| Coincidence detection between two curves is an important part of thecorresponding intersection algorithms, and it is also a key issue of the robustness ofthe intersection algorithm. Bézier curves of low degrees, such as cubics, quartics andquintics, are widely used in CAGD. There are few references discussing theconditions that two curves are coincident or partially coincident. In this paper, itdiscusses the coincidence condition between two quintic Bézier curves and tworational cubic Bézier curves, includes：1. The coincidence condition between two quintic Bézier curves. It proves that twonon-degenerated quintic Bézier curves are coincident with each other if and only iftheir control polygons are coincident. For the degenerated case that the given quinticBézier curves can be degenerated into another Bézier curve of a lower degree, thecorresponding coincidence condition is that either the two control polygons arecoincident or they can be degenerated into a same Bézier curve of a lower degree.Furthermore, simple but explicit formulae are provided for detecting the locationwhere two Bézier curves are partially coincided. Experimental results show that usingour conclusion, it is easy to judge whether two curves are coincident or not.2. The coincidence condition between two rational Bézier curves.It provides asufficient and necessary condition that two non-degenerated rational cubic Béziercurves are completely coincident with each other, i.e., the two curves are coincident ifand only if their control polygons and weights are coincident under the constraint thatthe first and last weighs are equal to1. The condition that two rational cubic Béziercurves are partially coincident and its detecting method on the coincident position arealso provided. |
