Projections

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.

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.

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.

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.