Are Two Curves the Same?

Abstract

We present an algorithm for answering the following fundamental question: Given two arbitrary (piecewise) polynomial curves, are they the same? This basic CAGD question is answered by first reducing the two curves into canonical irreducible forms. This is done by reversing the processes of knot refinement, degree raising, and composition. The two curves are then compared in their irreducible forms and their shared domains, if any, are identified. The ability to answer this fundamental identity question will be a boon for numerous applications. In this paper, we demonstrate a few such applications. The algorithm allows one to identify two boundary curves (shared as a common seam) between two different surfaces as an identical curve (or not) even when they are represented differently. Moreover, we show that reparameterization is insecure as a watermarking method, which invalidates the proposal of (8).