三角多项式曲线模型及曲面绘制方法的研究
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摘要
曲线曲面造型是CAD/CAM系统中的关键技术之一。NURBS作为一个统一的数学模型,既可以表示自由曲线,也可以表示传统的几何曲线,因而成为工业产品制造中的一个标准。但NURBS方法的权因子、参数化、曲线曲面连续性问题,至今没有完全得到解决。为了克服NURBS模型中的局限性,近年来,许多学者试图在三角函数空间中寻求新的曲线曲面造型方法。
本文介绍了论文的研究背景和意义,在分析和总结CAGD中曲线曲面造型已有成果的基础上,以一个曲面造型原型系统为主线,重点研究了三角多项式曲线模型和曲面绘制的理论与方法。主要工作及创新点如下:
(1)为了有效利用形状参数来调整曲线的形状,增强修改曲线的灵活性,研究了5种带形状参数的样条曲线的表示方法及性质。通过大量的公式推导和实验,分析了每种造型方法的形状参数对曲线形状的影响,给出了形状参数的适用范围,比较了5种造型方法的特点。提出了利用形状参数不同取值来表示一些自由曲线的新方法,并用实例进行了说明。
(2)为了从理论上探讨T-Bézier和T-B样条曲线模型的完整性,提出了n+1阶T-Bézier和T-B样条基函数的表达式和求解方法。提出了T-Bézier曲线间G~1/C~1拼接的几何条件,解决了多段T-Bézier曲线的拼接问题。提出了C-B样条曲线和T-B样条曲线间G~1/C~1拼接条件,利用T-B样条曲线表示半椭圆弧(半圆弧)的特点,并与C-B样条曲线进行G~1/C~1拼接,解决了C-B样条曲面造型中不能精确表示半椭圆弧(半圆弧)的问题。
(3)为了避免曲线数学模型的复杂度过高,以[1,sint,cost,sin~2t,cos~2t]为基构造了一种带形状参数λ的TC-Bézier曲线,讨论了基函数和曲线的性质。在一定范围内,可以通过调整λ的值来调整曲线的形状,并能精确表示椭圆(圆)等曲线。给出了3阶和4阶TC-Bézier曲线间的G~1/C~1拼接条件及应用的造型实例,所得结论具有明确的几何意义,可方便的应用于曲面造型中。
(4)为了提高曲面模型的精度,利用径向基函数神经网络(RBFNN)具有的非线性逼近能力和抗噪能力,建立了适合曲面重构的径向基函数网络模型,提出了用RBF神经网络模型去噪处理并重构自由曲面的方法,并进行了4阶TC-Bézier曲面的绘制实验。结果表明:该模型不仅能够对带有噪声的曲面进行去噪处理,而且学习速度快,得到的曲面光顺性好。
(5)为了减少三维物体存储和传输的数据量,实现多分辨率三维动态实时显示,提出了一种基于空间八叉剖分的面聚类网格简化方法,即建立空间八叉树,对同一空间内的三角面片进行面聚类。实验结果表明:与原有方法比较,采用新的快速面聚类方法,网格简化的速度有了明显提高。
(6)基于上述的研究工作,设计了曲面造型的系列算法,在Microsoft VisualC++6.0编程环境下,以OpenGL为图形库,开发了空间自由曲面造型的原型系统,构造了不同模型造型的统一平台,用以验证本文提出的相关算法。通过该系统,可以方便地生成旋转曲面和自由曲面,并可以通过添加光照和纹理来增加图形的真实感。
Curve and surface modeling is a key technique in CAD/CAM system. NURBS as a uniform model, which can represent both free curves and traditional geometrical curves, has become a criterion in the field of industrial manufacturing. However, some constraints of NURBS, including its weight, parameterization and continuity remain a problem. Recently researchers have been trying to seek a solution in space of trigonometric function, hoping to overcome the constraints of NURBS model.
After introducing the significance and reviewing previous research on curve and surface modeling in CAGD, this study focuses on the theory and method of trigonometric polynomial curve model and surface rendering. The main content and renovation include:
(1) In order to adjust effectively the shape of a curve by using shape parameters and boost up its flexibility, the study investigates the representations and properties of five types of spline curves with shape parameters. By means of a large amount of mathematical derivation, we analyze the effect of shape parameters on curve shape. Besides, through comparing five modeling approaches, the applicable scale of shape parameters is also derived. Moreover, we also propose a new approach to representing free curves by means of different values of the shape parameters and give an example of it.
