Error Analysis of the Bezier Approximation of Algebraic Surfaces

Abstract

For computer-aided geometric design or for computer-aided geometric representation, parametric curve segments and parametric surface patches such as the Bezier form or B-spline form have been frequently used. It is very convenient if we can use the same parametric representations for the approximation of algebraic curves or surfaces, such as circles, ellipses, spheres and tori. Then we can perform unified treatment for all free-form and nonfree-form curves and surfaces. For the design, machining and inspection of accurate geometry such as precision machine parts, it is very important to know the qualitative and quantitative error of approximation from exact geometry. But up to the present, little work has been done on this error. In this report, we treated the error of approximation and examined analytically and numerically the positions of control points and the magnitudes of the errors to approximate optimally algebraic curves and surfaces by cubic Bezier curves and by bicubic Bezier surfaces.