Projections

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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 $$(\mathbf y_\unicode{x25CB} \mathbin{\unicode{x22A2}} \mathbf x) \mathbin{\unicode{x22A3}} \mathbf y$$. Applying the definitions of the left and right interior products, this becomes

$$(\mathbf y_\unicode{x25CB} \mathbin{\unicode{x22A2}} \mathbf x) \mathbin{\unicode{x22A3}} \mathbf y = \left(\underline{\mathbf y_\smash{\unicode{x25CB}}} \wedge \mathbf x\right) \vee \mathbf y$$ .

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 $$\mathbf g$$.

$$\left(\underline{\mathbf g_\smash{\unicode{x25CB}}} \wedge \mathbf p\right) \vee \mathbf g = (g_x^2 + g_y^2 + g_z^2)\mathbf p - (g_xp_x + g_yp_y + g_zp_z + g_wp_w)(g_x \mathbf e_1 + g_y \mathbf e_2 + g_z \mathbf e_3)$$

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

$$\begin{split}\left(\underline{\boldsymbol l_\smash{\unicode{x25CB}}} \wedge \mathbf p\right) \vee \boldsymbol l =\, &(l_{vx} p_x + l_{vy} p_y + l_{vz} p_z)\mathbf v \\ +\, &(l_{vy} l_{mz} - l_{vz} l_{my})p_w \mathbf e_1 \\ +\, &(l_{vz} l_{mx} - l_{vx} l_{mz})p_w \mathbf e_2 \\ +\, &(l_{vx} l_{my} - l_{vy} l_{mx})p_w \mathbf e_3 \\ +\, &(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)p_w \mathbf e_4\end{split}$$

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

$$\begin{split}\left(\underline{\mathbf g_\smash{\unicode{x25CB}}} \wedge \boldsymbol l\right) \vee \mathbf g =\, &(g_x^2 + g_y^2 + g_z^2)(l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43}) \\ -\, &(g_x l_{vx} + g_y l_{vy} + g_z l_{vz})(g_x \mathbf e_{41} + g_y \mathbf e_{42} + g_z \mathbf e_{43}) \\ +\, &(g_x l_{mx} + g_y l_{my} + g_z l_{mz})(g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}) \\ -\, &(g_y l_{vz} - g_z l_{vy})g_w \mathbf e_{23} - (g_z l_{vx} - g_x l_{vz})g_w \mathbf e_{31} - (g_x l_{vy} - g_y l_{vx})g_w \mathbf e_{12}\end{split}$$

Line plane.svg

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

$$(\mathbf y_\unicode{x25CB} \mathbin{\unicode{x22A8}} \mathbf x) \mathbin{\unicode{x2AE4}} \mathbf y = \left(\underline{\mathbf y_\smash{\unicode{x25CB}}} \vee \mathbf x\right) \wedge \mathbf y$$ .

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 $$\mathbf p$$.

$$\left(\underline{\mathbf p_\smash{\unicode{x25CB}}} \vee \mathbf g\right) \wedge \mathbf p = g_xp_w^2 \mathbf e_{423} + g_yp_w^2 \mathbf e_{431} + g_zp_w^2 \mathbf e_{412} - (g_xp_x + g_yp_y + g_zp_z)p_w \mathbf e_{321}$$

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

$$\begin{split}\left(\underline{\mathbf p_\smash{\unicode{x25CB}}} \vee \boldsymbol l\right) \wedge \mathbf p =\, &l_{vx} p_w^2 \mathbf e_{41} + l_{vy} p_w^2 \mathbf e_{42} + l_{vz} p_w^2 \mathbf e_{43} \\ +\, &(p_y l_{vz} - p_z l_{vy})p_w \mathbf e_{23} + (p_z l_{vx} - p_x l_{vz})p_w \mathbf e_{31} + (p_x l_{vy} - p_y l_{vx})p_w \mathbf e_{12}\end{split}$$

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

$$\begin{split}\left(\underline{\boldsymbol l_\smash{\unicode{x25CB}}} \vee \mathbf g\right) \wedge \boldsymbol l =\, &(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)(g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412}) \\ -\, &(g_x l_{vx} + g_y l_{vy} + g_z l_{vz})(l_{vx} \mathbf e_{423} + l_{vy} \mathbf e_{431} + l_{vz} \mathbf e_{412}) \\ +\, &(g_x l_{my} l_{vz} - g_x l_{mz} l_{vy} + g_y l_{mz} l_{vx} - g_y l_{mx} l_{vz} + g_z l_{mx} l_{vy} - g_z l_{my} l_{vx}) \mathbf e_{321}\end{split}$$

Plane line.svg

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
$$\left(\underline{\mathbf g_\smash{\unicode{x25CB}}} \wedge \mathbf e_4\right) \vee \mathbf g = -g_xg_w \mathbf e_1 - g_yg_w \mathbf e_2 - g_zg_w \mathbf e_3 + (g_x^2 + g_y^2 + g_z^2)\mathbf e_4$$ Point closest to the origin on the plane $$\mathbf g$$.
$$\left(\underline{\boldsymbol l_\smash{\unicode{x25CB}}} \wedge \mathbf e_4\right) \vee \boldsymbol l = (l_{vy} l_{mz} - l_{vz} l_{my})\mathbf e_1 + (l_{vz} l_{mx} - l_{vx} l_{mz})\mathbf e_2 + (l_{vx} l_{my} - l_{vy} l_{mx})\mathbf e_3 + (l_{vx}^2 + l_{vy}^2 + l_{vz}^2)\mathbf e_4$$ 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 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.

Antiprojection Formula Description
$$\left(\underline{\mathbf p_\smash{\unicode{x25CF}}} \vee \mathbf e_{321}\right) \wedge \mathbf p = -p_xp_w \mathbf e_{423} - p_yp_w \mathbf e_{431} - p_zp_w \mathbf e_{412} + (p_x^2 + p_y^2 + p_z^2)\mathbf e_{321}$$ Plane farthest from the origin containing the point $$\mathbf p$$.
$$\left(\underline{\boldsymbol l_\smash{\unicode{x25CF}}} \vee \mathbf e_{321}\right) \wedge \boldsymbol l = (l_{my} l_{vz} - l_{mz} l_{vy})\mathbf e_{423} + (l_{mz} l_{vx} - l_{mx} l_{vz})\mathbf e_{431} + (l_{mx} l_{vy} - l_{my} l_{vx})\mathbf e_{412} + (l_{mx}^2 + l_{my}^2 + l_{mz}^2)\mathbf e_{321}$$ Plane farthest from the origin containing the line $$\boldsymbol l$$.

See Also