ray tracing algebraic surfaces
Many interesting surfaces can be written as polynomial functions of the spatial coordinates, often of low degree. We present a method based on a ray casting algorithm, extended to work in more than three dimensions, to produce pictures of these surfaces. The method uses a symbolic algebra system to automatically derive the equation of intersection between the ray and the surface and then solves this equation using an exact polynomial root finding algorithm. Included are illustrations of the cusp catastrophe surface, and two unusually shaped quartic surfaces, Kummer's quadruple and Steiner's surface.
ray tracing algebraic surfaces algebra surface polynomial spatial coordination method ray casting algorithm symbolic algebra systems intersection polynomial roots curved surface display