保形映射理论在几何造型中的某些应用研究
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摘要
几何造型是CAD/CAM和CG技术的核心,主要研究如何利用计算机描述、分析、调整和输出物体模型的几何外形信息。经过近半个世纪的发展,传统的几何造型技术已经比较成熟。但是,随着几何设计对象向着多样性、复杂性的靠拢以及图形工业和制造工业迈向一体化、集成化和网络化步伐的日益加快,几何造型技术面临着许多新的问题和挑战。保形映射理论是复分析的重要内容之一,它在许多科学技术领域中有着广泛的应用。保形映射具有保角性和伸缩率不变性。换句话说,几何对象经过保形变换后,它的整体外形发生改变,同时又局部地保持一定的刚体性质。鉴于此,保形变换能够满足几何造型中的一些保形性要求。本文对保形映射理论在几何造型中的某些应用问题进行了研究,提出了基于M(o|¨)bius变换的圆弧样条曲线构造方法、基于保形映射的变形造型方法以及利用保形映射进行三角网格曲面参数化的新方法。本文的主要工作及贡献如下:
1.本文首先提出了一种基于M(o|¨)bius变换的圆弧样条曲线构造方法。利用M(o|¨)bius变换的保交比性质和保圆性质,给出了一种快速、简便地生成平面上的一段精确圆弧曲线的新方法。该方法将平面上的一段圆弧曲线表示为参数复有理函数的形式。该表示形式中没有权因子和额外的控制参数,而且具有几何不变性、仿射不变性等良好性质。本文进一步构造了具有一定光滑性质的圆弧样条曲线。此外,本文定义了一种类似于deCasteljau算法的圆弧样条曲线的高阶推广算法。该算法形式上与de Casteljau算法类似,但是几何意义却完全不同:这里以圆弧代替de Casteljau算法中的直线段。
2.变形造型方法是近些年来几何造型领域中的一个热点研究领域,利用变形方法可以简化三维几何形体的构造和编辑。本文提出了一种基于保形映射的变形方法。该方法直接对原始几何模型进行操作,通过改变模型的基空间和高度场来实现变形的目的,是一种简便、有效的变形造型方法。同时,该方法能够给出原始模型与变形后的模型之间的变换关系式,因而可以利用变换的雅可比矩阵方便地分析变形后模型的一些拓扑性质和几何性质。此外,由于该方法很好地利用了保形映射的保形性质,所以它在使三维模型得到可观整体形变的同时,又局部地保持一定的刚体性质。文中给出了大量的数值实验,实验表明文中的方法是有效的。
3.三角网格的参数化是指从空间三角网格到参数区域的一个同构映射,参数区域的选择可以是平面区域、球面等。三角网格的参数化是对三角网格的几何和拓扑信息作进一步处理的基础,它在计算机图形学、计算机辅助几何设计和数字几何处理等领域有着广泛的应用。本文首先基于保形映射和微分几何相关理论,定义了一种统一衡量参数化的角度变形和面积变形的变形能量。接下来,结合区域增长算法的思想,通过迭代求解网格中每一层三角形的变形能量最小化问题得到网格的自由边界的参数化。最后,本文利用2D区域之间的保形变换将自由边界的参数化变换为规则边界的参数化,即得到最终的参数化结果。将文中方法得到的参数化结果以及一些经典的参数化方法得到的结果进行比较,无论是视觉上的直观效果(纹理映射)还是数据上的定量分析,都表明了本文方法具有参数化变形较小的特点。
Geometric modeling is the kernel of CAD/CAM and CG,whose major components are the description,analysis,adjustment and output of object models using computer. In the past fifty years,traditional geometric modeling techniques have been developed deeply.However,with the trends of variety and complexity of geometrical objects,and with the acceleration of the pace of the graphics industry and manufacture industry toward integration and network,geometric modeling faces many new problems and challenges. Conformal mapping theory is an important component of complex analysis,and it has been widely used in many areas of science and technology.Conformal mapping preserves angles and expands or contracts distances equally in all directions.In other words,conformal mapping has the ability to preserve local rigidity while allowing considerable global deformation.Hence,conformal mapping satisfies the demands on shape preserving in geometric modeling.Research on application of conformal mapping in geometric modeling is investigated in this paper.The main work and contributions are shown as follows:
1.A new method for constructing circular arc spline based on M(o|¨)bius transformation is firstly presented in the paper.Since cross ratio is an invariant of the group of all M(o|¨)bius transformations,and a M(o|¨)bius transformation maps a circle into another circle, it is convenient to represent a segment of circular arc with a parametric complex rational function.The representation has no weight factors or control parameters,and it is geometric and affine invariant.Furthermore,some circular arc spline curves which have C~0 or GC~1 continuity are constructed.In the end,a recursion algorithm which is similar to the de Casteljau algorithm is presented.The algorithm has a different geometric meaning with the de Casteljau algorithm.Precisely,it uses segments of circular arc in stead of sections of line.
2.One of the most important operations in computer graphics and computer-aided design is object deformation.This operation allows easy creation of plain shaped objects from regular shapes like spheres,and allows the deformation of exiting objects.In this paper,a solid model deformation method based on layered conformal mapping is presented. For a solid model represented by base patch and height field,the shape of the model can be deformed by interactive means,such as changing the base patch by conformal mappings and adjusting the height field by a pre-defined function.In our method, the deformation is predictable and the transformation function of the deformation can be expressed analytically by Schwarz-Christoffel formula.To perform a deformation of a cylinder to a desired solid of rotation hierarchically,a generalized Schwarz-Christoffel formula is also introduced.Numerical examples show that the proposed method is convenient and efficient to deform solid models,especially for solids with genus zero.
