7 - Orthogonal transformations as versors

Abstract

Reflection in a line is represented by a sandwiching construction involving the geometric product. Though that may have seemed like a curiosity in the previous chapter, it is crucial to the representation of operators in geometric algebra. Geometrically, all orthogonal transformations can be considered as multiple reflections. Algebraically, this leads to their representation as a geometric product of unit vectors. An even number of reflection gives a rotation, represented as a rotor—the geometric product of an even number of unit vectors. This chapter shows that 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. The distinction between subspaces and operators fades when it is realized that any subspace generates a reflection operator, which can act on any element. The concept of a versor (a product of vectors to be used as an operator in a sandwiching product) combines all these representations of orthogonal transformations. This chapter shows that 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. The chapter ends with a discussion of the difference between geometric algebra and Clifford algebra, and a preliminary consideration of issues in efficient implementation to convince users of the practical usability of the versor techniques in writing efficient code for geometry.