基于仿射变换的曲线曲面约束变形技术研究
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摘要
几何造型是CAD/CAM、CAGD和CG等领域的重要研究内容，曲线曲面变形技术作为几何造型的重要组成部分，是数字化产品快速开发和创新设计的关键技术之一，灵活简便的变形技术对缩短产品设计周期、提高建模效率具有重要意义。本文从变形的本质出发，结合仿射变换思想，对曲线曲面约束变形技术进行了深入研究，主要研究内容和成果如下：
针对参数曲线约束变形，提出了一种伸缩矩阵作用下的参数曲线约束变形算法。首先利用多项式定义了一种带参数的伸缩因子，它既保持了现有伸缩因子的特性，还具有区间峰值性。然后利用由伸缩因子构成的变形矩阵作用于待变形曲线方程，直接对待变形曲线进行仿射变换，从而得到约束变形和周期变形结果。该算法简单易操作，无需借助其它辅助工具，在控制变形范围、变形边界处的光滑度、变形方向和变形幅度等方面有较好的效果，且对Bézier、B样条、NURBS曲线的作用结果仍可表示成其原有形式。
提出了一种伸缩矩阵作用下的参数曲面约束变形算法，并进一步给出了参数曲面的周期变形算法。利用由多项式伸缩因子构成的变形矩阵直接作用于待变形曲面方程，实现了基于点控制的参数曲面一般约束变形。该算法中的各变形控制参数均具有明确的几何意义，能够直观有效地控制变形区域、峰值区域、伸缩幅度、边界光滑度等。同时将变形区域由圆域推广至矩形域，从而使曲面变形的效果更加丰富。通过引入脊曲线的概念，利用脊曲线对曲面进行控制变形，实现了参数曲面的线约束变形，并在此基础之上结合压缩平移的方法，实现了参数曲面的周期控制变形。实例表明，该算法简单直观且便于操作，无需任何辅助工具，可灵活地对参数曲面进行特征造型设计。
将伸缩矩阵作用下的参数曲面约束变形算法推广至网格曲面，给出了一种参数曲面驱动的网格模型局部约束变形算法。通过将待变形网格曲面参数化到规范参数域上，并对待变形网格曲面进行B样条曲面重建，建立网格曲面与参数曲面的映射，然后运用伸缩矩阵作用下的参数曲面约束变形方法对参数曲面进行约束变形，进而根据映射关系实现网格曲面的约束变形。通过引入不同的伸缩因子，实现了网格模型的点约束、线约束局部变形及周期变形。与传统的网格曲面变形方法相比，本文算法几何意义明确、易于操作，能精确控制变形范围和变形幅度，在变形区域与非变形区域之间实现较为光滑的过渡。
结合仿射变换思想和广义元球变形技术，研究了基于空间变换的网格模型约束变形算法。通过给定变形约束源和变形目标，并利用带势函数的约束变形模型作用于网格的骨架或直接作用于网格顶点，实现了骨架驱动的网格模型约束变形和基于测地距离的网格模型约束变形。前者通过创建与网格模型拓扑关系一致的骨架模型，进而由骨架模型的约束变形来带动网格模型发生变形；后者则是根据变形区域内网格顶点与约束源之间的测地距离，来确定各点处的变形权值，从而直接对网格模型进行约束变形。两者均可实现网格模型的旋转、平移、缩放变形及其复合变形，都能很好地满足预先设定的约束要求，且操作简单直观。前者更适用于具有明显骨架语义的网格模型大变形情形，但变形约束源只能是一点；而后者则对一般网格模型均适用，且变形约束源可以是点、线或面。
Geometric modeling is an important research content in areas of CAD/CAM, CAGD, CG and soon. As an important part of geometric modeling, deformation technique of curves and surfaces is oneof the key techniques in the rapid development and innovative design of digital products. A flexibleand handy deformation technique is extremely significant to shorten the design cycle of products andimprove the efficiency of modeling. Based on the essence of the deformation and the idea of affinetransformation, research on constrained deformation technology of curves and surfaces is deeplystudied in this paper. The major contents and achievements of the study are as follows:
In order to realize the constrained deformation of parametric curves, a new constraineddeformation algorithm based on scaling matrix is proposed for parametric curves. Firstly, constructinga new polynomial scaling factor with parameters, the new factor not only brings the excellence of theexisted scaling factors, but can also get its maximum in the region. And then, multiplying the equationof the curve by the scaling matrix, the constrained deformation and periodic deformation of the curvecan be achieved. The new algorithm is simple and easy to control without any auxiliary tools. It alsocan achieve good results in controlling the region of deformation, the amplitude of deformation, thedirection of deformation, the continuity and smoothness of junction between deformed andundeformed portions. The closure property can be kept in the deformations of Bézier curves, B-splinecurves and NURBS curves.
