Perturbing Bézier coefficients for best constrained degree reduction in the L2-norm |
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Authors: | Jianmin Zheng Guozhao Wang |
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Affiliation: | Department of Mathematics/Institute of Image and Computer Graphics, Zhejiang University, Hangzhou 310027, PR China |
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Abstract: | This paper first shows how the Bézier coefficients of a given degree n polynomial are perturbed so that it can be reduced to a degree m (<n) polynomial with the constraint that continuity of a prescribed order is preserved at the two endpoints. The perturbation vector, which consists of the perturbation coefficients, is determined by minimizing a weighted Euclidean norm. The optimal degree n−1 approximation polynomial is explicitly given in Bézier form. Next the paper proves that the problem of finding a best L2-approximation over the interval 0,1] for constrained degree reduction is equivalent to that of finding a minimum perturbation vector in a certain weighted Euclidean norm. The relevant weights are derived. This result is applied to computing the optimal constrained degree reduction of parametric Bézier curves in the L2-norm. |
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Keywords: | Degree reduction Endpoint constraints Bé zier form L2-norm Least squares perturbation |
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