C1 Quadratic Interpolation over Arbitrary Point Sets

Abstract

New formulas for generating smooth surfaces over arbitrarily spaced data points are developed. The formulas are based on quadratic polynomials for the construction of derivative continuous surfaces rather than on the cubic polynomials generally used. The technique is based on a subdivision procedure, dividing each triangle in a triangulation of the data points into six subtriangles and fitting a quadratic Bezier surface patch over each subtriangle. THe formulas require only function and first derivative values at the data points and are easily evaluated in terms of the Bezier coefficients. Since two-dimensional quadratic polynomials contain only six terms, while 10 terms are required to evaluate a cubic, the new procedure significantly improves the efficiency of algorithms for drawing surfaces in computer-aided geometric design.