(2) The study explores from the theoretical perspective the issue of completeness of T-Bézier and T-B spline curve model and puts forward their presentation and solution of (n+1)~(th)-order T-Bézier and T-B spline basic function. Besides, we propose G~1/C~1 geometrical continuity condition of T-Bézier curve, which offers a solution to the joining of several T-Bézier curves. At the same time, we also present G~1/C~1 continuity condition of C-B spline curve and T-B spline curve. Such G~1/C~1 continuity with C-B spline curves, which makes use of the property that T-B spline curves can represent semi-ellipses and semicircles, successfully makes up the constraint of C-B spline modeling, which cannot represent semi-ellipses and semi-circles accurately.
(3) We construct a TC-Bézier curve with shape parameterλ, with [1,sint,cost, sin~2t,cos~2t] as its base, so as to avoid the complexity of the mathematical model. We also discuss the basic function and property of such curves. Within a certain scope, curves like semi-ellipses and semi-circles can be accurately presented by adjusting the value of λ. Meanwhile, we also give some examples of G~1/C~1 continuity condition between third-order and fouth-order TC-Bézier surface. The result bears explicit geometrical significance and can be conveniently applied to surface modeling.
(4) In order to improve the accuracy of surface modeling and to increase its non-linear approaching capacity as well as the anti-noise capacity derived from radius basic function neural network (RBFNN), we establish an RBFNN model, which fits surface reconstruction. We also propose a method of using RBF neural network to eliminate noise and reform freedom surface. Besides, we conduct a rendering test of forth-order TC-Bézier surface. The test result shows that not only can the model eliminate the noise from surfaces but also learn fast and produce smooth surface.
(5) In order to reduce the amount of storage and transmission and to achieve real time and dynamic display of 3-D objects with multi-resolution, we propose a new method of face cluster mesh simplification, i.e. to establish an Octree, which clusters the triangle mesh in the same space. The experiment demonstrates that the new face clustering method exceeds the old one because it accelerates mesh simplification.
(6) Based on the exploration above, we design a series of algorithms of surface modeling. Under Microsoft Visual C++ 6.0, with OpenGL as graphics library, we also develop a Space Freedom Surface Modeling System, thus create a unified platform of different models in order to test the algorithms proposed in this study. The system can produce rotating surface and free surface in an easy way and make the graph look more real by adding illumination and texture.
本文介绍了论文的研究背景和意义,在分析和总结CAGD中曲线曲面造型已有成果的基础上,以一个曲面造型原型系统为主线,重点研究了三角多项式曲线模型和曲面绘制的理论与方法。主要工作及创新点如下:
(1)为了有效利用形状参数来调整曲线的形状,增强修改曲线的灵活性,研究了5种带形状参数的样条曲线的表示方法及性质。通过大量的公式推导和实验,分析了每种造型方法的形状参数对曲线形状的影响,给出了形状参数的适用范围,比较了5种造型方法的特点。提出了利用形状参数不同取值来表示一些自由曲线的新方法,并用实例进行了说明。
(2)为了从理论上探讨T-Bézier和T-B样条曲线模型的完整性,提出了n+1阶T-Bézier和T-B样条基函数的表达式和求解方法。提出了T-Bézier曲线间G~1/C~1拼接的几何条件,解决了多段T-Bézier曲线的拼接问题。提出了C-B样条曲线和T-B样条曲线间G~1/C~1拼接条件,利用T-B样条曲线表示半椭圆弧(半圆弧)的特点,并与C-B样条曲线进行G~1/C~1拼接,解决了C-B样条曲面造型中不能精确表示半椭圆弧(半圆弧)的问题。
(3)为了避免曲线数学模型的复杂度过高,以[1,sint,cost,sin~2t,cos~2t]为基构造了一种带形状参数λ的TC-Bézier曲线,讨论了基函数和曲线的性质。在一定范围内,可以通过调整λ的值来调整曲线的形状,并能精确表示椭圆(圆)等曲线。给出了3阶和4阶TC-Bézier曲线间的G~1/C~1拼接条件及应用的造型实例,所得结论具有明确的几何意义,可方便的应用于曲面造型中。
(4)为了提高曲面模型的精度,利用径向基函数神经网络(RBFNN)具有的非线性逼近能力和抗噪能力,建立了适合曲面重构的径向基函数网络模型,提出了用RBF神经网络模型去噪处理并重构自由曲面的方法,并进行了4阶TC-Bézier曲面的绘制实验。结果表明:该模型不仅能够对带有噪声的曲面进行去噪处理,而且学习速度快,得到的曲面光顺性好。
(5)为了减少三维物体存储和传输的数据量,实现多分辨率三维动态实时显示,提出了一种基于空间八叉剖分的面聚类网格简化方法,即建立空间八叉树,对同一空间内的三角面片进行面聚类。实验结果表明:与原有方法比较,采用新的快速面聚类方法,网格简化的速度有了明显提高。
(6)基于上述的研究工作,设计了曲面造型的系列算法,在Microsoft VisualC++6.0编程环境下,以OpenGL为图形库,开发了空间自由曲面造型的原型系统,构造了不同模型造型的统一平台,用以验证本文提出的相关算法。通过该系统,可以方便地生成旋转曲面和自由曲面,并可以通过添加光照和纹理来增加图形的真实感。
