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. You see most clearly what it does when you place the black point on one of the lines.
This concept is essential to the
composition of projections.
Created with Cinderella