Chapter 1 - Why Geometric Algebra?

Abstract

This chapter provides an introduction to geometric algebra, a powerful computational system used to describe and solve geometrical problems. The main features of geometric algebra includes that vectors can be used to represent aspects of geometry, but the precise correspondence is a modeling choice. Geometric algebra offers three increasingly powerful models for Euclidean geometry. Geometric algebra has products to combine vectors to new elements of computation. They represent oriented subspaces of any dimension, and they have rich geometric interpretations within the models. A linear transformation on the vector space dictates how subspaces transform; this augments the power of linear algebra in a structural manner to the extended elements. Geometric objects and operators are represented on a par and are exchangeable: objects can act as operators, and operators can be transformed such as geometrical objects. Geometric algebra focuses on the subspaces of a vector space as elements of computation. It constructs these systematically from the underlying vector space and extends the matrix techniques to transform them, even supplanting those completely when the transformations are orthogonal. In the vector space model and the conformal model, orthogonal transformations are used to represent basic motions of Euclidean geometry. This makes that type of linear transformation fundamental to doing geometry in the models.