基于细分的曲线曲面变形技术研究
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摘要
几何造型是CAD的核心内容,变形技术是几何造型的重要组成部分。细分建模因其具有任意拓扑适应性、样条曲线曲面的连续性等诸多优势,已成为几何造型领域的重要内容,基于细分的变形也成为变形研究的一个重要方向。本文主要研究内容和创新性成果如下:
提出了一种基本函数作用下基于细分的曲线变形方法和一种改进的基于细分的保弧长曲线变形算法。前者结合了基本函数作用下的自由变形以及插值细分的曲线“蒙皮”,解决了多点约束作用于多条相交曲线情况下变形的快速求解问题;后者将弧长增加的曲线简化方式和带调节参数的逼近型细分模式应用于保弧长曲线变形过程,直接从整体弧长不变考虑细分调节参数的选取,改善了原算法在曲线简化程度较高时出现的不光顺现象。
提出了一种基于细分曲面的泊松网格编辑方法,在此基础上进一步提出了一种基于细分曲面控制的网格变形方法。前者以待变形模型包围网格所决定的细分曲面构造变形控制曲面,通过修改包围网格,将对应细分曲面变化信息转化为对模型梯度场的操纵;后者在指定变形区域模型表面设计细分曲面作为变形控制曲面,根据编辑前后的细分控制曲面以及因需设计的参照和目标控制曲线共同对变形区域网格执行梯度场操纵。两种方法均有利于几何细节在编辑过程中的有效保持,同时前者具有以细分曲面张成中间变形空间的FFD方法的变形优势,后者克服了传统参数样条曲面作为变形控制曲面难以贴合任意拓扑物体外形的缺陷。
提出了两种细分曲面的自由变形算法:变形参考曲线(DRC)作用下的细分曲面的自由变形算法和势函数作用下的细分曲面的自由变形算法。前者将DRC作用下的简单几何约束变形应用于细分曲面的形状编辑,根据细分规则求解约束点、线、面以及交互划定的变形区域在各层次细分网格间的传递映射关系,变形区域内每个细分网格顶点由对应DRC得到变形后位置;后者将势函数作用下的网格约束变形应用于细分曲面的形状编辑,根据均匀细分网格更新细分后各顶点基于测地距离的势函数值,并以此作为各顶点的变形权值。两者得到的细分变形网格都满足预期的约束要求,前者变形求解速度更快,后者在变形质量和变形稳定性方面更具优势。
提出了插值曲线约束下的非均匀Doo-Sabin细分曲面的两种变形算法:基于最小二乘法的变形算法和基于离散PDE的变形算法。两种变形算法在基于非均匀Doo-Sabin细分方法构造曲线插值曲面的基础之上,遵循了插值曲线驱动变形的基本思路。前者根据对称网格带建立约束方程,变形求解基于对称网格带的控制顶点扰动量总和最小,适合局部变形,运算速度快;后者建立在前者变形基础之上,作用于细分到一定深度的非均匀Doo-Sabin细分网格,通过建立离散PDE方程求解所有自由顶点的理想平均曲率值并据此调整自由顶点位置,适合整体变形,得到的曲线插值细分曲面更为光滑。
Geometric modeling is the kernel of CAD. Deformation technology is an important component of geometric modeling. Subdivision modeling has become the important content in the domain of geometric modeling because of its good-sized advantages such as the arbitrary topological adaptability, the continuity of spline curves and surfaces, and so on. Deformation based on subdivision also becomes an important direction of deformation research. The main research contents and creative achievements are as follows:
A subdivision-based curve deformation method under basic functions and an improved arc-length preserving curve deformation algorithm based on subdivision are presented. The former combines the free-form deformation under basic functions with the interpolatory subdivision curve“skinning”, and resolves the fast solution to deformation in the case of multiple intersectant curves with multiple constrained points. The latter applies a curve simplification method which can add the arc-length and an approximating subdivision scheme with an adjustable parameter to the arc-length preserving curve deformation process, and selects directly the adjustable parameter according to the constant global arc-length. Compared with the existing arc-length preserving curve deformation algorithm, the improved algorithm can achieve better fairness, especially when a curve is simplified highly.
