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| Projections and antiprojections of one geometric object onto another can be accomplished using the connect and meet operations as described below.
| | ''Complements'' are unary operations in geometric algebra that perform a specific type of dualization. |
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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.
| | Every basis element $$\mathbf u$$ has a ''right complement'', which we denote by $$\overline{\mathbf u}$$, that satisfies the equation |
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| == Projection == | | :$$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ . |
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| The geometric projection of an object $$\mathbf x$$ onto an object $$\mathbf y$$ is given by the general formula
| | There is also a ''left complement'', which we denote by $$\underline{\mathbf u}$$, that satisfies the equation |
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| :$$(\mathbf y^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf x) \vee \mathbf y$$ . | | :$$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$ . |
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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.
| | Complements exchange full and empty dimensions, and the left and right complements can differ only by sign according to the relationship |
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| {| class="wikitable" | | :$$\underline{\mathbf u} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}\,\overline{\mathbf u}$$ . |
| ! Projection Formula !! Illustration
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| | style="padding: 12px;" | Projection of point $$\mathbf p$$ onto plane $$\mathbf g$$.
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| $$(\mathbf g^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf p) \vee \mathbf g = (g_x^2 + g_y^2 + g_z^2)\mathbf p - (g_xp_x + g_yp_y + g_zp_z + g_wp_w)(g_x \mathbf e_1 + g_y \mathbf e_2 + g_z \mathbf e_3)$$ | | This shows that the left and right complements of an element $$\mathbf u$$ are always the same if either its [[grade]] $$\operatorname{gr}(\mathbf u)$$ or its [[antigrade]] $$\operatorname{ag}(\mathbf u)$$ is even. If the number of dimensions is odd, then it is always true that one of these is even, so left and right complements are the same for all elements in an odd-dimensional algebra. As shown in the table below, applying the right or left complement twice can negate the operand in even numbers of dimensions. However, the right and left complements are inverse operations, so we always have $$\overline{\underline{\mathbf u}} = \mathbf u$$. |
| | style="padding: 2em;" | [[Image:point_onto_plane.svg|200px]]
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| | style="padding: 12px;" | Projection of point $$\mathbf p$$ onto line $$\boldsymbol l$$.
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| $$\begin{split}(\boldsymbol l^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf p) \vee \boldsymbol l =\, &(l_{vx} p_x + l_{vy} p_y + l_{vz} p_z)\mathbf v \\ +\, &(l_{vy} l_{mz} - l_{vz} l_{my})p_w \mathbf e_1 \\ +\, &(l_{vz} l_{mx} - l_{vx} l_{mz})p_w \mathbf e_2 \\ +\, &(l_{vx} l_{my} - l_{vy} l_{mx})p_w \mathbf e_3 \\ +\, &(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)p_w \mathbf e_4\end{split}$$
| | Taking the right or left complement twice causes the sign to change according to the formula |
| | style="padding: 2em;" | [[Image:point_onto_line.svg|200px]]
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| | style="padding: 12px;" | Projection of line $$\boldsymbol l$$ onto plane $$\mathbf g$$.
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| $$\begin{split}(\mathbf g^\unicode["segoe ui symbol"]{x2605} \wedge \boldsymbol l) \vee \mathbf g =\, &(g_x^2 + g_y^2 + g_z^2)(l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43}) \\ -\, &(g_x l_{vx} + g_y l_{vy} + g_z l_{vz})(g_x \mathbf e_{41} + g_y \mathbf e_{42} + g_z \mathbf e_{43}) \\ +\, &(g_x l_{mx} + g_y l_{my} + g_z l_{mz})(g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}) \\ -\, &(g_y l_{vz} - g_z l_{vy})g_w \mathbf e_{23} - (g_z l_{vx} - g_x l_{vz})g_w \mathbf e_{31} - (g_x l_{vy} - g_y l_{vx})g_w \mathbf e_{12}\end{split}$$ | | :$$\underline{\underline{\mathbf u}} = \overline{\overline{\mathbf u}} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}\,\mathbf u$$ . |
| | style="padding: 2em;" | [[Image:line_onto_plane.svg|200px]]
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| == Antiprojection ==
| | The right and left complements under the [[wedge product]] are also the right and left complements under the [[antiwedge product]], so we can write |
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| The geometric antiprojection of an object $$\mathbf x$$ onto an object $$\mathbf y$$ is given by the general formula
| | :$$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ |
| | :$$\underline{\mathbf u} \vee\mathbf u = \mathbf 1$$ . |
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| :$$(\mathbf y^\unicode["segoe ui symbol"]{x2605} \vee \mathbf x) \wedge \mathbf y$$ .
| | To extend the complements to all elements of an algebra, we simply require that they are linear operations. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the right complement, |
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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.
| | :$$\overline{(a\mathbf x + b\mathbf y)} = a\overline{\mathbf x} + b\overline{\mathbf y}$$ , |
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| {| class="wikitable"
| | and similarly for the left complement. |
| ! Antiprojection Formula !! Illustration
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| | style="padding: 12px;" | Antiprojection of plane $$\mathbf g$$ onto point $$\mathbf p$$.
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| $$(\mathbf p^\unicode["segoe ui symbol"]{x2605} \vee \mathbf g) \wedge \mathbf p = g_xp_w^2 \mathbf e_{423} + g_yp_w^2 \mathbf e_{431} + g_zp_w^2 \mathbf e_{412} - (g_xp_x + g_yp_y + g_zp_z)p_w \mathbf e_{321}$$ | | The following table lists the left and right complements for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$. |
| | style="padding: 2em;" | [[Image:plane_onto_point.svg|200px]]
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| | style="padding: 12px;" | Antiprojection of line $$\boldsymbol l$$ onto point $$\mathbf p$$.
