Degree reduction of Béezier curves using constrained Chebyshev polynomials of the second kind

Abstract

AbstractIn this paper a constrained Chebyshev polynomial of the second kind with C1-continuity is proposed as an error function for degree reduction of Bézier curves with a C1-constraint at both endpoints. A sharp upper bound of the L∞ norm for a constrained Chebyshev polynomial of the second kind with C1-continuity can be obtained explicitly along with its coefficients, while those of the constrained Chebyshev polynomial which provides the best C1-constrained degree reduction are obtained numerically. The representations in closed form for the coefficients and the error bound are very useful to the users of Computer Graphics or CAD/CAM systems. Using the error bound in the closed form, a simple subdivision scheme for C1-constrained degree reduction within a given tolerance is presented. As an illustration, our method is applied to C1-constrained degree reduction of a plane Bézier curve, and the numerical result is compared visually to that of the best degree reduction method.