Pab(x) = ( x - abxba )/2 and Pba(x) = ( x - baxab )/2
(where a and b are unit vectors).
Geometrically, the rotor projection is minus half the chord from x to its rotation.
(Alternatively, you can see it as minus the orthogonal projection of x onto its half-rotated carrier (i.e. the carrier rotated over the same angle as from a to b). A mouthful, but obvious from the picture below.)
Drag the black point to note that this construction is rotationally covariant, and a or b to find its dependency on the angles.
This concept is essential to the composition of projections.
Created with Cinderella