New Proper Reparameterization of Plane Rational Bézier Curves

Coincidence detection of two curves or two surfaces has wide application in computer-aided design (CAD) and computer-aided geometric design (CAGD). Proper reparameterization is the most complicated part in the detection. This paper presents and proves efficient and necessary coincidence condition for two rational Bézier curves in a new way. It also proposes an effective and efficient proper reparameterization method, Algorithm 1, for detecting a rational Bézier curve which can be degenerated into a new one of a lower degree. A numerical proper reparameterization method, Algorithm 2, and examples are also presented. Algorithm 1 is up to ten times faster than other prevailing methods, and Algorithm 2 is twice faster and half closer than other prevailing methods. New CAD systems using Algorithm 1 and Algorithm 2 will hold accuracy and little computation time.