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(Created page with "Projections and antiprojections of one geometric object onto another can be accomplished using interior products as described below. The formulas on this page are general and do not require the geometric objects to be unitized. Most of them become simpler if unitization can be assumed. == Projection == The geometric projection of an object $$\mathbf x$$ onto an object $$\mathbf y$$ is given by the general formula $$(\mathbf y_\unicode{x25CB} \mathbin{\unicode{...")
 
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Projections and antiprojections of one geometric object onto another can be accomplished using [[interior products]] as described below.
Projections and antiprojections of one geometric object onto another can be accomplished using the connect and meet operations as described below.


The formulas on this page are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.
The formulas on this page are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.
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== Projection ==
== Projection ==


The geometric projection of an object $$\mathbf x$$ onto an object $$\mathbf y$$ is given by the general formula $$(\mathbf y_\unicode{x25CB} \mathbin{\unicode{x22A2}} \mathbf x) \mathbin{\unicode{x22A3}} \mathbf y$$. Applying the definitions of the left and right [[interior products]], this becomes
The orthogonal projection of an object $$\mathbf a$$ onto an object $$\mathbf b$$ is given by the general formula


:$$(\mathbf y_\unicode{x25CB} \mathbin{\unicode{x22A2}} \mathbf x) \mathbin{\unicode{x22A3}} \mathbf y = \left(\underline{\mathbf y_\smash{\unicode{x25CB}}} \wedge \mathbf x\right) \vee \mathbf y$$ .
:$$\mathbf b \vee (\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2606})$$ ,
 
where the operation in parentheses is the [[weight expansion]] of $$\mathbf a$$ into $$\mathbf b$$.


Projections involving [[points]], [[lines]], and [[planes]] in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ are shown in the following table.
Projections involving [[points]], [[lines]], and [[planes]] in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ are shown in the following table.
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| style="padding: 12px;" | Projection of point $$\mathbf p$$ onto plane $$\mathbf g$$.
| style="padding: 12px;" | Projection of point $$\mathbf p$$ onto plane $$\mathbf g$$.


$$\left(\underline{\mathbf g_\smash{\unicode{x25CB}}} \wedge \mathbf p\right) \vee \mathbf g = (g_x^2 + g_y^2 + g_z^2)\mathbf p - (g_xp_x + g_yp_y + g_zp_z + g_wp_w)(g_x \mathbf e_1 + g_y \mathbf e_2 + g_z \mathbf e_3)$$
$$\mathbf g \vee (\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606}) = (g_x^2 + g_y^2 + g_z^2)\mathbf p - (g_xp_x + g_yp_y + g_zp_z + g_wp_w)(g_x \mathbf e_1 + g_y \mathbf e_2 + g_z \mathbf e_3)$$
| style="padding: 12px;" | [[Image:point_plane.svg|300px]]
| style="padding: 2em;" | [[Image:point_onto_plane.svg|200px]]
|-
|-
| style="padding: 12px;" | Projection of point $$\mathbf p$$ onto line $$\boldsymbol l$$.
| style="padding: 12px;" | Projection of point $$\mathbf p$$ onto line $$\boldsymbol l$$.


$$\begin{split}\left(\underline{\boldsymbol l_\smash{\unicode{x25CB}}} \wedge \mathbf p\right) \vee \boldsymbol l =\, &(l_{vx} p_x + l_{vy} p_y + l_{vz} p_z)\mathbf v \\ +\, &(l_{vy} l_{mz} - l_{vz} l_{my})p_w \mathbf e_1 \\ +\, &(l_{vz} l_{mx} - l_{vx} l_{mz})p_w \mathbf e_2 \\ +\, &(l_{vx} l_{my} - l_{vy} l_{mx})p_w \mathbf e_3 \\ +\, &(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)p_w \mathbf e_4\end{split}$$
$$\begin{split}\boldsymbol l \vee (\mathbf p \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606}) =\, &(l_{vx} p_x + l_{vy} p_y + l_{vz} p_z)(l_{vx} \mathbf e_1 + l_{vy} \mathbf e_2 + l_{vz} \mathbf e_3) \\ +\, &(l_{vy} l_{mz} - l_{vz} l_{my})p_w \mathbf e_1 \\ +\, &(l_{vz} l_{mx} - l_{vx} l_{mz})p_w \mathbf e_2 \\ +\, &(l_{vx} l_{my} - l_{vy} l_{mx})p_w \mathbf e_3 \\ +\, &(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)p_w \mathbf e_4\end{split}$$
| style="padding: 12px;" | [[Image:point_line.svg|300px]]
| style="padding: 2em;" | [[Image:point_onto_line.svg|200px]]
|-
|-
| style="padding: 12px;" | Projection of line $$\boldsymbol l$$ onto plane $$\mathbf g$$.
| style="padding: 12px;" | Projection of line $$\boldsymbol l$$ onto plane $$\mathbf g$$.


