From Conformal Geometric Algebra
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Any geometric object $$\mathbf a$$ can be projected onto another geometric object $$\mathbf b$$ of higher grade by first calculating the expansion of $$\mathbf a$$ onto $$\mathbf b$$ and then using the meet operation to intersect the result with $$\mathbf b$$. That is, the projection of $$\mathbf a$$ onto $$\mathbf b$$ is given by

$$\mathbf b \vee (\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2606})$$ .

This formula is general and works for flat points, lines, planes, round points, dipoles, circles, and spheres.

See Also