四次有理Bézier曲线曲面造型的研究
|
摘要
计算机辅助几何设计CAGD(Computer Aided Geometric Design)主要研究以复杂方式自由变化的曲线曲面,即所谓的自由型曲线曲面。Bézier曲线和曲面广泛应用于计算机图形学,并且在形状设计方面有很好的性质。
近年来Bézier曲线、曲面在工程设计应用中,有着极其重要的作用。如飞机、船体等外形曲面设计,首先是通过设计出经过一定型值点的曲线作为线型,再在这些线型上插值出相应的外形曲面。因此Bézier曲线曲面在实践中表现出强大的生命力,受到学术界的广泛重视。
本文首先从有理Bézier曲线的定义出发,阐述它的性质,通过权因子而不是控制顶点来修改有理四次Bézier曲线的形状,实现了相邻曲线段的G 2的连续拼接;进而实现了相邻三段曲线的G 2的连续拼接。
同时讨论了有理四次Bézier曲线参数化,应用重心坐标推导出有理四次Bézier曲线的表达式,通过给定5个控制顶点和位于这些顶点凸包内四次有理曲线上一点,反算出了该点的参数和内权因子,研究了在曲线形状不变的情况下,空间有理四次Bézier曲线权因子变换和参数变换的等效性。
通过改变有理四次贝齐尔曲线的控制顶点或权因子是实现形状修改的途径之一。实际应用中人们更希望能实现有预定目标的修改,譬如说使修改过的曲线经过某个点,本文也给出了既可以修改控制顶点,也可以修改权因子来实现这个目标的方法。
采用在公共边界处曲线连续和切平面光滑连续的性质,研究了有理Bézier曲面的拼接问题,给出了具有公共边界曲线的两张双四次有理Bézier曲面G1光滑拼接条件,从而得到更多的可调形状参数。
Computer aided geometric design(CAGD) mainly researches curves and surfaces which vary in the free and complex forms, that is to say, curves and surfaces of freedom forms. Bézier curves and surfaces are widely applied in computer graphics and have many good properties in shape design of industrial products.
In recent years, Bézier curves and surfaces have extremely important function in the applications of engineering design, such as air planes, ships and so on. Firstly a curve shape is obtained by designing a curve that crosses some data points. Then the corresponding shape surface is formed by interpolation on these curve shapes. So Bézier curves and surfaces show their strong vitality in practice and receive wide attentions.
Firstly, the definition of rational quartic Bézier curve is introduced and its properties are described. The shape of rational quartic Bézier curve is modified by changing the weights corresponding to control points instead of control points themselves. The G2 continuity for the joining between two adjacent rational quartic Bézier curves is realized. Based on this the G2continuity for the joining among three adjacent quartic rational Bézier curves are achieved further.
Secondly, the parameterization of rational quartic Bézier curve is discussed. The expression formula of rational quartic Bézier curve is deduced by barycentric coordinates. The parameter and inner weight factors are calculated inversely by given five control points and a point on the rational quartic Bézier curve within the convex hull formed by all the control points of the curve. The equivalent property between the weight transformation for spatial rational quartic Bézier curve and parametric transformation of the curve under the condition of keeping the shape unchanged is investigated.
Altering the control points or weights of rational quartic Bézier curve is one of important approaches in achieving the shape modification. In practical applications, the modification with predestinate target is more hoped to be realized, for example, to make the modified curve pass a given point. The method to make the modification is given both by modifying the control points and by modifying weight factors.
According to the theory of Bézier surface, by taking the property that the curve and its tangent plane is continuous on the common boundary, the joining between adjacent rational Bezier surface is studied and the G1 continuity conditions of two adjacent rational Bézier surfaces of double four degrees are given, therefore more adjustable shape parameters can be obtained.