$$\begin{split}\left(\underline{\mathbf g_\smash{\unicode{x25CB}}} \wedge \boldsymbol l\right) \vee \mathbf g =\, &(g_x^2 + g_y^2 + g_z^2)(l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43}) \\ -\, &(g_x l_{vx} + g_y l_{vy} + g_z l_{vz})(g_x \mathbf e_{41} + g_y \mathbf e_{42} + g_z \mathbf e_{43}) \\ +\, &(g_x l_{mx} + g_y l_{my} + g_z l_{mz})(g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}) \\ -\, &(g_y l_{vz} - g_z l_{vy})g_w \mathbf e_{23} - (g_z l_{vx} - g_x l_{vz})g_w \mathbf e_{31} - (g_x l_{vy} - g_y l_{vx})g_w \mathbf e_{12}\end{split}$$
$$\begin{split}\mathbf g \vee (\boldsymbol l \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606}) =\, &(g_x^2 + g_y^2 + g_z^2)(l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43}) \\ -\, &(g_x l_{vx} + g_y l_{vy} + g_z l_{vz})(g_x \mathbf e_{41} + g_y \mathbf e_{42} + g_z \mathbf e_{43}) \\ +\, &(g_x l_{mx} + g_y l_{my} + g_z l_{mz})(g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}) \\ -\, &(g_y l_{vz} - g_z l_{vy})g_w \mathbf e_{23} - (g_z l_{vx} - g_x l_{vz})g_w \mathbf e_{31} - (g_x l_{vy} - g_y l_{vx})g_w \mathbf e_{12}\end{split}$$
| style="padding: 12px;" | [[Image:line_plane.svg|300px]]
| style="padding: 2em;" | [[Image:line_onto_plane.svg|200px]]
|}
|}


== Antiprojection ==
== Antiprojection ==


The geometric antiprojection of an object $$\mathbf x$$ onto an object $$\mathbf y$$ is given by the general formula $$(\mathbf y_\unicode{x25CB} \mathbin{\unicode{x22A8}} \mathbf x) \mathbin{\unicode{x2AE4}} \mathbf y$$. Applying the definitions of the left and right [[interior antiproducts]], this becomes
The orthogonal antiprojection of an object $$\mathbf a$$ onto an object $$\mathbf b$$ is given by the general formula


:$$(\mathbf y_\unicode{x25CB} \mathbin{\unicode{x22A8}} \mathbf x) \mathbin{\unicode{x2AE4}} \mathbf y = \left(\underline{\mathbf y_\smash{\unicode{x25CB}}} \vee \mathbf x\right) \wedge \mathbf y$$ .
:$$\mathbf b \wedge (\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606})$$ ,
 
where the operation in parentheses is the [[weight contraction]] of $$\mathbf a$$ with $$\mathbf b$$.


Antiprojections involving [[points]], [[lines]], and [[planes]] in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ are shown in the following table.
Antiprojections involving [[points]], [[lines]], and [[planes]] in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ are shown in the following table.
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| style="padding: 12px;" | Antiprojection of plane $$\mathbf g$$ onto point $$\mathbf p$$.
| style="padding: 12px;" | Antiprojection of plane $$\mathbf g$$ onto point $$\mathbf p$$.


