Projections in Geometric algebra

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While the projection formula for vectors is well-known, in geometric algebra, this formula works for any geometric object!

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There's also the rejection, which is the exact same formula, but using the outer product rather than the inner product. It too works for a variety of different objects in GA and in many different flavors, though its behavior can be harder to predict than the projection.

angeldude
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I think another way to see this is writing V•U = P_U (V) U. (Projection of V onto U multiplied by U with the geometric product). Then the U^-1 removed the extra factor of U, leaving us with just P_U(V). Not sure if this alternate way of writing it is very useful, but I’d come across it before so thought I’d add it here

kikivoorburg
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Well this locus type is very special and my personal favourite and it is essentially the Apollonian definition of circle

questforeverything
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This formula works in every possible pair to every possible dimensionality

theodorostsilikis
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What is the projection of a bivector onto a vector?

wordsonplay