近年来Bézier曲线、曲面在工程设计应用中,有着极其重要的作用。如飞机、船体等外形曲面设计,首先是通过设计出经过一定型值点的曲线作为线型,再在这些线型上插值出相应的外形曲面。因此Bézier曲线曲面在实践中表现出强大的生命力,受到学术界的广泛重视。
本文首先从有理Bézier曲线的定义出发,阐述它的性质,通过权因子而不是控制顶点来修改有理四次Bézier曲线的形状,实现了相邻曲线段的G 2的连续拼接;进而实现了相邻三段曲线的G 2的连续拼接。
同时讨论了有理四次Bézier曲线参数化,应用重心坐标推导出有理四次Bézier曲线的表达式,通过给定5个控制顶点和位于这些顶点凸包内四次有理曲线上一点,反算出了该点的参数和内权因子,研究了在曲线形状不变的情况下,空间有理四次Bézier曲线权因子变换和参数变换的等效性。
通过改变有理四次贝齐尔曲线的控制顶点或权因子是实现形状修改的途径之一。实际应用中人们更希望能实现有预定目标的修改,譬如说使修改过的曲线经过某个点,本文也给出了既可以修改控制顶点,也可以修改权因子来实现这个目标的方法。
采用在公共边界处曲线连续和切平面光滑连续的性质,研究了有理Bézier曲面的拼接问题,给出了具有公共边界曲线的两张双四次有理Bézier曲面G1光滑拼接条件,从而得到更多的可调形状参数。
Computer aided geometric design(CAGD) mainly researches curves and surfaces which vary in the free and complex forms, that is to say, curves and surfaces of freedom forms. Bézier curves and surfaces are widely applied in computer graphics and have many good properties in shape design of industrial products.
In recent years, Bézier curves and surfaces have extremely important function in the applications of engineering design, such as air planes, ships and so on. Firstly a curve shape is obtained by designing a curve that crosses some data points. Then the corresponding shape surface is formed by interpolation on these curve shapes. So Bézier curves and surfaces show their strong vitality in practice and receive wide attentions.
Firstly, the definition of rational quartic Bézier curve is introduced and its properties are described. The shape of rational quartic Bézier curve is modified by changing the weights corresponding to control points instead of control points themselves. The G2 continuity for the joining between two adjacent rational quartic Bézier curves is realized. Based on this the G2continuity for the joining among three adjacent quartic rational Bézier curves are achieved further.
Secondly, the parameterization of rational quartic Bézier curve is discussed. The expression formula of rational quartic Bézier curve is deduced by barycentric coordinates. The parameter and inner weight factors are calculated inversely by given five control points and a point on the rational quartic Bézier curve within the convex hull formed by all the control points of the curve. The equivalent property between the weight transformation for spatial rational quartic Bézier curve and parametric transformation of the curve under the condition of keeping the shape unchanged is investigated.
Altering the control points or weights of rational quartic Bézier curve is one of important approaches in achieving the shape modification. In practical applications, the modification with predestinate target is more hoped to be realized, for example, to make the modified curve pass a given point. The method to make the modification is given both by modifying the control points and by modifying weight factors.
According to the theory of Bézier surface, by taking the property that the curve and its tangent plane is continuous on the common boundary, the joining between adjacent rational Bezier surface is studied and the G1 continuity conditions of two adjacent rational Bézier surfaces of double four degrees are given, therefore more adjustable shape parameters can be obtained.
引文
[1] PIEGL L. Modifying of the Shape of Rational B spline. Part 1: Curves[J]. Computer Aided Design, 1989, 21(8): 509-518.
[2] FARIN T. From Conics to NURBS[J]. A tutorial Graphics and Applications, 1992, 1(5): 78-86.
[3] PIEGL L. A Geometric Investigation of the Rational Bézier Scheme of Computer Aided Design[J]. Computer in industry, 1986, 7(6): 44-47.
[4] KANG B S, ZHAO L G. Automatic Fairing Algorithm for Planar Cubic NURBS Curves [J]. Journal of Computer Aided Design & Computer Graphics, 2002, 14 (3): 225 - 227.