$$\left(\underline{\mathbf p_\smash{\unicode{x25CB}}} \vee \mathbf g\right) \wedge \mathbf p = g_xp_w^2 \mathbf e_{423} + g_yp_w^2 \mathbf e_{431} + g_zp_w^2 \mathbf e_{412} - (g_xp_x + g_yp_y + g_zp_z)p_w \mathbf e_{321}$$
$$\mathbf p \wedge (\mathbf g \vee \mathbf p^\unicode["segoe ui symbol"]{x2606}) = p_w^2(g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412}) - (g_xp_x + g_yp_y + g_zp_z)p_w \mathbf e_{321}$$
| style="padding: 12px;" | [[Image:plane_point.svg|300px]]
| style="padding: 2em;" | [[Image:plane_onto_point.svg|200px]]
|-
|-
| style="padding: 12px;" | Antiprojection of line $$\boldsymbol l$$ onto point $$\mathbf p$$.
| style="padding: 12px;" | Antiprojection of line $$\boldsymbol l$$ onto point $$\mathbf p$$.


$$\begin{split}\left(\underline{\mathbf p_\smash{\unicode{x25CB}}} \vee \boldsymbol l\right) \wedge \mathbf p =\, &l_{vx} p_w^2 \mathbf e_{41} + l_{vy} p_w^2 \mathbf e_{42} + l_{vz} p_w^2 \mathbf e_{43} \\ +\, &(p_y l_{vz} - p_z l_{vy})p_w \mathbf e_{23} + (p_z l_{vx} - p_x l_{vz})p_w \mathbf e_{31} + (p_x l_{vy} - p_y l_{vx})p_w \mathbf e_{12}\end{split}$$
$$\begin{split}\mathbf p \wedge (\boldsymbol l \vee \mathbf p^\unicode["segoe ui symbol"]{x2606}) =\, &p_w^2(l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43}) \\ +\, &(p_y l_{vz} - p_z l_{vy})p_w \mathbf e_{23} + (p_z l_{vx} - p_x l_{vz})p_w \mathbf e_{31} + (p_x l_{vy} - p_y l_{vx})p_w \mathbf e_{12}\end{split}$$
| style="padding: 12px;" | [[Image:line_point.svg|300px]]
| style="padding: 2em;" | [[Image:line_onto_point.svg|200px]]
|-
|-
| style="padding: 12px;" | Antiprojection of plane $$\mathbf g$$ onto line $$\boldsymbol l$$.
| style="padding: 12px;" | Antiprojection of plane $$\mathbf g$$ onto line $$\boldsymbol l$$.


$$\begin{split}\left(\underline{\boldsymbol l_\smash{\unicode{x25CB}}} \vee \mathbf g\right) \wedge \boldsymbol l =\, &(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)(g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412}) \\ -\, &(g_x l_{vx} + g_y l_{vy} + g_z l_{vz})(l_{vx} \mathbf e_{423} + l_{vy} \mathbf e_{431} + l_{vz} \mathbf e_{412}) \\ +\, &(g_x l_{my} l_{vz} - g_x l_{mz} l_{vy} + g_y l_{mz} l_{vx} - g_y l_{mx} l_{vz} + g_z l_{mx} l_{vy} - g_z l_{my} l_{vx}) \mathbf e_{321}\end{split}$$
$$\begin{split}\boldsymbol l \wedge (\mathbf g \vee \boldsymbol l^\unicode["segoe ui symbol"]{x2606}) =\, &(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)(g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412}) \\ -\, &(g_x l_{vx} + g_y l_{vy} + g_z l_{vz})(l_{vx} \mathbf e_{423} + l_{vy} \mathbf e_{431} + l_{vz} \mathbf e_{412}) \\ +\, &(g_x l_{my} l_{vz} - g_x l_{mz} l_{vy} + g_y l_{mz} l_{vx} - g_y l_{mx} l_{vz} + g_z l_{mx} l_{vy} - g_z l_{my} l_{vx}) \mathbf e_{321}\end{split}$$
| style="padding: 12px;" | [[Image:plane_line.svg|300px]]
| style="padding: 2em;" | [[Image:plane_onto_line.svg|200px]]
|}
|}


