Ray-Affine Functions: A General Dual Form to Describe Curves, Surfaces, and Volumes

Abstract

In computer graphics modeling, two different forms are used to represent curves and surfaces: implicit and parametric. Functions that can be expressed both in implicit and parametric forms are called dual forms. To date, the only known dual forms are monoids and superquadrics. In this paper, we introduce a new dual form: ray-affine functions. Ray-affines include both monoids and superquadrics and provide a wide range of other modeling functions including exponentials and sinusoidals. Ray-affines are closed under operations that implement morphing, union, and interpolation. This feature of ray-affine functions lets the user construct a ray-affine function to model a shape as a smooth aproximation of a control shape given by set union or set intersection of shapes defined by simpler ray-affine functions.