Brief Introduction to Conformal Geometric Algebra

In words of Pablo Colapinto: “modern day graphics systems are an amalgam of matrix, vector, and tensor algebras“. These algebras run fast in computers, but the intricacies of their algebraic representations make it difficult to realize their truly geometric interpretation. Within the last decade, Geometric Algebra has emerged as a powerful alternative to the classical matrix, vector and tensor algebras as a comprehensive conceptual language and computational system for computer science. In his Master’s thesis Colapinto states that “great research has been done demonstrating its powers of synthesis, especially with regards to forms and control systems generated by twist groups and motor algebras“. However, it is known that little research focuses on their application to computer graphics, geometric modeling and animation. The aim of this post is to encourage the reader for use geometric algebra in geometric  modeling applications and computer graphics.

Geometric Algebra can be defined as the subset of Clifford Algebra that has geometric interpretation. In Conformal Geometric Algebra (CGA), rigid body motions can be represented as compact (multi-)vectors called Motors. Unlike homogeneous matrices, motors are linear transformations. All multivectors can be transformed by motors in a covariant and linear way. CGA motors can be efficiently interpolated for generating continuously deformed motions, and that is one of the main features that we should exploit in computer graphics. On the other hand, the interpolation of matrices is difficult and expensive. The logarithm of a matrix can be defined but is not elementary and not in closed form. A straightforward way to compute it is to take the eigenvalue decomposition of the rigid body motion matrix in the homogeneous coordinate framework, and take the Nth root of the diagonal matrix. Such numerical techniques makes the matrix logarithm expensive to compute and hard to analyze.