A subdivision surface-based Poisson mesh editing approach is proposed, and based on it, a mesh deformation method based on subdivision surface control is further proposed. The former constructs the deformation control surface using the subdivision surface determined by a bounding mesh of the deformable model. When the bounding mesh is modified, the change information of the corresponding subdivision surface is transformed into the alteration of the mesh gradient field. The latter designs a subdivision surface attached to a specified mesh deformation region as the deformation control surface. A gradient field modification is performed for the deformation region mesh according to the subdivision control surfaces before and after editing as well as the reference and target controlling curves designed for need. Both can effectively preserve the geometric details of an object. The former has the advantage of the FFD method using the subdivision surface spanning an intermediate deformation space, and the latter overcomes the shortcoming that traditional parametric spline surfaces as the deformation controlling surfaces are difficult to attach those objects with arbitrary topologies.
Two free-form deformation algorithms of subdivision surfaces are put forward. One is named as the free-form deformation algorithm of subdivision surfaces under Deformation Reference Curve (DRC), and the other is named as the free-form deformation algorithm of subdivision surfaces under field functions. The former applies the simple geometric constrained deformation under DRC to the shape editing of subdivision surfaces. According to the subdivision algorithm, the image of the constrained points, constrained curves, constrained surfaces, and the deformation region determined interactively at successive subdivision levels is resolved. For each subdivision mesh vertex within the deformation region, its new position after deformation is obtained according to the corresponding DRC. The latter applies the mesh constrained deformation under field functions to the shape editing of subdivision surfaces. The geodesic-based field value of each vertex after subdivision as the deformation weight of this vertex is updated according to the equal subdivision mesh. The former has the advantage of fast solution to deformation, and the latter outgoes the former in terms of the deformation quality and the deformation stability.
Two deformation algorithms of non-uniform Doo-Sabin subdivision surfaces with interpolated curve constraints are presented. One is based on least-square, and the other is based on discrete PDE. Acting on curve interpolation surfaces constructed based on the non-uniform Doo-Sabin subdivision scheme, both algorithms follow the basic idea of interpolated curve driving deformation. The former establishes constraint equations according to symmetric polygonal complexes, and resolves them to minimize the total disturbance of control vertices of symmetric polygonal complexes. It is suitable for local deformation and has the advantage of fast calculation speed. The latter is based on the former, and fit for the non-uniform Doo-Sabin subdivision mesh at a deep level. For each free vertex, the ideal average curvature value as the criterion of adjusting its position is obtained by establishing and resolving the discrete PDE equation. The latter is suitable for global deformation and the resulting deformation surface has better fairness.
提出了一种基本函数作用下基于细分的曲线变形方法和一种改进的基于细分的保弧长曲线变形算法。前者结合了基本函数作用下的自由变形以及插值细分的曲线“蒙皮”,解决了多点约束作用于多条相交曲线情况下变形的快速求解问题;后者将弧长增加的曲线简化方式和带调节参数的逼近型细分模式应用于保弧长曲线变形过程,直接从整体弧长不变考虑细分调节参数的选取,改善了原算法在曲线简化程度较高时出现的不光顺现象。
提出了一种基于细分曲面的泊松网格编辑方法,在此基础上进一步提出了一种基于细分曲面控制的网格变形方法。前者以待变形模型包围网格所决定的细分曲面构造变形控制曲面,通过修改包围网格,将对应细分曲面变化信息转化为对模型梯度场的操纵;后者在指定变形区域模型表面设计细分曲面作为变形控制曲面,根据编辑前后的细分控制曲面以及因需设计的参照和目标控制曲线共同对变形区域网格执行梯度场操纵。两种方法均有利于几何细节在编辑过程中的有效保持,同时前者具有以细分曲面张成中间变形空间的FFD方法的变形优势,后者克服了传统参数样条曲面作为变形控制曲面难以贴合任意拓扑物体外形的缺陷。
提出了两种细分曲面的自由变形算法:变形参考曲线(DRC)作用下的细分曲面的自由变形算法和势函数作用下的细分曲面的自由变形算法。前者将DRC作用下的简单几何约束变形应用于细分曲面的形状编辑,根据细分规则求解约束点、线、面以及交互划定的变形区域在各层次细分网格间的传递映射关系,变形区域内每个细分网格顶点由对应DRC得到变形后位置;后者将势函数作用下的网格约束变形应用于细分曲面的形状编辑,根据均匀细分网格更新细分后各顶点基于测地距离的势函数值,并以此作为各顶点的变形权值。两者得到的细分变形网格都满足预期的约束要求,前者变形求解速度更快,后者在变形质量和变形稳定性方面更具优势。
提出了插值曲线约束下的非均匀Doo-Sabin细分曲面的两种变形算法:基于最小二乘法的变形算法和基于离散PDE的变形算法。两种变形算法在基于非均匀Doo-Sabin细分方法构造曲线插值曲面的基础之上,遵循了插值曲线驱动变形的基本思路。前者根据对称网格带建立约束方程,变形求解基于对称网格带的控制顶点扰动量总和最小,适合局部变形,运算速度快;后者建立在前者变形基础之上,作用于细分到一定深度的非均匀Doo-Sabin细分网格,通过建立离散PDE方程求解所有自由顶点的理想平均曲率值并据此调整自由顶点位置,适合整体变形,得到的曲线插值细分曲面更为光滑。
