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 $$\mathbf a$$ onto an object $$\mathbf b$$ is given by the general formula $$(\mathbf b_\unicode{x25CB} \mathbin{\unicode{x22A2}} \mathbf a) \mathbin{\unicode{x22A3}} \mathbf b$$. Applying the definitions of the left and right interior products, this becomes

$$(\mathbf b_\unicode{x25CB} \mathbin{\unicode{x22A2}} \mathbf a) \mathbin{\unicode{x22A3}} \mathbf b = \left(\underline{\mathbf b_\smash{\unicode{x25CB}}} \wedge \mathbf a\right) \vee \mathbf b$$ .

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

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

$$\left(\underline{\mathbf f_\smash{\unicode{x25CB}}} \wedge \mathbf p\right) \vee \mathbf f = (f_x^2 + f_y^2 + f_z^2)\mathbf p - (f_xp_x + f_yp_y + f_zp_z + f_wp_w)(f_x \mathbf e_1 + f_y \mathbf e_2 + f_z \mathbf e_3)$$

Point plane.png
Projection of point $$\mathbf p$$ onto line $$\mathbf L$$.

$$\begin{split}\left(\underline{\mathbf L_\smash{\unicode{x25CB}}} \wedge \mathbf p\right) \vee \mathbf L =\, &(v_xp_x + v_yp_y + v_zp_z)\mathbf v \\ +\, &(v_ym_z - v_zm_y)p_w \mathbf e_1 \\ +\, &(v_zm_x - v_xm_z)p_w \mathbf e_2 \\ +\, &(v_xm_y - v_ym_x)p_w \mathbf e_3 \\ +\, &(v_x^2 + v_y^2 + v_z^2)p_w \mathbf e_4\end{split}$$

Point line.png
Projection of line $$\mathbf L$$ onto plane $$\mathbf f$$.

$$\begin{split}\left(\underline{\mathbf f_\smash{\unicode{x25CB}}} \wedge \mathbf L\right) \vee \mathbf f =\, &(f_x^2 + f_y^2 + f_z^2)(v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43}) \\ -\, &(f_xv_x + f_yv_y + f_zv_z)(f_x \mathbf e_{41} + f_y \mathbf e_{42} + f_z \mathbf e_{43}) \\ +\, &(f_xm_x + f_ym_y + f_zm_z)(f_x \mathbf e_{23} + f_y \mathbf e_{31} + f_z \mathbf e_{12}) \\ -\, &(f_yv_z - f_zv_y)f_w \mathbf e_{23} - (f_zv_x - f_xv_z)f_w \mathbf e_{31} - (f_xv_y - f_yv_x)f_w \mathbf e_{12}\end{split}$$

Line plane.png

Antiprojection

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

$$(\mathbf b_\unicode{x25CB} \mathbin{\unicode{x22A8}} \mathbf a) \mathbin{\unicode{x2AE4}} \mathbf b = \left(\underline{\mathbf b_\smash{\unicode{x25CB}}} \vee \mathbf a\right) \wedge \mathbf b$$ .

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

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

$$\left(\underline{\mathbf p_\smash{\unicode{x25CB}}} \vee \mathbf f\right) \wedge \mathbf p = f_xp_w^2 \mathbf e_{234} + f_yp_w^2 \mathbf e_{314} + f_zp_w^2 \mathbf e_{124} - (f_xp_x + f_yp_y + f_zp_z)p_w \mathbf e_{321}$$

Plane point.png
Antiprojection of line $$\mathbf L$$ onto point $$\mathbf p$$.

$$\begin{split}\left(\underline{\mathbf p_\smash{\unicode{x25CB}}} \vee \mathbf L\right) \wedge \mathbf p =\, &v_xp_w^2 \mathbf e_{41} + v_yp_w^2 \mathbf e_{42} + v_zp_w^2 \mathbf e_{43} \\ +\, &(p_yv_z - p_zv_y)p_w \mathbf e_{23} + (p_zv_x - p_xv_z)p_w \mathbf e_{31} + (p_xv_y - p_yv_x)p_w \mathbf e_{12}\end{split}$$

Line point.png
Antiprojection of plane $$\mathbf f$$ onto line $$\mathbf L$$.

$$\begin{split}\left(\underline{\mathbf L_\smash{\unicode{x25CB}}} \vee \mathbf f\right) \wedge \mathbf L =\, &(v_x^2 + v_y^2 + v_z^2)(f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124}) \\ -\, &(f_xv_x + f_yv_y + f_zv_z)(v_x \mathbf e_{234} + v_y \mathbf e_{314} + v_z \mathbf e_{124}) \\ +\, &(f_xm_yv_z - f_xm_zv_y + f_ym_zv_x - f_ym_xv_z + f_zm_xv_y - f_zm_yv_x) \mathbf e_{321}\end{split}$$

Plane line.png

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
$$\left(\underline{\mathbf f_\smash{\unicode{x25CB}}} \wedge \mathbf e_4\right) \vee \mathbf f = -f_xf_w \mathbf e_1 - f_yf_w \mathbf e_2 - f_zf_w \mathbf e_3 + (f_x^2 + f_y^2 + f_z^2)\mathbf e_4$$ Point closest to the origin on the plane $$\mathbf f$$.
$$\left(\underline{\mathbf L_\smash{\unicode{x25CB}}} \wedge \mathbf e_4\right) \vee \mathbf L = (v_ym_z - v_zm_y)\mathbf e_1 + (v_zm_x - v_xm_z)\mathbf e_2 + (v_xm_y - v_ym_x)\mathbf e_3 + (v_x^2 + v_y^2 + v_z^2)\mathbf e_4$$ Point closest to the origin on the line $$\mathbf L$$.

Antiprojection of Plane at Infinity

Symmetrically to the projection of the origin, the plane at infinity $$\mathbf f = \mathbf e_{321}$$ 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
$$\left(\underline{\mathbf p_\smash{\unicode{x25CF}}} \vee \mathbf e_{321}\right) \wedge \mathbf p = -p_xp_w \mathbf e_{234} - p_yp_w \mathbf e_{314} - p_zp_w \mathbf e_{124} + (p_x^2 + p_y^2 + p_z^2)\mathbf e_{321}$$ Plane farthest from the origin containing the point $$\mathbf p$$.
$$\left(\underline{\mathbf L_\smash{\unicode{x25CF}}} \vee \mathbf e_{321}\right) \wedge \mathbf L = (m_yv_z - m_zv_y)\mathbf e_{234} + (m_zv_x - m_xv_z)\mathbf e_{314} + (m_xv_y - m_yv_x)\mathbf e_{124} + (m_x^2 + m_y^2 + m_z^2)\mathbf e_{321}$$ Plane farthest from the origin containing the line $$\mathbf L$$.

See Also