Exploring Novel Surface Representations via an Experimental Ray-Tracer in CGA

Abstract

Conformal Geometric Algebra (CGA) provides a unified representation of both geometric primitives and conformal transformations, and as such holds significant promise in the field of computer graphics. In this paper we implement a simple ray tracer in CGA with a Blinn–Phong lighting model, before putting it to use to examine ray intersections with surfaces generated from the direct interpolation of geometric primitives. General surfaces formed from these interpolations are rendered using analytic normals. In addition, special cases of point-pair interpolation, which might find use in graphics applications, are described and rendered. A closed form expression is found for the derivative of the square root of a scalar plus 4-vector element with respect to a scalar parameter. This square root derivative is used to construct an expression for the derivative of a pure-grade multivector projected to the blade manifold. The blade manifold projection provides an analytical method for finding the normal line to the interpolated surfaces and its use is shown in lighting calculations for the ray tracer and in generating vertex normals for exporting the evolved surfaces as polygonal meshes.