Topologically Faithful Fitting of Simple Closed Curves

Abstract

Implicit representations of curves have certain advantages over explicit representation, one of them being the ability to determine with ease whether a point is inside or outside the curve (inside-outside functions). However, save for some special cases, it is not known how to construct implicit representations which are guaranteed to preserve the curve's topology. As a result, points may be erroneously classified with respect to the curve. The paper offers to overcome this problem by using a representation which is guaranteed to yield the correct topology of a simple closed curve by using homeomorphic mappings of the plane to itself. If such a map carries the curve onto the unit circle, then a point is inside the curve if and only if its image is inside the unit circle.