Performance and elegance of five models of 3D Euclidean geometry in a ray tracing application

Computations of 3D Euclidean geometry can be performed using various computational models of different effectiveness. In this paper we compare five alternatives: 3D linear algebra, 3D geometric algebra, a mix of 4D homogeneous coordinates and Plücker coordinates, a 4D homogeneous model using geometric algebra, and the 5D conformal model using geometric algebra. Higher dimensional models and models using geometric algebra can express geometric primitives, computations and constructions more elegantly, but this elegance may come at a performance penalty. We explore these issues using the implementation of a simple ray tracer as our practical goal and guide. We show how to implement the most important geometric computations of the ray tracing algorithm using each of the five models and benchmark each implementation. 1