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! Projection Formula !! Description
! Projection Formula !! Description
|-
|-
| style="padding: 12px;" | $$\left(\underline{\mathbf g_\smash{\unicode{x25CB}}} \wedge \mathbf e_4\right) \vee \mathbf g = -g_xg_w \mathbf e_1 - g_yg_w \mathbf e_2 - g_zg_w \mathbf e_3 + (g_x^2 + g_y^2 + g_z^2)\mathbf e_4$$
| style="padding: 12px;" | $$\mathbf g \vee (\mathbf e_4 \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606}) = -g_xg_w \mathbf e_1 - g_yg_w \mathbf e_2 - g_zg_w \mathbf e_3 + (g_x^2 + g_y^2 + g_z^2)\mathbf e_4$$
| style="padding: 12px;" | Point closest to the origin on the plane $$\mathbf g$$.
| style="padding: 12px;" | Point closest to the origin on the plane $$\mathbf g$$.
|-
|-
| style="padding: 12px;" | $$\left(\underline{\boldsymbol l_\smash{\unicode{x25CB}}} \wedge \mathbf e_4\right) \vee \boldsymbol l = (l_{vy} l_{mz} - l_{vz} l_{my})\mathbf e_1 + (l_{vz} l_{mx} - l_{vx} l_{mz})\mathbf e_2 + (l_{vx} l_{my} - l_{vy} l_{mx})\mathbf e_3 + (l_{vx}^2 + l_{vy}^2 + l_{vz}^2)\mathbf e_4$$
| style="padding: 12px;" | $$\boldsymbol l \vee (\mathbf e_4 \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606}) = (l_{vy} l_{mz} - l_{vz} l_{my})\mathbf e_1 + (l_{vz} l_{mx} - l_{vx} l_{mz})\mathbf e_2 + (l_{vx} l_{my} - l_{vy} l_{mx})\mathbf e_3 + (l_{vx}^2 + l_{vy}^2 + l_{vz}^2)\mathbf e_4$$
| style="padding: 12px;" | Point closest to the origin on the line $$\boldsymbol l$$.
| style="padding: 12px;" | Point closest to the origin on the line $$\boldsymbol l$$.
|}
|}
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== Antiprojection of Horizon ==
== Antiprojection of Horizon ==


Symmetrically to the projection of the origin, the horizon $$\mathbf g = \mathbf e_{321}$$ (the plane at infinity) can be antiprojected onto a point or line using [[interior antiproducts]] with the [[bulk]] of the point or line instead of the [[weight]]. This operation finds the plane containing the geometry that is farthest from the origin. Specific formulas are listed in the following table.
Symmetrically to the projection of the origin, the horizon $$\mathbf g = \mathbf e_{321}$$ (the plane at infinity) can be antiprojected onto a point or line using the [[bulk contraction]]. This operation finds the plane containing the geometry that is farthest from the origin. Specific formulas are listed in the following table.


{| class="wikitable"
{| class="wikitable"
! Antiprojection Formula !! Description
! Antiprojection Formula !! Description
|-
|-
| style="padding: 12px;" | $$\left(\underline{\mathbf p_\smash{\unicode{x25CF}}} \vee \mathbf e_{321}\right) \wedge \mathbf p = -p_xp_w \mathbf e_{423} - p_yp_w \mathbf e_{431} - p_zp_w \mathbf e_{412} + (p_x^2 + p_y^2 + p_z^2)\mathbf e_{321}$$
| style="padding: 12px;" | $$\mathbf p \wedge (\mathbf e_{321} \vee \mathbf p^\unicode["segoe ui symbol"]{x2605}) = p_xp_w \mathbf e_{423} + p_yp_w \mathbf e_{431} + p_zp_w \mathbf e_{412} - (p_x^2 + p_y^2 + p_z^2)\mathbf e_{321}$$
| style="padding: 12px;" | Plane farthest from the origin containing the point $$\mathbf p$$.
| style="padding: 12px;" | Plane farthest from the origin containing the point $$\mathbf p$$.
|-
|-
| style="padding: 12px;" | $$\left(\underline{\boldsymbol l_\smash{\unicode{x25CF}}} \vee \mathbf e_{321}\right) \wedge \boldsymbol l = (l_{my} l_{vz} - l_{mz} l_{vy})\mathbf e_{423} + (l_{mz} l_{vx} - l_{mx} l_{vz})\mathbf e_{431} + (l_{mx} l_{vy} - l_{my} l_{vx})\mathbf e_{412} + (l_{mx}^2 + l_{my}^2 + l_{mz}^2)\mathbf e_{321}$$
| style="padding: 12px;" | $$\boldsymbol l \wedge (\mathbf e_{321} \vee \boldsymbol l^\unicode["segoe ui symbol"]{x2605}) = (l_{my} l_{vz} - l_{mz} l_{vy})\mathbf e_{423} + (l_{mz} l_{vx} - l_{mx} l_{vz})\mathbf e_{431} + (l_{mx} l_{vy} - l_{my} l_{vx})\mathbf e_{412} + (l_{mx}^2 + l_{my}^2 + l_{mz}^2)\mathbf e_{321}$$
| style="padding: 12px;" | Plane farthest from the origin containing the line $$\boldsymbol l$$.
| style="padding: 12px;" | Plane farthest from the origin containing the line $$\boldsymbol l$$.
|}
|}
== In the Book ==
* Projections are discussed in Section 2.13.6.


