本科生毕业论文-贝塞尔曲线降阶逼近

湖 南 农 业 大 学 全日制普通本科生毕业论文

Bézier曲线的降阶逼近

Degree reduction of Bézier curves

学生姓名： 李齐放

学 号： 200740204202

年级专业及班级： 2007级信息与计算科学(2)班 指导老师及职称： 刘凯 讲师 学 院： 理学院

湖南·长沙

提交日期： 2011 年 6 月

湖南农业大学全日制普通本科生毕业论文

诚 信 声 明

本人郑重声明：所呈交的本科毕业论文是本人在指导老师的指导下，进行研究工作所取得的成果，成果不存在知识产权争议。除文中已经注明引用的内容外，本论文不含任何其他个人或集体已经发表或撰写过的作品成果。对本文的研究做出重要贡献的个人和集体在文中均作了明确的说明并表示了谢意。本人完全意识到本声明的法律结果由本人承担。

毕业论文作者签名：

2011 年 月 日

目 录

摘 要 ························································································································································· 1 关键词 ························································································································································· 1 1 前 言 ···················································································································································· 1 2 Bezier曲线的应用 ································································································································· 2 3 贝塞尔曲线拆分算法 ····························································································································· 3 4 Bezier曲线的数学论证 ·························································································································· 9 5 与其他算法的比较 ································································································································ 11 5 结 论 ··················································································································································· 13 6 结束语 ··················································································································································· 14 参考文献 ····················································································································································· 14 致 谢 ····················································································································································· 16

Bézier曲线的降阶逼近

学 生：李齐放 指导老师：刘 凯 (湖南农业大学理学院，长沙 410128)

摘 要：研究了两端点具有任意阶插值条件的Bézier曲线降多阶逼近的问题，对于给定的首末端点的

各阶插值条件，给出了一种新的一次降多阶逼近算法，应用Chebyshev多项式逼近理论达到了满足端点插值条件下的近似最佳一致逼近。此算法易于实现，误差计算简单，且所得降阶曲线具有很好的逼近效果，结合分割算法，可获得相当高的误差收敛速度。（五号宋体）

关键词：Bézier曲线，降多阶，端点插值,逼近，分割

1 前 言

计算机辅助几何设计始兴于上世纪60年代，最初始于飞机、船舶的外形放样（Lofting）工艺。在当时计算机发展的影响下，为了利用计算机更高效地进行设计，人们开始寻找研究曲线曲面的各种表示方法，其中最著名、最实用的技术当是由法国雷诺（Renault）汽车公司的工程师提出的Bezier技术和美国机械工程师教授 Coons 提出的 Coons 技术（本文只涉及 Bezier 曲线，故只讨论 Bezier技术）。在大多情况下，描述产品外形的曲线只有大概形状或者只知道它所通过的一系列空间点列，这些点称为型值点，这类曲线叫自由曲线；而计算机辅助几何设计就是研究自由曲线的表示、设计、显示、分析与综合以及处理等问题。在Bezier 曲线的表示中，预先给定一批控制顶点，通过这些控制顶点生成 Bezier 曲线，其形状当然由控制顶点来控制，当然形状的改变也受这些控制顶点位置改变的影响，因此我们可以通过这些顶点的位置改变来调控曲线的形状。

在工程设计和科学实验当中，我们经常要设计或描绘一些不规则的曲线，当利用计算机对这些不规则曲线进行表达、分析和研究时首要的任务就是对特定的不规则曲线要建立一个数学模型去描述它，因为在计算机内部图是以二进制的形式存贮着。一般，碰到的实际问题有以下三种情况：

(1)由已知的一系列准确的数据点来定义曲线，也就是说，要求曲线必须经过所有的数据点。一般，经常采用多项式插值法，例如样条方程来解决这个问题。

(2)给定一些离散的数据点，这些数据点仅是某些未知真实值的近似数据。要求用一条曲线来指出这些数据点的正确趋势。这条曲线可能只通过一部分数据点，或根本不通过任何

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