Chapter 7 - Orthogonal Transformations as Versors

Abstract

The concept of a versor (a product of vectors to be used as an operator in a sandwiching product) combines all the representations of orthogonal transformations. The versors preserve the structure of geometric constructions and can be universally applied to any geometrical element. This is a unique feature of geometric algebra, and it can simplify code considerably. An even number of reflections gives a rotation, represented as a rotor—the geometric product of an even number of unit vectors. The rotors encompass and extend complex numbers and quaternions and present a real 3-D visualization of the quaternion product. Rotors transcend quaternions in that they can be applied to elements of any grade, in a space of any dimension. Two reflections make a rotation, even in R3. Since an even number of reflections absorbs any sign, these reflections are either line reflections or both (dual) hyperplane reflections, whichever feels most natural or is easiest to visualize. The composition of rotations follows automatically from their representation as a geometric product—the rotor of successive rotations, first R1 then R2, is their geometric product R2 R1.