[5] WALTON D J, MEEK D S. Planar G 2 Curve Design with Spiral Segments[J]. Computer Aided Design, 1998, 30(7): 525-538.
[6] Flower B R. Constrained Based Curve Manipulation. IEEE Computer Graphics and Application, 1993, 13(5): 43-49.
[7] JIWEN ZHANG. C-curves: An Extension of Cubic Curves [J]. Computer Aided Geometric Design, 1996, 13(3): 199-217.
[8] B A. The Beta-Saline: A Local Representation Based on Shape Parameters and Fundamental Geometric Measures[J]. University of Utah Salt Lake City, Utah, ph, D Thesis, 1981, 24(6): 442-447.
[9] BOEHM W. Inserting New Knots into B Spine Curve[J]. Computer Aided Design, 1980, 12(4): 199-201.
[10] JU I. Weight Based Modification of NURBS Curves[J]. Computer Aided Geometric Design, 1999, 16(5): 377-383.
[11] SANCHEZ R J. A Simple Technique for NURBS Shape Modification[J]. IEEE Computer Graphics and Application, 1997, 17(1): 52-59.
[12] AU C K, YUEN M F. Unified Approach to NURBS Curve Manipulation [J]. Computer Aided Design, 1995, 27(2): 85-93.
[13] JAVTER S R. A Simple Technique for NURBS Shape Modification [J]. IEEE
[14] FORREST A R. The Twisted Cubic Curve: A Computer Aided Geometric Design Approach [J]. CAD, 1980, 12(4): 165-172.
[15]沈海鸥,陈淑珍,孙晓安.曲线的二次有理Bézier曲线拟合[J].武汉大学学报:自然科学版, 1997, 43(1): 119-123.
[16]王哲,王知行. NURBS曲线、曲面造型过程的可视化设计[J].机械设计, 1999, 5(5): 13-16.
[17]朱心雄.自由形曲线曲面造型技术[M].北京:科学出版社, 2000: 61-71.
[18]贾红丽,汤正诠.双三次Bézier曲面片的光滑拼接[J].应用数学与计算数学学报, 2001, 15(2): 91-96.
[19]陈锦辉.基于有理二次Bézier曲线段的G 2连续闭曲线插值[J]. 2001, 28(1): 227-230.
[20]陈宝平,尹志凌.基于有理二次Bézier曲线的G 2连续的插值曲线[J].内蒙古大学学报:自然科学版, 2004, 35(4): 464-467.
[21]严兰兰,宋来忠.双三次有理Bézier曲面G 1光滑拼接算法[J].宁波职业技术学院学报, 2005, 9(5): 69-71.
[22]王继贵,王仁宏. Bézier曲面片的光滑拼接[J].辽宁师范大学学报:自然科学版, 2005, 28(1): 16-18.
[23]严兰兰,宋来忠,李军成.双二次有理Bézier曲面G 1光滑拼接算法[J].三峡大学学报:自然科学版, 2007, 27(6): 568-570.
[24]卢殿军. Bézier曲线的拼接及其连续性[J].青海大学学报:自然科学版, 2004, 22(6): 84-86.
[25]施法中.计算机辅助几何设计与非均匀有理B样条[M].北京:北京航空航天大学出版社, 2001: 230-236.
[26]樊建华,张纪文. C-Bézier曲线的形状修改[J].软件学报, 2002, 13(11): 2194-2199.
[27]张艳君,闵淑辉,陈飞.三角Bézier曲面光滑拼接研究[J].企收技术开发, 2005, 24(7): 40-42.
[28]曹毓秀,孙延奎.三角域Bézier曲面若干算法研究[J].清华大学学报:自然科学版, 2001, 41(7): 83-86.
[29]程永福,朱功勤.三次有理Bézier曲线G 1光滑拼接条件[J].合肥工业大学学报:自然科学版, 2007, 30(3): 397-399.