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| $$\begin{split}(\mathbf p^\unicode["segoe ui symbol"]{x2605} \vee \boldsymbol l) \wedge \mathbf p =\, &l_{vx} p_w^2 \mathbf e_{41} + l_{vy} p_w^2 \mathbf e_{42} + l_{vz} p_w^2 \mathbf e_{43} \\ +\, &(p_y l_{vz} - p_z l_{vy})p_w \mathbf e_{23} + (p_z l_{vx} - p_x l_{vz})p_w \mathbf e_{31} + (p_x l_{vy} - p_y l_{vx})p_w \mathbf e_{12}\end{split}$$
| | [[Image:Complements.svg|720px]] |
| | style="padding: 2em;" | [[Image:line_onto_point.svg|200px]]
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| | style="padding: 12px;" | Antiprojection of plane $$\mathbf g$$ onto line $$\boldsymbol l$$.
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| $$\begin{split}(\boldsymbol l^\unicode["segoe ui symbol"]{x2605} \vee \mathbf g) \wedge \boldsymbol l =\, &(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)(g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412}) \\ -\, &(g_x l_{vx} + g_y l_{vy} + g_z l_{vz})(l_{vx} \mathbf e_{423} + l_{vy} \mathbf e_{431} + l_{vz} \mathbf e_{412}) \\ +\, &(g_x l_{my} l_{vz} - g_x l_{mz} l_{vy} + g_y l_{mz} l_{vx} - g_y l_{mx} l_{vz} + g_z l_{mx} l_{vy} - g_z l_{my} l_{vx}) \mathbf e_{321}\end{split}$$
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| | style="padding: 2em;" | [[Image:plane_onto_line.svg|200px]]
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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;" | $$(\mathbf g^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf e_4) \vee \mathbf g = -g_xg_w \mathbf e_1 - g_yg_w \mathbf e_2 - g_zg_w \mathbf e_3 + (g_x^2 + g_y^2 + g_z^2)\mathbf e_4$$
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| | style="padding: 12px;" | Point closest to the origin on the plane $$\mathbf g$$.
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| | style="padding: 12px;" | $$(\boldsymbol l^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf e_4) \vee \boldsymbol l = (l_{vy} l_{mz} - l_{vz} l_{my})\mathbf e_1 + (l_{vz} l_{mx} - l_{vx} l_{mz})\mathbf e_2 + (l_{vx} l_{my} - l_{vy} l_{mx})\mathbf e_3 + (l_{vx}^2 + l_{vy}^2 + l_{vz}^2)\mathbf e_4$$
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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 the connect operation with the antidual instead of the dual. 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;" | $$(\mathbf p^\unicode["segoe ui symbol"]{x2606} \vee \mathbf e_{321}) \wedge \mathbf p = p_xp_w \mathbf e_{423} + p_yp_w \mathbf e_{431} + p_zp_w \mathbf e_{412} - (p_x^2 + p_y^2 + p_z^2)\mathbf e_{321}$$
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| | style="padding: 12px;" | Plane farthest from the origin containing the point $$\mathbf p$$.
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| | style="padding: 12px;" | $$(\boldsymbol l^\unicode["segoe ui symbol"]{x2606} \vee \mathbf e_{321}) \wedge \boldsymbol l = (l_{my} l_{vz} - l_{mz} l_{vy})\mathbf e_{423} + (l_{mz} l_{vx} - l_{mx} l_{vz})\mathbf e_{431} + (l_{mx} l_{vy} - l_{my} l_{vx})\mathbf e_{412} + (l_{mx}^2 + l_{my}^2 + l_{mz}^2)\mathbf e_{321}$$
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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 == | | == See Also == |
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| * [[Join and meet]] | | * [[Duals]] |
| | * [[Grade and antigrade]] |
| | * [[Bulk and weight]] |
| | * [[Reverses]] |
| | * [[Duality]] |
Complements are unary operations in geometric algebra that perform a specific type of dualization.
Every basis element $$\mathbf u$$ has a right complement, which we denote by $$\overline{\mathbf u}$$, that satisfies the equation
- $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ .
There is also a left complement, which we denote by $$\underline{\mathbf u}$$, that satisfies the equation
- $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$ .
Complements exchange full and empty dimensions, and the left and right complements can differ only by sign according to the relationship
- $$\underline{\mathbf u} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}\,\overline{\mathbf u}$$ .
This shows that the left and right complements of an element $$\mathbf u$$ are always the same if either its grade $$\operatorname{gr}(\mathbf u)$$ or its antigrade $$\operatorname{ag}(\mathbf u)$$ is even. If the number of dimensions is odd, then it is always true that one of these is even, so left and right complements are the same for all elements in an odd-dimensional algebra. As shown in the table below, applying the right or left complement twice can negate the operand in even numbers of dimensions. However, the right and left complements are inverse operations, so we always have $$\overline{\underline{\mathbf u}} = \mathbf u$$.
Taking the right or left complement twice causes the sign to change according to the formula
- $$\underline{\underline{\mathbf u}} = \overline{\overline{\mathbf u}} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}\,\mathbf u$$ .
The right and left complements under the wedge product are also the right and left complements under the antiwedge product, so we can write
- $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$
- $$\underline{\mathbf u} \vee\mathbf u = \mathbf 1$$ .
To extend the complements to all elements of an algebra, we simply require that they are linear operations. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the right complement,
- $$\overline{(a\mathbf x + b\mathbf y)} = a\overline{\mathbf x} + b\overline{\mathbf y}$$ ,
and similarly for the left complement.
The following table lists the left and right complements for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$.
See Also