== See Also ==
== See Also ==


* [[Interior product]]
* [[Join and meet]]
* [[Join and meet]]

Latest revision as of 23:28, 13 April 2024

Projections and antiprojections of one geometric object onto another can be accomplished using the connect and meet operations as described below.

The formulas on this page are general and do not require the geometric objects to be unitized. Most of them become simpler if unitization can be assumed.

Projection

The orthogonal projection of an object $$\mathbf a$$ onto an object $$\mathbf b$$ is given by the general formula

$$\mathbf b \vee (\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2606})$$ ,

where the operation in parentheses is the weight expansion of $$\mathbf a$$ into $$\mathbf b$$.

Projections involving points, lines, and planes in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ are shown in the following table.

Projection Formula Illustration
Projection of point $$\mathbf p$$ onto plane $$\mathbf g$$.

$$\mathbf g \vee (\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606}) = (g_x^2 + g_y^2 + g_z^2)\mathbf p - (g_xp_x + g_yp_y + g_zp_z + g_wp_w)(g_x \mathbf e_1 + g_y \mathbf e_2 + g_z \mathbf e_3)$$

Projection of point $$\mathbf p$$ onto line $$\boldsymbol l$$.

$$\begin{split}\boldsymbol l \vee (\mathbf p \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606}) =\, &(l_{vx} p_x + l_{vy} p_y + l_{vz} p_z)(l_{vx} \mathbf e_1 + l_{vy} \mathbf e_2 + l_{vz} \mathbf e_3) \\ +\, &(l_{vy} l_{mz} - l_{vz} l_{my})p_w \mathbf e_1 \\ +\, &(l_{vz} l_{mx} - l_{vx} l_{mz})p_w \mathbf e_2 \\ +\, &(l_{vx} l_{my} - l_{vy} l_{mx})p_w \mathbf e_3 \\ +\, &(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)p_w \mathbf e_4\end{split}$$

Projection of line $$\boldsymbol l$$ onto plane $$\mathbf g$$.

$$\begin{split}\mathbf g \vee (\boldsymbol l \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606}) =\, &(g_x^2 + g_y^2 + g_z^2)(l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43}) \\ -\, &(g_x l_{vx} + g_y l_{vy} + g_z l_{vz})(g_x \mathbf e_{41} + g_y \mathbf e_{42} + g_z \mathbf e_{43}) \\ +\, &(g_x l_{mx} + g_y l_{my} + g_z l_{mz})(g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}) \\ -\, &(g_y l_{vz} - g_z l_{vy})g_w \mathbf e_{23} - (g_z l_{vx} - g_x l_{vz})g_w \mathbf e_{31} - (g_x l_{vy} - g_y l_{vx})g_w \mathbf e_{12}\end{split}$$

Antiprojection

The orthogonal antiprojection of an object $$\mathbf a$$ onto an object $$\mathbf b$$ is given by the general formula

$$\mathbf b \wedge (\mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606})$$ ,

where the operation in parentheses is the weight contraction of $$\mathbf a$$ with $$\mathbf b$$.

Antiprojections involving points, lines, and planes in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ are shown in the following table.

