In the preceding chapters of this book we have dealt entirely with a single geometric interpretation of the elements of a geometric algebra. But the relationship between algebra and geometry is seldom unique. Geometric problems can be studied using a variety of algebraic techniques, and the same algebraic result can typically be pictured in a variety of different ways. In this chapter, we explore a range of alternative geometric systems, and discover how geometric algebra can be applied to each of them. We will find that there is no unique interpretation forced on the multivectors of a given grade. For example, to date we have viewed bivectors solely as directed plane segments. But in projective geometry a bivector represents a line, and in conformal geometry a bivector can represent a pair of points.
Ideas from geometry have always been a prime motivating factor in the development of mathematics. By the nineteenth century mathematicians were familiar with affine, Euclidean, spherical, hyperbolic, projective and inversive geometries. The unifying framework for studying these geometries was provided by the Kleinian viewpoint. Under this view a geometry consists of a space of points, together with a group of transformations mapping the points onto themselves. Any property of a particular geometry must be invariant under the action of the associated symmetry group.