[30]朱承学,李崧.四次有理Bézier曲线的形状调整[J].工程图学报, 2005, 12(2): 73-76.
[31]耿紫星.三次Bézier曲线与三次均匀B样条曲线的光滑拼接[J].甘肃联合大学学报:自然科学版, 2007, 21(3): 58-61.
[32]郑建民.三角域上有理Bézier曲面的曲率连续拼接[J].浙江大学学报:自然科学版, 1993, 21(5): 621-633.
[33]严兰兰,宋来忠,李军成.有理Bézier曲线的拼接[J].三峡大学学报:自然科学版, 2005, 27(5): 469-471.
[34]冯仁忠,王仁宏.三次B样条曲线间G 2连续条件[J].大连理工大学学报, 2003, 43(4): 407-411.
[35]姜献峰,梁友栋.有理Bézier曲线几何连续条件及应用[J].高等数学计算数学学报, 1992, 25(4): 340-353.
[36]杨莉,晁翠华,贾晓. G 2连续的三次有理Bézier样条插值曲线[J].机械科学与技术, 2000, 19(3): 512-513.
[37]张宏鑫,王国瑾.保持几何连续性的曲线形状调配[J].高校应用数学学报A辑, 2001, 16(2): 187-194.
[38]李睿,张贵仓.基于矩形域的Bézier曲面及其拼接技术[J].兰州理工大学学报, 2005, 31(2): 96-98.
[39]车翔玖.三次均匀有理B样条曲线G 2光滑拼接条件[J].辽宁师范大学学报:自然科学版, 2002, 25(3): 225-228.
[40]车翔玖,刘大有,王钲旋.两张NURBS曲面间G 1光滑过渡曲面的构造[J].吉林大学学报:工学版, 2007, 37(4): 839-841.
[41]车翔玖,梁学章.两邻接B样条曲面的G 1连续条件[J].应用数学MATHEMATICA APPLICATA, 2004, 17(3): 410-416.
[42]陈锦辉,张三元. G~3连续的有理三次Bézier样条曲线造型[J].自然科学进展, 2001, 11(7): 747-753.
[43]张三元.基于代数曲线段的G 2连续的曲线造型方法[J].计算机学报, 2000, 2(23): 153-157.
[44]李军,廖群云.非均匀有理B样条曲面G 2光滑拼接算法[J].湖北民族学院学报:自然科学版, 2000, 18(1): 65-68.
[45]程少华,吴华.两相邻3次B样条曲面G 1连续充分条件[J].河南师范大学学报:自然科学版, 2006, 34(3): 48-50.
[46]赵岩,施锡泉.双五次B-样条曲面的G 2连续条件[J].数值计算与计算机应用, 2004, 22(1): 20-35.
[47]张三元.一种G 2连续的二次曲线样条插值方法[J].计算机辅助设计与图形学学报, 2000, 12(6): 420-422.
[48]张同琦.三角域上邻接有理Bézier曲面片的G 1和G 2光滑[J].工程数学学报, 2000, 17(1): 22-26.
[49]苏本跃,余宏杰.一类G 2连续的C2 Bézier保凸插值曲线[J].安徽技术师范学院学报, 2003 ,17 (2): 165-169.
[50]王烈,叶正麟,王树勋. G 1和G 2连续的双参数型B样条曲线[J].西北工业大学学报, 2006, 35(6): 1-3.
[51]梁锡坤. B样条类曲线曲面理论及其应用研究[D].安徽省:合肥工业大学, 2003: 15-16.
[52]夏飞海. Bézier曲线和曲面形状修改奇异点问题的研究[D].浙江省:浙江大学, 2006: 1-4.
[2] FARIN T. From Conics to NURBS[J]. A tutorial Graphics and Applications, 1992, 1(5): 78-86.
[3] PIEGL L. A Geometric Investigation of the Rational Bézier Scheme of Computer Aided Design[J]. Computer in industry, 1986, 7(6): 44-47.