Antiprojection Formula Illustration
Antiprojection of plane $$\mathbf g$$ onto point $$\mathbf p$$.

$$\mathbf p \wedge (\mathbf g \vee \mathbf p^\unicode["segoe ui symbol"]{x2606}) = p_w^2(g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412}) - (g_xp_x + g_yp_y + g_zp_z)p_w \mathbf e_{321}$$

Antiprojection of line $$\boldsymbol l$$ onto point $$\mathbf p$$.

$$\begin{split}\mathbf p \wedge (\boldsymbol l \vee \mathbf p^\unicode["segoe ui symbol"]{x2606}) =\, &p_w^2(l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43}) \\ +\, &(p_y l_{vz} - p_z l_{vy})p_w \mathbf e_{23} + (p_z l_{vx} - p_x l_{vz})p_w \mathbf e_{31} + (p_x l_{vy} - p_y l_{vx})p_w \mathbf e_{12}\end{split}$$

Antiprojection of plane $$\mathbf g$$ onto line $$\boldsymbol l$$.

$$\begin{split}\boldsymbol l \wedge (\mathbf g \vee \boldsymbol l^\unicode["segoe ui symbol"]{x2606}) =\, &(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)(g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412}) \\ -\, &(g_x l_{vx} + g_y l_{vy} + g_z l_{vz})(l_{vx} \mathbf e_{423} + l_{vy} \mathbf e_{431} + l_{vz} \mathbf e_{412}) \\ +\, &(g_x l_{my} l_{vz} - g_x l_{mz} l_{vy} + g_y l_{mz} l_{vx} - g_y l_{mx} l_{vz} + g_z l_{mx} l_{vy} - g_z l_{my} l_{vx}) \mathbf e_{321}\end{split}$$

Projection of Origin

When a point $$\mathbf p$$ is projected onto another geometry, the result can be interpreted as the point on that geometry that is closest to the original point $$\mathbf p$$. In the particular case that $$\mathbf p = \mathbf e_4$$, which is the unitized origin, the projection finds the point on a geometry that is closest to the origin. Specific formulas are listed in the following table.

Projection Formula Description
$$\mathbf g \vee (\mathbf e_4 \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606}) = -g_xg_w \mathbf e_1 - g_yg_w \mathbf e_2 - g_zg_w \mathbf e_3 + (g_x^2 + g_y^2 + g_z^2)\mathbf e_4$$ Point closest to the origin on the plane $$\mathbf g$$.
$$\boldsymbol l \vee (\mathbf e_4 \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606}) = (l_{vy} l_{mz} - l_{vz} l_{my})\mathbf e_1 + (l_{vz} l_{mx} - l_{vx} l_{mz})\mathbf e_2 + (l_{vx} l_{my} - l_{vy} l_{mx})\mathbf e_3 + (l_{vx}^2 + l_{vy}^2 + l_{vz}^2)\mathbf e_4$$ Point closest to the origin on the line $$\boldsymbol l$$.

Antiprojection of Horizon

Symmetrically to the projection of the origin, the horizon $$\mathbf g = \mathbf e_{321}$$ (the plane at infinity) can be antiprojected onto a point or line using the bulk contraction. This operation finds the plane containing the geometry that is farthest from the origin. Specific formulas are listed in the following table.

Antiprojection Formula Description
$$\mathbf p \wedge (\mathbf e_{321} \vee \mathbf p^\unicode["segoe ui symbol"]{x2605}) = p_xp_w \mathbf e_{423} + p_yp_w \mathbf e_{431} + p_zp_w \mathbf e_{412} - (p_x^2 + p_y^2 + p_z^2)\mathbf e_{321}$$ Plane farthest from the origin containing the point $$\mathbf p$$.
$$\boldsymbol l \wedge (\mathbf e_{321} \vee \boldsymbol l^\unicode["segoe ui symbol"]{x2605}) = (l_{my} l_{vz} - l_{mz} l_{vy})\mathbf e_{423} + (l_{mz} l_{vx} - l_{mx} l_{vz})\mathbf e_{431} + (l_{mx} l_{vy} - l_{my} l_{vx})\mathbf e_{412} + (l_{mx}^2 + l_{my}^2 + l_{mz}^2)\mathbf e_{321}$$ Plane farthest from the origin containing the line $$\boldsymbol l$$.


In the Book

  • Projections are discussed in Section 2.13.6.

See Also

Retrieved from "https://rigidgeometricalgebra.org/wiki/index.php?title=Projections&oldid=352"