On the conditions for the coincidence of two cubic Bézier curves

Abstract

In a recent article, Wang et al. [2] derive a necessary and sufficient condition for the coincidence of two cubic Bézier curves with non-collinear control points. The condition reads that their control points must be either coincident or in reverse order. We point out that this uniqueness of the control points for polynomial cubics is a straightforward consequence of a previous and more general result of Barry and Patterson, namely the uniqueness of the control points for rational Bézier curves. Moreover, this uniqueness applies to properly parameterized polynomial curves of arbitrary degree.