Constructing curves and triangular patches by Beta functions

Abstract

A class of new basis functions for curve and triangular patch modeling is constructed by means of Beta functions. Based on these basis functions, a new scheme of generating curves and triangular patches is proposed. First we demonstrate that these basis functions have similar properties as those of the Bernstein–Bézier basis functions, such as non-negativity, partition of unity and others. Thus, these basis functions give rise to curve and triangular patch representations with affine invariance, convex hull, symmetry and endpoint interpolation, as well as an evaluation algorithm, which is similar to the de Casteljau’s algorithm for Bézier curves and surfaces. In addition, these basis functions have a shape parameter. The shape of the curve or triangular patch can be modified by changing the value of the shape parameter under the same control polygon or control net. The modeling examples illustrate the validity of the new methods.