Conditions for Determining the Regularity of Bézier Curve and Surface

Regularity is an important algebraic property of parametric curve and surface, which depends on the parameterization of parametric curve and surface. In computer-aided manufacturing, the processed parametric curve and surface should be regular, so the parametric curve and surface generated by computer-aided design should be regular first. However, the computation of determining the regularity of parametric curve and surface by solving equation or system of equations induced by the definition of regularity is considerably complex, and is actually infeasible. In this paper, by transforming the parametric representations of derivative vector curve (of Bezier curve) and normal vector surface (of Bezier surface) to their implicit representations, a simple and practical sufficient condition for determining the regularity of Bezier curve and surface is presented.