Conformal Geometry, Euclidean Space and Geometric Algebra

TL;DR

This paper explores how conformal geometry and geometric algebra can improve Euclidean space representations in graphics, providing a unified framework that addresses distance measurement issues and enables efficient geometric operations.

Contribution

It introduces a conformal geometric algebra approach that enhances Euclidean geometry handling, including a formula for reflecting lines off spherical surfaces.

Findings

Unified framework for Euclidean geometry and conformal transformations

Efficient formula for reflecting lines on spherical surfaces

Addresses distance measurement limitations in projective geometry

Abstract

Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both 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 conformal 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.