[4] KANG B S, ZHAO L G. Automatic Fairing Algorithm for Planar Cubic NURBS Curves [J]. Journal of Computer Aided Design & Computer Graphics, 2002, 14 (3): 225 - 227.
[5] WALTON D J, MEEK D S. Planar G 2 Curve Design with Spiral Segments[J]. Computer Aided Design, 1998, 30(7): 525-538.
[6] Flower B R. Constrained Based Curve Manipulation. IEEE Computer Graphics and Application, 1993, 13(5): 43-49.
[7] JIWEN ZHANG. C-curves: An Extension of Cubic Curves [J]. Computer Aided Geometric Design, 1996, 13(3): 199-217.
[8] B A. The Beta-Saline: A Local Representation Based on Shape Parameters and Fundamental Geometric Measures[J]. University of Utah Salt Lake City, Utah, ph, D Thesis, 1981, 24(6): 442-447.
[9] BOEHM W. Inserting New Knots into B Spine Curve[J]. Computer Aided Design, 1980, 12(4): 199-201.
[10] JU I. Weight Based Modification of NURBS Curves[J]. Computer Aided Geometric Design, 1999, 16(5): 377-383.
[11] SANCHEZ R J. A Simple Technique for NURBS Shape Modification[J]. IEEE Computer Graphics and Application, 1997, 17(1): 52-59.
[12] AU C K, YUEN M F. Unified Approach to NURBS Curve Manipulation [J]. Computer Aided Design, 1995, 27(2): 85-93.
[13] JAVTER S R. A Simple Technique for NURBS Shape Modification [J]. IEEE
[14] FORREST A R. The Twisted Cubic Curve: A Computer Aided Geometric Design Approach [J]. CAD, 1980, 12(4): 165-172.
[15]沈海鸥,陈淑珍,孙晓安.曲线的二次有理Bézier曲线拟合[J].武汉大学学报:自然科学版, 1997, 43(1): 119-123.
[16]王哲,王知行. NURBS曲线、曲面造型过程的可视化设计[J].机械设计, 1999, 5(5): 13-16.
[17]朱心雄.自由形曲线曲面造型技术[M].北京:科学出版社, 2000: 61-71.
[18]贾红丽,汤正诠.双三次Bézier曲面片的光滑拼接[J].应用数学与计算数学学报, 2001, 15(2): 91-96.
[19]陈锦辉.基于有理二次Bézier曲线段的G 2连续闭曲线插值[J]. 2001, 28(1): 227-230.
[20]陈宝平,尹志凌.基于有理二次Bézier曲线的G 2连续的插值曲线[J].内蒙古大学学报:自然科学版, 2004, 35(4): 464-467.
[21]严兰兰,宋来忠.双三次有理Bézier曲面G 1光滑拼接算法[J].宁波职业技术学院学报, 2005, 9(5): 69-71.
[22]王继贵,王仁宏. Bézier曲面片的光滑拼接[J].辽宁师范大学学报:自然科学版, 2005, 28(1): 16-18.
[23]严兰兰,宋来忠,李军成.双二次有理Bézier曲面G 1光滑拼接算法[J].三峡大学学报:自然科学版, 2007, 27(6): 568-570.
[24]卢殿军. Bézier曲线的拼接及其连续性[J].青海大学学报:自然科学版, 2004, 22(6): 84-86.
[25]施法中.计算机辅助几何设计与非均匀有理B样条[M].北京:北京航空航天大学出版社, 2001: 230-236.
[26]樊建华,张纪文. C-Bézier曲线的形状修改[J].软件学报, 2002, 13(11): 2194-2199.
[27]张艳君,闵淑辉,陈飞.三角Bézier曲面光滑拼接研究[J].企收技术开发, 2005, 24(7): 40-42.
