基于几何约束的细分曲面造型基础技术研究
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摘要
细分曲面因其对任意拓扑的适应性正逐渐成为几何建模的强有力工具。几何约束是提高复杂曲面建模精度的重要手段,其中点与曲线是最有效的几何约束。本文主要研究任意拓扑曲线网和顶点约束下的细分曲面造型技术,旨在进一步提高细分曲面的造型能力,使其能更有效地应用于CAD概念设计中。本文的主要研究内容和创新性成果如下:
(1)给出了曲线网的定义后,提出了曲线网的编辑和构造方法,并由此对具有曲线网插值功能的联合细分模式进行了扩展,使其具备任意拓扑曲线网的插值能力。
(2)分析了联合细分模式的特性以及产生不光顺现象的原因后,从改进细分规则入手提出了非均匀联合细分模式,与原有联合细分模式相比,由非均匀联合细分模式生成的极限曲面更为光顺,插值曲面至少达到G1连续。
(3)提出了联合细分曲面形状修改的两种方法:曲线网编辑法和基于PDE优化的点约束方法。前者适用于联合细分曲面的初期形状修改,可实现插值曲面的局部和全局修改,后者可以对远离插值曲线的内部曲面进行修改,具有计算速度快、修改曲面平均曲率分布更均匀的优点。两者分别作用于联合细分曲面的不同区域和形状修改的不同阶段,互为补充,可较好地完成联合细分曲面的形状修改。
(4)提出了两种基于插值曲线构造特征的方法:一种根据给定的特征角,通过细分过程中跨界二阶偏导的调整,而在插值曲线上生成尖锐特征;另一种将定义特征的曲线与定义基曲面的曲线相分离,构造出复合曲线网,然后通过改进的联合细分模式进行细分,生成具有特征的细分曲面,特征的形状可通过改变特征曲线对应的尖锐系数来调节。前者操作简单,实现方便,而后者具有更大的设计自由度,并且生成的特征效果较好。
(5)提出了一种顶点和法向约束下的细分曲面构造方法。该方法通过将待插网格剖分一次后不断迭代优化而得到插值曲面的控制网格,无需求解全局线性方程组,具有计算效率高和插值曲面更光顺的优点,并且能够在一定程度上对插值曲面的形状进行灵活调节,给设计者提供了更多的形状表达自由度。
The subdivision surface is gradually becoming a powerful geometric modeling tool because of its advantage of arbitrary topological adaptability. Geometric constraints are important means to improve the precision of complex surface modeling. Points and curves are the most effective ones among geometric constraints. In order to improve the modeling abilities of subdivision surfaces and make it applied in conceptual design of CAD more efficiently, this dissertation mainly studies subdivision modeling technologies based on constraints of vertices and arbitrary topological curve networks. The main research contents and creative achievements are as fellows:
(1) After introducing the definition of curve networks, we present the constructing and editing methods of them. Based on these methods, the combined subdivision scheme is extended to have the ability of interpolating arbitrary topological curve networks.
(2) Based on the analyses of the feature and the unfair reasons of combined subdivision surfaces, we propose the non-uniform combined subdivision scheme by improving subdivision rules. The improved one can produce interpolating limit surface of better fairness. The surface generated by the non-uniform combined subdivision scheme reaches at least G1 continuity.
(3) Two shape modification methods of combined subdivision surfaces are put forward by editing curve networks and applying point constraints based on PDE optimum method. The former is suitable to be used at the initial stage of combined subdivision surfaces’modification, and can be used to modify the local and the global regions of combined subdivision surfaces. The latter can modify the region far away from interpolated curve, and has the advantages of high calculation efficiency and more uniform curvature distribution of modified surfaces. Two methods work on different regions of combined subdivision surfaces and at different stages of shape modification. They supplement for each other to modify combined subdivision surfaces.
(4) Two methods of constructing feature based on interpolated curves are presented. The first one is implemented by adjusting second derivatives on cross curve according to the given feature angle during subdivision and generating sharp feature on interpolated curves. The second one is to separate the curves defining features from the curves defining base surfaces and build compound curve networks, and then combined subdivision surface with sharp feature is generated after subdividing compound curve networks with the improved rules. The shape of features can be adjusted by changing the sharp coefficients related with the curves. The former has advantages of operating easily and realizing conveniently, and the latter can be used to design surfaces freely and produce features of better effect.
