Conformal Geometry, Euclidean Space and Geometric Algebra

Abstract

Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are linear transformations in projective geometry, which helps when it comes to programming complicated geometrical operations. But there is a fundamental weakness in this approach — the Euclidean distance between points is not handled in a straightforward manner. Here we discuss a solution to this problem, based on conformai geometry. The language of geometric algebra is best suited to exploiting this geometry, as it handles the interior and exterior products in a single, unified framework. A number of applications are discussed, including a compact formula for reflecting a line off a general spherical surface.KeywordsGeometric algebraClifford algebraconformal geometryprojective geometryhomogeneous coordinatessphere geometrystereographic projection