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| Projections and antiprojections of one geometric object onto another can be accomplished using [[interior products]] as described below.
| | #REDIRECT [[Geometric norm]] |
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| 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 ==
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| 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
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| :$$(\mathbf y_\unicode{x25CB} \mathbin{\unicode{x22A2}} \mathbf x) \mathbin{\unicode{x22A3}} \mathbf y = \left(\underline{\mathbf y_\smash{\unicode{x25CB}}} \wedge \mathbf x\right) \vee \mathbf y$$ .
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| 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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| {| class="wikitable"
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| ! Projection Formula !! Illustration
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| | style="padding: 12px;" | Projection of point $$\mathbf p$$ onto plane $$\mathbf g$$.
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| $$\left(\underline{\mathbf g_\smash{\unicode{x25CB}}} \wedge \mathbf p\right) \vee \mathbf g = (g_x^2 + g_y^2 + g_z^2)\mathbf p - (g_xp_x + g_yp_y + g_zp_z + g_wp_w)(g_x \mathbf e_1 + g_y \mathbf e_2 + g_z \mathbf e_3)$$
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| | style="padding: 12px;" | [[Image:point_plane.svg|300px]]
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| | style="padding: 12px;" | Projection of point $$\mathbf p$$ onto line $$\boldsymbol l$$.
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| $$\begin{split}\left(\underline{\boldsymbol l_\smash{\unicode{x25CB}}} \wedge \mathbf p\right) \vee \boldsymbol l =\, &(l_{vx} p_x + l_{vy} p_y + l_{vz} p_z)\mathbf v \\ +\, &(l_{vy} l_{mz} - l_{vz} l_{my})p_w \mathbf e_1 \\ +\, &(l_{vz} l_{mx} - l_{vx} l_{mz})p_w \mathbf e_2 \\ +\, &(l_{vx} l_{my} - l_{vy} l_{mx})p_w \mathbf e_3 \\ +\, &(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)p_w \mathbf e_4\end{split}$$
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| | style="padding: 12px;" | [[Image:point_line.svg|300px]]
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| | style="padding: 12px;" | Projection of line $$\boldsymbol l$$ onto plane $$\mathbf g$$.
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| $$\begin{split}\left(\underline{\mathbf g_\smash{\unicode{x25CB}}} \wedge \boldsymbol l\right) \vee \mathbf g =\, &(g_x^2 + g_y^2 + g_z^2)(l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43}) \\ -\, &(g_x l_{vx} + g_y l_{vy} + g_z l_{vz})(g_x \mathbf e_{41} + g_y \mathbf e_{42} + g_z \mathbf e_{43}) \\ +\, &(g_x l_{mx} + g_y l_{my} + g_z l_{mz})(g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}) \\ -\, &(g_y l_{vz} - g_z l_{vy})g_w \mathbf e_{23} - (g_z l_{vx} - g_x l_{vz})g_w \mathbf e_{31} - (g_x l_{vy} - g_y l_{vx})g_w \mathbf e_{12}\end{split}$$
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| | style="padding: 12px;" | [[Image:line_plane.svg|300px]]
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| |}
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| == Antiprojection ==
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| 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
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| :$$(\mathbf y_\unicode{x25CB} \mathbin{\unicode{x22A8}} \mathbf x) \mathbin{\unicode{x2AE4}} \mathbf y = \left(\underline{\mathbf y_\smash{\unicode{x25CB}}} \vee \mathbf x\right) \wedge \mathbf y$$ .
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| 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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| {| class="wikitable"
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| ! Antiprojection Formula !! Illustration
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| | style="padding: 12px;" | Antiprojection of plane $$\mathbf g$$ onto point $$\mathbf p$$.
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| $$\left(\underline{\mathbf p_\smash{\unicode{x25CB}}} \vee \mathbf g\right) \wedge \mathbf p = g_xp_w^2 \mathbf e_{423} + g_yp_w^2 \mathbf e_{431} + g_zp_w^2 \mathbf e_{412} - (g_xp_x + g_yp_y + g_zp_z)p_w \mathbf e_{321}$$
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| | style="padding: 12px;" | [[Image:plane_point.svg|300px]]
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| | style="padding: 12px;" | Antiprojection of line $$\boldsymbol l$$ onto point $$\mathbf p$$.
