Rendering Implicit Curves With Isolated Points Through Binary Space Partitioning

Graphics systems usually have primitives for rendering straight-line segments, circles, Bézier curves and surfaces. However, they do not have generalized primitives for implicit curves and surfaces. This is so because current algorithms have serious difficulties in dealing with differential singularities (e.g. self-intersections). This paper is about a generic algorithm capable of rendering any implicit curve in the plane, no matter whether or not it has singularities. For that, a BSP (Binary Space Partitioning) algorithm subdivides the ambient space Ω ⊆ R recursively by bisection lines. Each partition step involves the computation of zero or more curve points that result from intersecting the current bisection line and the curve. Each point is determined by a numerical approximation without derivatives. Besides, unlike other space decomposition algorithms, the algorithm here proposed can also compute isolated points of the curve.