3.Parameterizing a 3D mesh means finding an isomorphic mapping between a discrete surface patch and a planar mesh.Parametrization is a crucial problem in computer graphics,computer aided geometric design and digital geometric processing.Based on conformal mapping and differential geometry theory,a triangular mesh parametrization method by optimizing deformation energy is proposed in the paper.The method is an iterative procedure that incrementally flattens 3D triangular mesh by growing region to obtain a parametrization result with free boundary.Application of the method to texture mapping is presented.Experiments show that the proposed method can obtain better results than some classical parametrization methods.
1.本文首先提出了一种基于M(o|¨)bius变换的圆弧样条曲线构造方法。利用M(o|¨)bius变换的保交比性质和保圆性质,给出了一种快速、简便地生成平面上的一段精确圆弧曲线的新方法。该方法将平面上的一段圆弧曲线表示为参数复有理函数的形式。该表示形式中没有权因子和额外的控制参数,而且具有几何不变性、仿射不变性等良好性质。本文进一步构造了具有一定光滑性质的圆弧样条曲线。此外,本文定义了一种类似于deCasteljau算法的圆弧样条曲线的高阶推广算法。该算法形式上与de Casteljau算法类似,但是几何意义却完全不同:这里以圆弧代替de Casteljau算法中的直线段。
2.变形造型方法是近些年来几何造型领域中的一个热点研究领域,利用变形方法可以简化三维几何形体的构造和编辑。本文提出了一种基于保形映射的变形方法。该方法直接对原始几何模型进行操作,通过改变模型的基空间和高度场来实现变形的目的,是一种简便、有效的变形造型方法。同时,该方法能够给出原始模型与变形后的模型之间的变换关系式,因而可以利用变换的雅可比矩阵方便地分析变形后模型的一些拓扑性质和几何性质。此外,由于该方法很好地利用了保形映射的保形性质,所以它在使三维模型得到可观整体形变的同时,又局部地保持一定的刚体性质。文中给出了大量的数值实验,实验表明文中的方法是有效的。
3.三角网格的参数化是指从空间三角网格到参数区域的一个同构映射,参数区域的选择可以是平面区域、球面等。三角网格的参数化是对三角网格的几何和拓扑信息作进一步处理的基础,它在计算机图形学、计算机辅助几何设计和数字几何处理等领域有着广泛的应用。本文首先基于保形映射和微分几何相关理论,定义了一种统一衡量参数化的角度变形和面积变形的变形能量。接下来,结合区域增长算法的思想,通过迭代求解网格中每一层三角形的变形能量最小化问题得到网格的自由边界的参数化。最后,本文利用2D区域之间的保形变换将自由边界的参数化变换为规则边界的参数化,即得到最终的参数化结果。将文中方法得到的参数化结果以及一些经典的参数化方法得到的结果进行比较,无论是视觉上的直观效果(纹理映射)还是数据上的定量分析,都表明了本文方法具有参数化变形较小的特点。
Geometric modeling is the kernel of CAD/CAM and CG,whose major components are the description,analysis,adjustment and output of object models using computer. In the past fifty years,traditional geometric modeling techniques have been developed deeply.However,with the trends of variety and complexity of geometrical objects,and with the acceleration of the pace of the graphics industry and manufacture industry toward integration and network,geometric modeling faces many new problems and challenges. Conformal mapping theory is an important component of complex analysis,and it has been widely used in many areas of science and technology.Conformal mapping preserves angles and expands or contracts distances equally in all directions.In other words,conformal mapping has the ability to preserve local rigidity while allowing considerable global deformation.Hence,conformal mapping satisfies the demands on shape preserving in geometric modeling.Research on application of conformal mapping in geometric modeling is investigated in this paper.The main work and contributions are shown as follows:
1.A new method for constructing circular arc spline based on M(o|¨)bius transformation is firstly presented in the paper.Since cross ratio is an invariant of the group of all M(o|¨)bius transformations,and a M(o|¨)bius transformation maps a circle into another circle, it is convenient to represent a segment of circular arc with a parametric complex rational function.The representation has no weight factors or control parameters,and it is geometric and affine invariant.Furthermore,some circular arc spline curves which have C~0 or GC~1 continuity are constructed.In the end,a recursion algorithm which is similar to the de Casteljau algorithm is presented.The algorithm has a different geometric meaning with the de Casteljau algorithm.Precisely,it uses segments of circular arc in stead of sections of line.
2.One of the most important operations in computer graphics and computer-aided design is object deformation.This operation allows easy creation of plain shaped objects from regular shapes like spheres,and allows the deformation of exiting objects.In this paper,a solid model deformation method based on layered conformal mapping is presented. For a solid model represented by base patch and height field,the shape of the model can be deformed by interactive means,such as changing the base patch by conformal mappings and adjusting the height field by a pre-defined function.In our method, the deformation is predictable and the transformation function of the deformation can be expressed analytically by Schwarz-Christoffel formula.To perform a deformation of a cylinder to a desired solid of rotation hierarchically,a generalized Schwarz-Christoffel formula is also introduced.Numerical examples show that the proposed method is convenient and efficient to deform solid models,especially for solids with genus zero.
3.Parameterizing a 3D mesh means finding an isomorphic mapping between a discrete surface patch and a planar mesh.Parametrization is a crucial problem in computer graphics,computer aided geometric design and digital geometric processing.Based on conformal mapping and differential geometry theory,a triangular mesh parametrization method by optimizing deformation energy is proposed in the paper.The method is an iterative procedure that incrementally flattens 3D triangular mesh by growing region to obtain a parametrization result with free boundary.Application of the method to texture mapping is presented.Experiments show that the proposed method can obtain better results than some classical parametrization methods.
引文
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