ANZIAM  J.  45 (2003), 195-205
Degree reduction of Bézier curves using constrained Chebyshev polynomials of the second kind

Young Joon Ahn
  Department of Mathematics Education
  Chosun University
  Gwangju 501--759

In this paper a constrained Chebyshev polynomial of the second kind with $C^1$-continuity is proposed as an error function for degree reduction of Bézier curves with a $C^1$-constraint at both endpoints. A sharp upper bound of the $L_\infty$ norm for a constrained Chebyshev polynomial of the second kind with $C^1$-continuity can be obtained explicitly along with its coefficients, while those of the constrained Chebyshev polynomial which provides the best $C^1$-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 $C^1$-constrained degree reduction within a given tolerance is presented. As an illustration, our method is applied to $C^1$-constrained degree reduction of a plane Bézier curve, and the numerical result is compared visually to that of the best degree reduction method.
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