The conditions for the coincidence or overlapping of two Bézier curves

Abstract

In a recent article, Chen et al. (2013) derive the condition for the coincidence of two quartic Bézier curves different from straight lines. The condition reads that either their control polygons coincide, or the curves are degenerate representations of the same quadratic curve. We point out that this uniqueness of the control points for properly parameterized polynomial curves is already well-known, not only for quartics but for curves of arbitrary degree. This is a direct consequence of a previous, more general result by Berry and Patterson, namely the uniqueness of the control points for rational Bézier curves. In addition, two conjectures posed in Chen et al. (2013) also derive as straightforward corollaries. Finally, the condition for overlapping of two properly parameterized curves is also derived. Unsurprisingly, it amounts to the coincidence of the Bézier polygons for the overlapping segments of the given curves, corresponding to the overlapping portions of their domains.