Ron Goldman (CS, Rice University)
In contemporary 3-dimensional computer graphics, the graphics pipeline consists of a sequence of affine transformations composed at the end
with a single perspective projection. Typically, affine and projective transformations are represented by 4׳4 matrices, and 3-dimensional points and
vectors are represented with four homogeneous coordinates. In this framework, quadric surfaces are also usually modeled by (symmetric) 4׳4 matrices.
Recently, however, several authors (Dorst, Mann, Fontijne, Hildenbrand, Perwass, Rockwood, Vince) have suggested that Clifford algebra might provide
a more natural, more powerful, more elegant algebraic framework for computer graphics than matrix algebra.
We begin by presenting a brief introduction to Clifford algebra. We shall then investigate several standard models of Clifford algebra for 3-dimensions,
including various homogeneous models as well as the conformal model and we will see that these algebras fail to include the full range of transformations used
in the graphics pipeline. Furthermore, all of these models fail to incorporate general quadric surfaces as elements of the algebra. Based, however, on
insights gained from investigating these basic models, we will show that there is a more universal variant of Clifford algebra for 3-dimensions that does indeed
incorporate the full range of affine and projective transformations used in the graphics pipeline as well as the full suite of quadric surfaces.