According to scaling matrix, a new constrained deformation algorithm for parametric surfaces isproposed. On this basis, a periodic deformation algorithm is presented. By applying scaling matrix tothe pending equation of the surface, the algorithm can achieve the point-constraint deformation of thesurface. Every certain control parameter in the algorithm has its own geometric property. The newalgorithm can intuitively and effectively control the region of deformation, the region of peak value,the amplitude of deformation, the smoothness of boundary, etc. In order to obtain more deformationeffects of parametric surfaces, the region of deformation can be extended from circle area to rectanglearea. With the introduction of the ridge curves, the line-constraint deformation of parametric surfacescan be achieved by the ridge curves. Besides, the periodic deformation of parametric surfaces can alsobe achieved. The example proves that the algorithm is simple, intuitive and easy to control withoutany auxiliary tools. The deformation algorithm can flexibly realize the feature modeling design ofparametric surfaces.
Extending from the constrained deformation algorithm for parametric surfaces, a new localconstrained deformation algorithm driven by parametric surfaces is proposed for triangular meshsurfaces. The mapping between the mesh surface and the parametric surface can be created after thetriangular mesh surface is parameterized to a specification parameter domain. And then, a B-splinesurface could be reconstructed from the triangular mesh surface. The constrained deformation of themesh surface can be achieved through the mapping and the constrained deformation method ofparametric surfaces. Based on the above method, the periodic deformation method and localconstrained deformation method with point-constraint and line-constraint are fulfilled by introducingsome different scaling factors. Compared with traditional deformation methods of triangular meshsurfaces, the new algorithm has definitely geometric meanings, and is easy to operate. At the sametime, it can exactly control the region of deformation, the amplitude of deformation and thesmoothness of boundary, etc.
Combining with the affine transformation and the generalized metaball deformation technology,a constrained deformation algorithm for mesh models based on space transformation is carried out. Byspecifying a series of constraint sources and targets, applying the constrained deformable model withpotential function to the skeleton model or vertexes of the mesh, two constrained deformationalgorithms can be achieved. One is based on skeleton-driven, and the other is based on geodesicdistance. The former constructs a skeleton model which has the same topological relation to the meshmodel, and then the deformation of the mesh model can be obtained by what has built. The latterobtains the deformation weights of the vertices in the deformation region, which can be determined bythe geodesic distances between the vertices and the constraint sources, and then the constraineddeformation of the mesh model can be achieved directly. Both of the two algorithms are simple andintuitive, and can commendably meet the need of specified geometric constraints and flexibly realizethe model deformation such as rotation, translation, scaling and other operations. The former is moresuitable for the large deformation of the mesh models with obvious skeleton features, but theconstraint source is just limited to one point. The latter is suitable for all mesh models, and theconstraint sources may consist of points, lines and faces.
针对参数曲线约束变形，提出了一种伸缩矩阵作用下的参数曲线约束变形算法。首先利用多项式定义了一种带参数的伸缩因子，它既保持了现有伸缩因子的特性，还具有区间峰值性。然后利用由伸缩因子构成的变形矩阵作用于待变形曲线方程，直接对待变形曲线进行仿射变换，从而得到约束变形和周期变形结果。该算法简单易操作，无需借助其它辅助工具，在控制变形范围、变形边界处的光滑度、变形方向和变形幅度等方面有较好的效果，且对Bézier、B样条、NURBS曲线的作用结果仍可表示成其原有形式。
提出了一种伸缩矩阵作用下的参数曲面约束变形算法，并进一步给出了参数曲面的周期变形算法。利用由多项式伸缩因子构成的变形矩阵直接作用于待变形曲面方程，实现了基于点控制的参数曲面一般约束变形。该算法中的各变形控制参数均具有明确的几何意义，能够直观有效地控制变形区域、峰值区域、伸缩幅度、边界光滑度等。同时将变形区域由圆域推广至矩形域，从而使曲面变形的效果更加丰富。通过引入脊曲线的概念，利用脊曲线对曲面进行控制变形，实现了参数曲面的线约束变形，并在此基础之上结合压缩平移的方法，实现了参数曲面的周期控制变形。实例表明，该算法简单直观且便于操作，无需任何辅助工具，可灵活地对参数曲面进行特征造型设计。
将伸缩矩阵作用下的参数曲面约束变形算法推广至网格曲面，给出了一种参数曲面驱动的网格模型局部约束变形算法。通过将待变形网格曲面参数化到规范参数域上，并对待变形网格曲面进行B样条曲面重建，建立网格曲面与参数曲面的映射，然后运用伸缩矩阵作用下的参数曲面约束变形方法对参数曲面进行约束变形，进而根据映射关系实现网格曲面的约束变形。通过引入不同的伸缩因子，实现了网格模型的点约束、线约束局部变形及周期变形。与传统的网格曲面变形方法相比，本文算法几何意义明确、易于操作，能精确控制变形范围和变形幅度，在变形区域与非变形区域之间实现较为光滑的过渡。
结合仿射变换思想和广义元球变形技术，研究了基于空间变换的网格模型约束变形算法。通过给定变形约束源和变形目标，并利用带势函数的约束变形模型作用于网格的骨架或直接作用于网格顶点，实现了骨架驱动的网格模型约束变形和基于测地距离的网格模型约束变形。前者通过创建与网格模型拓扑关系一致的骨架模型，进而由骨架模型的约束变形来带动网格模型发生变形；后者则是根据变形区域内网格顶点与约束源之间的测地距离，来确定各点处的变形权值，从而直接对网格模型进行约束变形。两者均可实现网格模型的旋转、平移、缩放变形及其复合变形，都能很好地满足预先设定的约束要求，且操作简单直观。前者更适用于具有明显骨架语义的网格模型大变形情形，但变形约束源只能是一点；而后者则对一般网格模型均适用，且变形约束源可以是点、线或面。
