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中文摘要：
为了保持有理Bézier曲线权因子的正性,提出一种有理Bézier曲线带端点约束条件的一次降多阶算法.通过给出有理Bézier曲线的降阶误差估计,揭示了原曲线权因子和降阶误差之间的关系;利用Mbius变换对权因子优化,通过缩小原曲线权因子之间的比值来缩小降阶误差;利用已有的Bézier曲线降阶算法和有理Bézier曲线的齐次形式,分别求得降阶曲线的控制顶点和权因子.通过数值实例将该算法与已有算法比较,结果表明：该算法具有保端点高阶插值、一次降多阶、显式表示、保权因子正性、逼近误差小等优点.
英文摘要：
An algorithm for constrained multi-degree reduction of rational Bézier curves at endpoints was presented to preserve the positive property of the weights of rational Bézier curves.Based on the approximation error estimation of the degree-reduced rational Bézier curves,the relation between the weights of original curves and the approximation error was revealed.The weights were optimized by using Mbius transformation.The ratio of the weights was minimized to minimize the approximation error.Based on the existing algorithm for degree reduction of Bézier curves and the homogeneous form of rational Bézier curves,the control points and the weights of the degree-reduced curves were obtained.The algorithm was compared with some existing algorithms through some numerical examples,and the results suggest that the algorithm can preserve high interpolation at endpoints,do multi-degree reduction at one time,use explicit approximation expression,and preserve positive weights and low approximation error.
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