Comparing Bézier Curves and Surfaces for Coincidence

Abstract

It is known that Bézier curves and surfaces may have multiple representations by different control polygons. The polygons may have different number of control points and may even be disjoint. This phenomenon causes difficulties in variety of applications where it is important to recognize cases where different representations define same curve (surface) or partially coincident curves (surfaces). The problem of finding whether two arbitrary parametric polynomial curves are the same has been addressed in Pekerman et al. (Are two curves the same? Comput.-Aided Geom. Des. Appl. 2(1–4):85–94, 2005). There the curves are reduced into canonical irreducible forms using the monomial basis, then they are compared and their shared domains, if any, are identified. Here we present an alternative geometric algorithm based on subdivision that compares two input control polygons and reports the coincidences between the corresponding Bézier curves if they are present. We generalize the algorithm for tensor product Bézier surfaces. The algorithms are implemented and tested using Mathematica package. The experimental results are presented.