Geometric modeling is an important research content in areas of CAD/CAM, CAGD, CG and soon. As an important part of geometric modeling, deformation technique of curves and surfaces is oneof the key techniques in the rapid development and innovative design of digital products. A flexibleand handy deformation technique is extremely significant to shorten the design cycle of products andimprove the efficiency of modeling. Based on the essence of the deformation and the idea of affinetransformation, research on constrained deformation technology of curves and surfaces is deeplystudied in this paper. The major contents and achievements of the study are as follows:
In order to realize the constrained deformation of parametric curves, a new constraineddeformation algorithm based on scaling matrix is proposed for parametric curves. Firstly, constructinga new polynomial scaling factor with parameters, the new factor not only brings the excellence of theexisted scaling factors, but can also get its maximum in the region. And then, multiplying the equationof the curve by the scaling matrix, the constrained deformation and periodic deformation of the curvecan be achieved. The new algorithm is simple and easy to control without any auxiliary tools. It alsocan achieve good results in controlling the region of deformation, the amplitude of deformation, thedirection of deformation, the continuity and smoothness of junction between deformed andundeformed portions. The closure property can be kept in the deformations of Bézier curves, B-splinecurves and NURBS curves.
According to scaling matrix, a new constrained deformation algorithm for parametric surfaces isproposed. On this basis, a periodic deformation algorithm is presented. By applying scaling matrix tothe pending equation of the surface, the algorithm can achieve the point-constraint deformation of thesurface. Every certain control parameter in the algorithm has its own geometric property. The newalgorithm can intuitively and effectively control the region of deformation, the region of peak value,the amplitude of deformation, the smoothness of boundary, etc. In order to obtain more deformationeffects of parametric surfaces, the region of deformation can be extended from circle area to rectanglearea. With the introduction of the ridge curves, the line-constraint deformation of parametric surfacescan be achieved by the ridge curves. Besides, the periodic deformation of parametric surfaces can alsobe achieved. The example proves that the algorithm is simple, intuitive and easy to control withoutany auxiliary tools. The deformation algorithm can flexibly realize the feature modeling design ofparametric surfaces.
Extending from the constrained deformation algorithm for parametric surfaces, a new localconstrained deformation algorithm driven by parametric surfaces is proposed for triangular meshsurfaces. The mapping between the mesh surface and the parametric surface can be created after thetriangular mesh surface is parameterized to a specification parameter domain. And then, a B-splinesurface could be reconstructed from the triangular mesh surface. The constrained deformation of themesh surface can be achieved through the mapping and the constrained deformation method ofparametric surfaces. Based on the above method, the periodic deformation method and localconstrained deformation method with point-constraint and line-constraint are fulfilled by introducingsome different scaling factors. Compared with traditional deformation methods of triangular meshsurfaces, the new algorithm has definitely geometric meanings, and is easy to operate. At the sametime, it can exactly control the region of deformation, the amplitude of deformation and thesmoothness of boundary, etc.
Combining with the affine transformation and the generalized metaball deformation technology,a constrained deformation algorithm for mesh models based on space transformation is carried out. Byspecifying a series of constraint sources and targets, applying the constrained deformable model withpotential function to the skeleton model or vertexes of the mesh, two constrained deformationalgorithms can be achieved. One is based on skeleton-driven, and the other is based on geodesicdistance. The former constructs a skeleton model which has the same topological relation to the meshmodel, and then the deformation of the mesh model can be obtained by what has built. The latterobtains the deformation weights of the vertices in the deformation region, which can be determined bythe geodesic distances between the vertices and the constraint sources, and then the constraineddeformation of the mesh model can be achieved directly. Both of the two algorithms are simple andintuitive, and can commendably meet the need of specified geometric constraints and flexibly realizethe model deformation such as rotation, translation, scaling and other operations. The former is moresuitable for the large deformation of the mesh models with obvious skeleton features, but theconstraint source is just limited to one point. The latter is suitable for all mesh models, and theconstraint sources may consist of points, lines and faces.
引文
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