具有混合权函数的有理Bézier曲线的极限性质  

Limit properties of rational Bézier curves with mixed weights

作　　者：魏琦文 刘向云 吴金明[2] 朱春钢[1] WEI Qiwen;LIU Xiangyun;WU Jinming;ZHU Chungang(School of Mathematical Sciences,Dalian University of Technology,Dalian 116024,Liaoning Province,China;School of Statistics and Mathematics,Zhejiang Gongshang University,Hangzhou 310018,China)

机构地区：[1]大连理工大学数学科学学院,辽宁大连116024 [2]浙江工商大学统计与数学学院,浙江杭州310018

出　　处：《浙江大学学报(理学版)》2025年第1期110-121,共12页Journal of Zhejiang University(Science Edition)

基　　金：国家自然科学基金资助项目(12071057,11671068);中央基本科研业务费项目(DUT23LAB302)。

摘　　要：权因子是调整有理Bézier曲线形状的重要手段。当单个权因子趋于无穷时,有理Bézier曲线趋向于相应的控制顶点。已有研究表明,当所有权因子都以幂函数或指数函数形式趋于无穷时,有理Bézier曲线的极限曲线为其正则控制曲线。基于此,提出了一个新模型,结合幂权函数与指数权函数的转换关系,定义了具有混合权函数的有理Bézier曲线的正则控制曲线;结合toric退化理论,证明了当所有权因子都以混合函数形式趋于无穷时,有理Bézier曲线的极限曲线恰为其正则控制曲线;最后通过实例验证了结论的正确性。The weights of a rational Bézier curve play a crucial role in shape modification of the curve.As the weight of a rational Bézier curve approaches infinity,the curve tends towards the corresponding control point.Previous research has shown that when all weights of a rational Bézier curve tend to infinity in the form of power or exponential functions,the curve approaches its regular control curve.In this paper,we propose a new model,combining the transformation relationship between power functions and exponential functions,and then define the regular control curve of a rational Bézier curve with mixed weight functions.Based the toric degeneration theory,we prove that the limit of a rational Bézier curve is exactly its regular control curve when all weights tend to infinity in the form of mixed functions.

关 键 词：有理BÉZIER曲线 权因子 混合权函数 toric退化 

分 类 号：TP391[自动化与计算机技术—计算机应用技术]