Curve and surface modeling is a key technique in CAD/CAM system. NURBS as a uniform model, which can represent both free curves and traditional geometrical curves, has become a criterion in the field of industrial manufacturing. However, some constraints of NURBS, including its weight, parameterization and continuity remain a problem. Recently researchers have been trying to seek a solution in space of trigonometric function, hoping to overcome the constraints of NURBS model.
After introducing the significance and reviewing previous research on curve and surface modeling in CAGD, this study focuses on the theory and method of trigonometric polynomial curve model and surface rendering. The main content and renovation include:
(1) In order to adjust effectively the shape of a curve by using shape parameters and boost up its flexibility, the study investigates the representations and properties of five types of spline curves with shape parameters. By means of a large amount of mathematical derivation, we analyze the effect of shape parameters on curve shape. Besides, through comparing five modeling approaches, the applicable scale of shape parameters is also derived. Moreover, we also propose a new approach to representing free curves by means of different values of the shape parameters and give an example of it.
(2) The study explores from the theoretical perspective the issue of completeness of T-Bézier and T-B spline curve model and puts forward their presentation and solution of (n+1)~(th)-order T-Bézier and T-B spline basic function. Besides, we propose G~1/C~1 geometrical continuity condition of T-Bézier curve, which offers a solution to the joining of several T-Bézier curves. At the same time, we also present G~1/C~1 continuity condition of C-B spline curve and T-B spline curve. Such G~1/C~1 continuity with C-B spline curves, which makes use of the property that T-B spline curves can represent semi-ellipses and semicircles, successfully makes up the constraint of C-B spline modeling, which cannot represent semi-ellipses and semi-circles accurately.
(3) We construct a TC-Bézier curve with shape parameterλ, with [1,sint,cost, sin~2t,cos~2t] as its base, so as to avoid the complexity of the mathematical model. We also discuss the basic function and property of such curves. Within a certain scope, curves like semi-ellipses and semi-circles can be accurately presented by adjusting the value of λ. Meanwhile, we also give some examples of G~1/C~1 continuity condition between third-order and fouth-order TC-Bézier surface. The result bears explicit geometrical significance and can be conveniently applied to surface modeling.
(4) In order to improve the accuracy of surface modeling and to increase its non-linear approaching capacity as well as the anti-noise capacity derived from radius basic function neural network (RBFNN), we establish an RBFNN model, which fits surface reconstruction. We also propose a method of using RBF neural network to eliminate noise and reform freedom surface. Besides, we conduct a rendering test of forth-order TC-Bézier surface. The test result shows that not only can the model eliminate the noise from surfaces but also learn fast and produce smooth surface.
(5) In order to reduce the amount of storage and transmission and to achieve real time and dynamic display of 3-D objects with multi-resolution, we propose a new method of face cluster mesh simplification, i.e. to establish an Octree, which clusters the triangle mesh in the same space. The experiment demonstrates that the new face clustering method exceeds the old one because it accelerates mesh simplification.
(6) Based on the exploration above, we design a series of algorithms of surface modeling. Under Microsoft Visual C++ 6.0, with OpenGL as graphics library, we also develop a Space Freedom Surface Modeling System, thus create a unified platform of different models in order to test the algorithms proposed in this study. The system can produce rotating surface and free surface in an easy way and make the graph look more real by adding illumination and texture.
引文
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