)\mathbf e_{321}}\end{split}$$
|}
|}


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* [[Translation]]
* [[Translation]]
* [[Rotation]]
* [[Rotation]]
* [[Reflection]]
* [[Inversion]]
* [[Inversion]]
* [[Transflection]]

Latest revision as of 05:56, 15 July 2023

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

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

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

See Also

Retrieved from "https://rigidgeometricalgebra.org/wiki/index.php?title=Transflection&oldid=31"