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| $$\begin{split}\left(\underline{\mathbf p_\smash{\unicode{x25CB}}} \vee \boldsymbol l\right) \wedge \mathbf p =\, &l_{vx} p_w^2 \mathbf e_{41} + l_{vy} p_w^2 \mathbf e_{42} + l_{vz} p_w^2 \mathbf e_{43} \\ +\, &(p_y l_{vz} - p_z l_{vy})p_w \mathbf e_{23} + (p_z l_{vx} - p_x l_{vz})p_w \mathbf e_{31} + (p_x l_{vy} - p_y l_{vx})p_w \mathbf e_{12}\end{split}$$
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| | style="padding: 12px;" | [[Image:line_point.svg|300px]]
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| | style="padding: 12px;" | Antiprojection of plane $$\mathbf g$$ onto line $$\boldsymbol l$$.
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| $$\begin{split}\left(\underline{\boldsymbol l_\smash{\unicode{x25CB}}} \vee \mathbf g\right) \wedge \boldsymbol l =\, &(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)(g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412}) \\ -\, &(g_x l_{vx} + g_y l_{vy} + g_z l_{vz})(l_{vx} \mathbf e_{423} + l_{vy} \mathbf e_{431} + l_{vz} \mathbf e_{412}) \\ +\, &(g_x l_{my} l_{vz} - g_x l_{mz} l_{vy} + g_y l_{mz} l_{vx} - g_y l_{mx} l_{vz} + g_z l_{mx} l_{vy} - g_z l_{my} l_{vx}) \mathbf e_{321}\end{split}$$
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| | style="padding: 12px;" | [[Image:plane_line.svg|300px]]
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| |}
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| == Projection of Origin ==
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| 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.
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| {| class="wikitable"
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| ! Projection Formula !! Description
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| | style="padding: 12px;" | $$\left(\underline{\mathbf g_\smash{\unicode{x25CB}}} \wedge \mathbf e_4\right) \vee \mathbf g = -g_xg_w \mathbf e_1 - g_yg_w \mathbf e_2 - g_zg_w \mathbf e_3 + (g_x^2 + g_y^2 + g_z^2)\mathbf e_4$$
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| | style="padding: 12px;" | Point closest to the origin on the plane $$\mathbf g$$.
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| | style="padding: 12px;" | $$\left(\underline{\boldsymbol l_\smash{\unicode{x25CB}}} \wedge \mathbf e_4\right) \vee \boldsymbol l = (l_{vy} l_{mz} - l_{vz} l_{my})\mathbf e_1 + (l_{vz} l_{mx} - l_{vx} l_{mz})\mathbf e_2 + (l_{vx} l_{my} - l_{vy} l_{mx})\mathbf e_3 + (l_{vx}^2 + l_{vy}^2 + l_{vz}^2)\mathbf e_4$$
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| | style="padding: 12px;" | Point closest to the origin on the line $$\boldsymbol l$$.
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| |}
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| == Antiprojection of Horizon ==
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| 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.
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| {| class="wikitable"
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| ! Antiprojection Formula !! Description
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| | style="padding: 12px;" | $$\left(\underline{\mathbf p_\smash{\unicode{x25CF}}} \vee \mathbf e_{321}\right) \wedge \mathbf p = -p_xp_w \mathbf e_{423} - p_yp_w \mathbf e_{431} - p_zp_w \mathbf e_{412} + (p_x^2 + p_y^2 + p_z^2)\mathbf e_{321}$$
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| | style="padding: 12px;" | Plane farthest from the origin containing the point $$\mathbf p$$.
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| | style="padding: 12px;" | $$\left(\underline{\boldsymbol l_\smash{\unicode{x25CF}}} \vee \mathbf e_{321}\right) \wedge \boldsymbol l = (l_{my} l_{vz} - l_{mz} l_{vy})\mathbf e_{423} + (l_{mz} l_{vx} - l_{mx} l_{vz})\mathbf e_{431} + (l_{mx} l_{vy} - l_{my} l_{vx})\mathbf e_{412} + (l_{mx}^2 + l_{my}^2 + l_{mz}^2)\mathbf e_{321}$$
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| | style="padding: 12px;" | Plane farthest from the origin containing the line $$\boldsymbol l$$.
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| |}
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| == See Also ==
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| * [[Interior product]]
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| * [[Join and meet]]
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