Conformal Geometry and Geometric Algebra for Euclidean Space

Title: Conformal Geometry and Geometric Algebra for Euclidean Space

Abstract: This research explores the use of conformal geometry and geometric algebra as a framework for representing Euclidean space in computer graphics applications. The authors discuss the advantages of using projective geometry, which is linear and handles translations and rotations, but lacks a straightforward method for calculating the Euclidean distance between points. They propose a solution using conformal geometry, which is a subset of projective geometry that preserves the Euclidean distance between points. The language of geometric algebra is then introduced as a unified framework for handling the interior and exterior products, which simplifies the implementation of complex geometric operations. The authors provide examples and discuss the implications of their findings, including a compact formula for reflecting a line off a general spherical surface.

Keywords: Conformal Geometry, Geometric Algebra, Projective Geometry, Homogeneous Coordinates, Sphere Geometry, Sterecopic Projection

Introduction: In computer graphics programming, the standard framework for modeling points in space is through a projective representation. This means that points in Euclidean space are represented as rays or vectors in a four-dimensional space, known as projective space. While this approach allows for the linear handling of translations and rotations, it lacks a straightforward method for calculating the Euclidean distance between points.

The authors propose a solution to this problem using conformal geometry, a subset of projective geometry that preserves the Euclidean distance between points. They argue that the language of geometric algebra is best suited to exploit this geometry, as it handles the interior and exterior products in a unified framework.

Methods:

1. Projective Geometry: The authors begin by discussing the advantages of using projective geometry in computer graphics applications. They explain how points in Euclidean space are represented as projective vectors, and how the group of Euclidean transformations (translations, reflections, and rotations) is represented by a set of linear transformations of projective vectors.

2. Conformal Geometry: The authors then introduce conformal geometry, which is a subset of projective geometry that preserves the Euclidean distance between points. They explain how this geometry can be used to calculate the Euclidean distance between points in a more straightforward manner than in projective geometry.

3. Geometric Algebra: The authors introduce the language of geometric algebra as a unified framework for handling the interior and exterior products in a single, unified manner. They explain how this language simplifies the implementation of complex geometric operations.

Results:

The authors provide examples and discuss the implications of their findings, including a compact formula for reflecting a line off a general spherical surface.

Conclusion:

In conclusion, the authors argue that the combination of conformal geometry and geometric algebra provides a powerful framework for representing Euclidean space in computer graphics applications. This framework not only handles the Euclidean distance between points in a straightforward manner, but also simplifies the implementation of complex geometric operations through the unified language of geometric algebra.

Link to Article: https://arxiv.org/abs/0203026v1 Authors: arXiv ID: 0203026v1