Geometries

The goal of this chapter is to show how geometric algebra can be used to represent geometry. This includes the representation of geometric entities, incidence relations, and transformations. The algebraic entities by themselves have no a priori geometric interpretation. They are simply mathematical objects that have particular algebraic properties. In order to associate the algebra with geometry, a particular representation has to be dened. The goal is, of course, to dene a representation where the standard algebraic operations relate to geometrically meaningful and useful operations. Given such a representation, geometric problems can be translated into algebraic expressions, whence they can be solved using algebraic operations. In this sense, the algebra is then an algebra of geometry, a geometric algebra.

There are many examples of these types of algebras, some of which we described in Sect. 3.8. The geometric algebra discussed in this text encompasses all the algebras described in Sect. 3.8, which is the justication for calling it the geometric algebra.

Note that the field of algebraic geometry is closely related to this method of representing geometry. In algebraic geometry, geometric entities are represented through the null space of a set of polynomials, an affne variety (see e.g. [38]). The representations of geometric entities discussed in this chapter can all be regarded as affine varieties or as intersections of affine varieties. In this context, geometric algebra offers a convenient way to work with certaintypes of affine varieties.