Geometric modeling is the kernel of CAD. Deformation technology is an important component of geometric modeling. Subdivision modeling has become the important content in the domain of geometric modeling because of its good-sized advantages such as the arbitrary topological adaptability, the continuity of spline curves and surfaces, and so on. Deformation based on subdivision also becomes an important direction of deformation research. The main research contents and creative achievements are as follows:
A subdivision-based curve deformation method under basic functions and an improved arc-length preserving curve deformation algorithm based on subdivision are presented. The former combines the free-form deformation under basic functions with the interpolatory subdivision curve“skinning”, and resolves the fast solution to deformation in the case of multiple intersectant curves with multiple constrained points. The latter applies a curve simplification method which can add the arc-length and an approximating subdivision scheme with an adjustable parameter to the arc-length preserving curve deformation process, and selects directly the adjustable parameter according to the constant global arc-length. Compared with the existing arc-length preserving curve deformation algorithm, the improved algorithm can achieve better fairness, especially when a curve is simplified highly.
A subdivision surface-based Poisson mesh editing approach is proposed, and based on it, a mesh deformation method based on subdivision surface control is further proposed. The former constructs the deformation control surface using the subdivision surface determined by a bounding mesh of the deformable model. When the bounding mesh is modified, the change information of the corresponding subdivision surface is transformed into the alteration of the mesh gradient field. The latter designs a subdivision surface attached to a specified mesh deformation region as the deformation control surface. A gradient field modification is performed for the deformation region mesh according to the subdivision control surfaces before and after editing as well as the reference and target controlling curves designed for need. Both can effectively preserve the geometric details of an object. The former has the advantage of the FFD method using the subdivision surface spanning an intermediate deformation space, and the latter overcomes the shortcoming that traditional parametric spline surfaces as the deformation controlling surfaces are difficult to attach those objects with arbitrary topologies.
Two free-form deformation algorithms of subdivision surfaces are put forward. One is named as the free-form deformation algorithm of subdivision surfaces under Deformation Reference Curve (DRC), and the other is named as the free-form deformation algorithm of subdivision surfaces under field functions. The former applies the simple geometric constrained deformation under DRC to the shape editing of subdivision surfaces. According to the subdivision algorithm, the image of the constrained points, constrained curves, constrained surfaces, and the deformation region determined interactively at successive subdivision levels is resolved. For each subdivision mesh vertex within the deformation region, its new position after deformation is obtained according to the corresponding DRC. The latter applies the mesh constrained deformation under field functions to the shape editing of subdivision surfaces. The geodesic-based field value of each vertex after subdivision as the deformation weight of this vertex is updated according to the equal subdivision mesh. The former has the advantage of fast solution to deformation, and the latter outgoes the former in terms of the deformation quality and the deformation stability.
Two deformation algorithms of non-uniform Doo-Sabin subdivision surfaces with interpolated curve constraints are presented. One is based on least-square, and the other is based on discrete PDE. Acting on curve interpolation surfaces constructed based on the non-uniform Doo-Sabin subdivision scheme, both algorithms follow the basic idea of interpolated curve driving deformation. The former establishes constraint equations according to symmetric polygonal complexes, and resolves them to minimize the total disturbance of control vertices of symmetric polygonal complexes. It is suitable for local deformation and has the advantage of fast calculation speed. The latter is based on the former, and fit for the non-uniform Doo-Sabin subdivision mesh at a deep level. For each free vertex, the ideal average curvature value as the criterion of adjusting its position is obtained by establishing and resolving the discrete PDE equation. The latter is suitable for global deformation and the resulting deformation surface has better fairness.
引文
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