@Ziriax, You’re right, we didn’t include that on the cheat sheet. Thanks for bringing it up, it’s a nice formula that reveals the “polymorphicity” of PGA.

The product of a normalized euclidean plane p and a normalized euclidean point P yields a normalized bivector \ell and a multiple of the pseudoscalar \bf{I}: pP = \ell + d\bf{I} where \ell is the line through P perpendicular to p, and d is the signed distance from the point to the plane. Furthermore, \ell is normalized and its orientation is determined by the side of p on which P lies. (I. e., switching the order of the product to Pp doesn’t change the orientation of \ell.)

If the point is a normalized ideal point, then the bivector part vanishes and the weight of the pseudoscalar gives the sine of the angle that the ideal point (think “direction”) makes to the plane: pP = \sin(\alpha)\bf{I}.

This result is exactly analogous to the product of a point and a line in 2D.

Connection to motors: The result {m}:=pP is an element of the even subalgebra that satisfies {m}\widetilde{{m}}=1 hence is a motor. Which motor? The fact that there is no scalar part, only a pseudoscalar part, is a tip that it is a translation. Which translation? Turns out to be the translation in the direction normal to p that translates by a distance of 2d, so that P is translated across p to its mirror point. By taking \sqrt{m} = \dfrac{(1+m)}2, we obtain the translation that takes P onto the plane p.