Degree reduction of Bézier curves and its error analysis

Abstract

A process for generating an (n-1)st degree approximate to a Bezier curve of degree n is proposed. This process is called degree reduction. The necessity to determine the degree reduced curve by approximation is manifest since generally the degree reduction is not exactly possible in contrast to the reversed question of degree elevation. Doing so, the degree reduction can be accomplished in number of ways. Forrest[12] proposed a geometrical algorithm preserving the tangent at endpoints. Farin[9] considered a degree reduction scheme for developing the rational case. Watkins and Worsey[24] suggested the use of the Chebyshev polynomials in degree reduction process. Nearly simultaneously, Lachance[16] described the same scheme whereby his algorithm needs the transformation to monomial series only. Eck[6] generalized the Farin``s method by using constrained Chebyshev polynomials. The methods of Forrest and Farin were mentioned without error analysis in the previous work. In this work, we investigate a geometric property of Bezier curves and describe a simple idea of degree reduction. The approximation does not coincide in general at the two boundaries. This problem can be overcome by introducing new factors. The approximation agrees at the two endpoints up to a preselected smoothness order. The presented scheme allows a detailed error analysis providing a priori bounds of the pointwise approximation error. The error analysis for the other scheme is also presented by applying the described scheme.