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毕业论文-B样条曲面拼接处满足容差的k阶几何连续性调整算法，共71页，28946字
中文摘要
曲面质量的优化是计算机辅助几何设计中的一个重要问题，在工业产品的外
形设计中具有广泛的应用．研究人员使用曲面的几何连续性作为曲面质量的评价
标准之一，并提出了一系列曲面几何连续性优化方法，可以有效地对单张曲面表
示的形体表面进行优化．但由于工业产品固有的复杂性，大量的几何外形是由多
张曲面拼接而成，从而很难用现有的曲面质量优化方法进行处理．
本文提出了一种在非裁剪 B 样条曲面与其他曲面的拼接处，对非裁剪 B 样条
曲面进行调整的方法．该方法将通过曲面的局部调整，使得两曲面在公共边界处
达到给定容差的 G k 连续性：即 0 阶至 k 阶几何连续性误差满足给定的容差向量
．对于两张曲面均为完整边界的情况，B 样条曲面边界拼接处还满足无误差的G 0 连续．
本文首先根据几何连续性的定义，提出了 k 阶几何连续性拼接的目标函数，
并构造出最小二乘误差下的求解表达式．随后通过在曲面中迭代地插入中间节点
的方法，本文算法可以不断地减小最小二乘误差的值，直至误差满足给定的容差
条件为止．本文给出了算法的实现效果和测试用例，分析了曲面调整算法的效率、
效果，误差或其他因素对算法性能的影响．实验表明，该算法能有效地优化曲面
边界处的几何形状，对于一阶至三阶的几何连续约束均能很好地满足给定的容差
要求．
关键词：B 样条曲面；曲面拼接；曲面调整；容差；几何连续性；

ABSTRACT
Automated surface refinement is an important goal of Computer-Aided Geometric
Design, which meets widely application in shape design of industrial products.
Geometric continuity is one of the evaluation criteria of surface quality, and based on
which a variety of surface optimization techniques have been proposed. These
techniques apply efficiently to shapes composed of a single surface. However, due to
the intrinsic complexity of industrial products, the shapes, in most cases, are made up
of multiple surface patches. Consequently, existing surface optimization methods are
hardly applicable to them.
In this paper we propose a new untrimmed B-spline surface-adjustment technique
along the topological border between an untrimmed B-spline surface and an arbitrary
surface. The algorithm makes the common boundary satisfy the G k continuity
condition with a given tolerance using local adjustment of the surfaces: the common
edge of the B-spline surfaces satisfies the zeroth- to kth-order geometric continuity to
the given tolerance vector ? . If neither of the surfaces is trimmed, the algorithm
ensures error-free zeroth-order geometric continuity.
We first establish an objective function for kth-order geometric continuity on the
basis of its definition, and construct the solving expression with the least squares error.
Then we illustrate that the least squares error will gradually reduce, by iteratively
subdividing the control grid using knot insertion, until the error meets the tolerance
and halts the iteration. In this paper we also show the test samples and results of the
implementation, discuss the efficiency and performance, and analyze the error and
other factors which may impact the results of our algorithm. The experiment indicates
that our method can efficiently optimize the geometry appearance along the common
boundary, and may easily satisfy the first- to third-order geometric continuity
constraints to the given tolerance.
Keywords：B-spline Surface; Surface Blending; Surface Adjustment; Tolerance;
Geometric Continuity

目录
第 1 章
引言 ............ 1
1.1 研究背景与意义 . 1
1.2 B 样条曲线与曲面基础 ....... 2
1.2.1 B 样条基函数 . 2
1.2.2 B 样条曲线 ..... 2
1.2.3 B 样条曲面 ..... 4
1.3 曲面的几何连续性问题 ...... 5
1.4 研究现状 ............. 6
1.5 本文主要研究内容 .............. 6
第 2 章 B 样条曲面拼接处调整算法 ....... 8
2.1 问题描述 ............. 8
2.2 算法原理 ............. 9
2.2.1 完整边界调整算法 ......... 9
2.2.2 裁剪边界处理 ............... 21
2.2.3 公共边界匹配 ............... 23
2.3 算法实现 ........... 24
2.3.1 输入输出描述 ............... 24
2.3.2 算法流程描述 ............... 24
第 3 章 算法结果与分析 ....... 27
3.1 实验结果 ........... 27
3.2 误差分析 ........... 34
3.3 算法效率与算法复杂度 .... 36
3.4 算法的扩展 ....... 36
3.5 算法优缺点分析 ................ 37
第 4 章 总结与展望 ............... 39
4.1 总结 .. 39
4.2 展望 .. 40
插图索引
表格索引
算法索引
参考文献
致 谢....... 47
声 明....... 49
附录 A 外文资料的调研阅读报告（或书面翻译） ... 51