(5) Method of constructing subdivision surfaces under constraints of vertices and normal vectors is proposed. The control net of the interpolating surface is obtained by repeatly modifying the positions of vertices according to the optimum result after refining mesh of constrained vertices once. This method avoids solving global equations so it has the advantages of high calculation efficiency and better fairness of interpolating surface. The shape of interpolating surface can be adjusted flexibly, and more freedom is given to designer.
(1)给出了曲线网的定义后,提出了曲线网的编辑和构造方法,并由此对具有曲线网插值功能的联合细分模式进行了扩展,使其具备任意拓扑曲线网的插值能力。
(2)分析了联合细分模式的特性以及产生不光顺现象的原因后,从改进细分规则入手提出了非均匀联合细分模式,与原有联合细分模式相比,由非均匀联合细分模式生成的极限曲面更为光顺,插值曲面至少达到G1连续。
(3)提出了联合细分曲面形状修改的两种方法:曲线网编辑法和基于PDE优化的点约束方法。前者适用于联合细分曲面的初期形状修改,可实现插值曲面的局部和全局修改,后者可以对远离插值曲线的内部曲面进行修改,具有计算速度快、修改曲面平均曲率分布更均匀的优点。两者分别作用于联合细分曲面的不同区域和形状修改的不同阶段,互为补充,可较好地完成联合细分曲面的形状修改。
(4)提出了两种基于插值曲线构造特征的方法:一种根据给定的特征角,通过细分过程中跨界二阶偏导的调整,而在插值曲线上生成尖锐特征;另一种将定义特征的曲线与定义基曲面的曲线相分离,构造出复合曲线网,然后通过改进的联合细分模式进行细分,生成具有特征的细分曲面,特征的形状可通过改变特征曲线对应的尖锐系数来调节。前者操作简单,实现方便,而后者具有更大的设计自由度,并且生成的特征效果较好。
(5)提出了一种顶点和法向约束下的细分曲面构造方法。该方法通过将待插网格剖分一次后不断迭代优化而得到插值曲面的控制网格,无需求解全局线性方程组,具有计算效率高和插值曲面更光顺的优点,并且能够在一定程度上对插值曲面的形状进行灵活调节,给设计者提供了更多的形状表达自由度。
The subdivision surface is gradually becoming a powerful geometric modeling tool because of its advantage of arbitrary topological adaptability. Geometric constraints are important means to improve the precision of complex surface modeling. Points and curves are the most effective ones among geometric constraints. In order to improve the modeling abilities of subdivision surfaces and make it applied in conceptual design of CAD more efficiently, this dissertation mainly studies subdivision modeling technologies based on constraints of vertices and arbitrary topological curve networks. The main research contents and creative achievements are as fellows:
(1) After introducing the definition of curve networks, we present the constructing and editing methods of them. Based on these methods, the combined subdivision scheme is extended to have the ability of interpolating arbitrary topological curve networks.
(2) Based on the analyses of the feature and the unfair reasons of combined subdivision surfaces, we propose the non-uniform combined subdivision scheme by improving subdivision rules. The improved one can produce interpolating limit surface of better fairness. The surface generated by the non-uniform combined subdivision scheme reaches at least G1 continuity.
(3) Two shape modification methods of combined subdivision surfaces are put forward by editing curve networks and applying point constraints based on PDE optimum method. The former is suitable to be used at the initial stage of combined subdivision surfaces’modification, and can be used to modify the local and the global regions of combined subdivision surfaces. The latter can modify the region far away from interpolated curve, and has the advantages of high calculation efficiency and more uniform curvature distribution of modified surfaces. Two methods work on different regions of combined subdivision surfaces and at different stages of shape modification. They supplement for each other to modify combined subdivision surfaces.
(4) Two methods of constructing feature based on interpolated curves are presented. The first one is implemented by adjusting second derivatives on cross curve according to the given feature angle during subdivision and generating sharp feature on interpolated curves. The second one is to separate the curves defining features from the curves defining base surfaces and build compound curve networks, and then combined subdivision surface with sharp feature is generated after subdividing compound curve networks with the improved rules. The shape of features can be adjusted by changing the sharp coefficients related with the curves. The former has advantages of operating easily and realizing conveniently, and the latter can be used to design surfaces freely and produce features of better effect.
(5) Method of constructing subdivision surfaces under constraints of vertices and normal vectors is proposed. The control net of the interpolating surface is obtained by repeatly modifying the positions of vertices according to the optimum result after refining mesh of constrained vertices once. This method avoids solving global equations so it has the advantages of high calculation efficiency and better fairness of interpolating surface. The shape of interpolating surface can be adjusted flexibly, and more freedom is given to designer.
引文
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