[28]曹毓秀,孙延奎.三角域Bézier曲面若干算法研究[J].清华大学学报:自然科学版, 2001, 41(7): 83-86.
[29]程永福,朱功勤.三次有理Bézier曲线G 1光滑拼接条件[J].合肥工业大学学报:自然科学版, 2007, 30(3): 397-399.
[30]朱承学,李崧.四次有理Bézier曲线的形状调整[J].工程图学报, 2005, 12(2): 73-76.
[31]耿紫星.三次Bézier曲线与三次均匀B样条曲线的光滑拼接[J].甘肃联合大学学报:自然科学版, 2007, 21(3): 58-61.
[32]郑建民.三角域上有理Bézier曲面的曲率连续拼接[J].浙江大学学报:自然科学版, 1993, 21(5): 621-633.
[33]严兰兰,宋来忠,李军成.有理Bézier曲线的拼接[J].三峡大学学报:自然科学版, 2005, 27(5): 469-471.
[34]冯仁忠,王仁宏.三次B样条曲线间G 2连续条件[J].大连理工大学学报, 2003, 43(4): 407-411.
[35]姜献峰,梁友栋.有理Bézier曲线几何连续条件及应用[J].高等数学计算数学学报, 1992, 25(4): 340-353.
[36]杨莉,晁翠华,贾晓. G 2连续的三次有理Bézier样条插值曲线[J].机械科学与技术, 2000, 19(3): 512-513.
[37]张宏鑫,王国瑾.保持几何连续性的曲线形状调配[J].高校应用数学学报A辑, 2001, 16(2): 187-194.
[38]李睿,张贵仓.基于矩形域的Bézier曲面及其拼接技术[J].兰州理工大学学报, 2005, 31(2): 96-98.
[39]车翔玖.三次均匀有理B样条曲线G 2光滑拼接条件[J].辽宁师范大学学报:自然科学版, 2002, 25(3): 225-228.
[40]车翔玖,刘大有,王钲旋.两张NURBS曲面间G 1光滑过渡曲面的构造[J].吉林大学学报:工学版, 2007, 37(4): 839-841.
[41]车翔玖,梁学章.两邻接B样条曲面的G 1连续条件[J].应用数学MATHEMATICA APPLICATA, 2004, 17(3): 410-416.
[42]陈锦辉,张三元. G~3连续的有理三次Bézier样条曲线造型[J].自然科学进展, 2001, 11(7): 747-753.
[43]张三元.基于代数曲线段的G 2连续的曲线造型方法[J].计算机学报, 2000, 2(23): 153-157.
[44]李军,廖群云.非均匀有理B样条曲面G 2光滑拼接算法[J].湖北民族学院学报:自然科学版, 2000, 18(1): 65-68.
[45]程少华,吴华.两相邻3次B样条曲面G 1连续充分条件[J].河南师范大学学报:自然科学版, 2006, 34(3): 48-50.
[46]赵岩,施锡泉.双五次B-样条曲面的G 2连续条件[J].数值计算与计算机应用, 2004, 22(1): 20-35.
[47]张三元.一种G 2连续的二次曲线样条插值方法[J].计算机辅助设计与图形学学报, 2000, 12(6): 420-422.
[48]张同琦.三角域上邻接有理Bézier曲面片的G 1和G 2光滑[J].工程数学学报, 2000, 17(1): 22-26.
[49]苏本跃,余宏杰.一类G 2连续的C2 Bézier保凸插值曲线[J].安徽技术师范学院学报, 2003 ,17 (2): 165-169.
[50]王烈,叶正麟,王树勋. G 1和G 2连续的双参数型B样条曲线[J].西北工业大学学报, 2006, 35(6): 1-3.
[51]梁锡坤. B样条类曲线曲面理论及其应用研究[D].安徽省:合肥工业大学, 2003: 15-16.
[52]夏飞海. Bézier曲线和曲面形状修改奇异点问题的研究[D].浙江省:浙江大学, 2006: 1-4.