准均匀B样条函数在地形模型中的应用
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摘要
20世纪50年代后期,DEM开始被采用并受到了极大的关注,它在测绘、土木工程、地质、矿山工程、景观建筑、道路设计、防洪、农业、规划、军事工程、飞行器与战场仿真等领域得到了广泛的应用。
DEM的理论和技术范畴涉及到数据采集的技术和方法、数据结构的设计和处理、模型构建与内插算法的研究及算法的实现、DEM的三维可视化技术、DEM的精度评价及其质量控制以及计算机技术发展水平等。长期以来,人们一直对DEM表示地形的精度、数据采集、粗差探测、质量控制、数据压缩、可视化、DEM的应用以及不规则三角网TIN的建立与内插等技术方面进行着不断的研究。
随着计算技术的迅速发展和DEM的广泛应用,寻找快速建模、内插精度高、可视化程度高的建模方法是非常必要的。
DEM可以使用一个或多个数学函数对地形进行表示,这样的数学函数通常被称为内插函数。当DEM表面建模后,模型上任意一点的高程信息就可以从DEM表面上获得,这与曲面几何造型非常相似。因此,DEM表面建模可视为地形表面的曲面造型。正是由于以上原因,我们可以将自由曲面的构造方法应用于DEM表面建模之中,同时还可以将自由曲线构造法用于构建地形模型的断面线。
本文主要对地形表面模型的建模及其断面图生成的方法进行了研究,主要研究内容如下:
1、结合国内外对地形表面建模的研究现状,阐述了地形表面建模研究的意义。
2、介绍了现有的基于规则格网(GRID)和基于不规则三角网(TriangulatedIrregular Network—TIN)两种数据格式的地形建模方法,总结了三维地形表面建模的表现方法及特点。
3、由于外业采集的数据常为离散点数据,需先构建不规则三角网再经过内插得到规则格网数据。因此,本文深入研究了不规则三角网的构网方法,对TIN的构网方法进行了改进,并用VB编程实现,以便提高其效率。
4、对几种常用的DEM内插方法进行了研究,并从内插精度、适用范围、计算量等方面进行了比较分析。
5、对自由曲线曲面的构造方法和三维数据场可视化理论进行了研究,总结了B样条基函数的优点和性质。
6、将自由曲线构造法中的准均匀B样条方法用于地形模型断面图的生成以及对隧道断面测量的粗糙度分析与评价。
7、基于准均匀B样条构造曲线曲面的方法,在MATLAB的编程环境下对一组规则格网数据进行了三维地形表面建模,并实现其可视化。
From 20 century 50 age later, Digital Elevation Model (DEM) have been adopting and getting infinite attention. It is broadly used in many fields, such as Surveying, Civil Engineering, Geology, Mine engineering, Sight Architecture, Road Design, Prevent or Control Flood, Agriculture, Programming, Military Affairs Engineering, Aircraft and Battlefield Emulate and so on.
The theory and technology of the DEM include the technique and means of gathering data, designing data structure, making model, research and putting into practice arithmetic of interpolation, 3D visualization, estimating accuracy, quality control about the DEM, and interrelated computer technology, etc. So far, the above-mentioned theory and technology about the DEM are researched continually.
With the rapid development of computer and extensive application of DEM, it is very necessary to find out a modeling method with faster modeling, higher precision and better visualization quality.
DEM uses one or more mathematical functions to portray the terrain. This function is usually called interpolation function. If DEM has been built, elevation information of everypoint on the model can be easily acquired. It is similar to geometric modeling very much, so that the DEM surface models can be regarded as surface modeling of terrain. Therefore, we can use the construction methods of free-surface to DEM modeling. At the same time, we also use the construction methods of free-curve in cross-section.
The contents researched in this thesis are as follows:
1. Combine with the study of terrain surface modeling status in the world, explain the significance of the terrain surface modeling's study.
2. Introduce the means of terrain modeling base on two data format—GRID and TIN, summarize the methods and characteristics of terrain surface modeling with three dimensions.
3. Because of the data which collected from field works is discrete, it need convert into GRID. So this paper studys the network construction methods of Triangulated Irregular Network earnestly, improves the arithmetic of TIN and realizes the new method based on the VB platform.
4. Compare and analyze the interpolation arithmetic which has been put into practice from interpolation accuracy, scope of application, computational complexity and so on.
5. Learn construction methods of free-curve (surface) and theoretics of three-dimensional datafield visualization, summarize the merits and properties of B-spline basic function.
6. Put the quasi-uniform B-spline curve's construction method into sectional drawing and analysis and evaluation of tunnel cross-section's roughness. Quasi-uniform B-spline is one kind of the free-curve construction methods.
7. Using the MATLAB software to carry out the quasi-uniform B-spline, realize the three-dimensional terrain surface modeling and visualization.
DEM的理论和技术范畴涉及到数据采集的技术和方法、数据结构的设计和处理、模型构建与内插算法的研究及算法的实现、DEM的三维可视化技术、DEM的精度评价及其质量控制以及计算机技术发展水平等。长期以来,人们一直对DEM表示地形的精度、数据采集、粗差探测、质量控制、数据压缩、可视化、DEM的应用以及不规则三角网TIN的建立与内插等技术方面进行着不断的研究。
随着计算技术的迅速发展和DEM的广泛应用,寻找快速建模、内插精度高、可视化程度高的建模方法是非常必要的。
DEM可以使用一个或多个数学函数对地形进行表示,这样的数学函数通常被称为内插函数。当DEM表面建模后,模型上任意一点的高程信息就可以从DEM表面上获得,这与曲面几何造型非常相似。因此,DEM表面建模可视为地形表面的曲面造型。正是由于以上原因,我们可以将自由曲面的构造方法应用于DEM表面建模之中,同时还可以将自由曲线构造法用于构建地形模型的断面线。
本文主要对地形表面模型的建模及其断面图生成的方法进行了研究,主要研究内容如下:
1、结合国内外对地形表面建模的研究现状,阐述了地形表面建模研究的意义。
2、介绍了现有的基于规则格网(GRID)和基于不规则三角网(TriangulatedIrregular Network—TIN)两种数据格式的地形建模方法,总结了三维地形表面建模的表现方法及特点。
3、由于外业采集的数据常为离散点数据,需先构建不规则三角网再经过内插得到规则格网数据。因此,本文深入研究了不规则三角网的构网方法,对TIN的构网方法进行了改进,并用VB编程实现,以便提高其效率。
4、对几种常用的DEM内插方法进行了研究,并从内插精度、适用范围、计算量等方面进行了比较分析。
5、对自由曲线曲面的构造方法和三维数据场可视化理论进行了研究,总结了B样条基函数的优点和性质。
6、将自由曲线构造法中的准均匀B样条方法用于地形模型断面图的生成以及对隧道断面测量的粗糙度分析与评价。
7、基于准均匀B样条构造曲线曲面的方法,在MATLAB的编程环境下对一组规则格网数据进行了三维地形表面建模,并实现其可视化。
From 20 century 50 age later, Digital Elevation Model (DEM) have been adopting and getting infinite attention. It is broadly used in many fields, such as Surveying, Civil Engineering, Geology, Mine engineering, Sight Architecture, Road Design, Prevent or Control Flood, Agriculture, Programming, Military Affairs Engineering, Aircraft and Battlefield Emulate and so on.
The theory and technology of the DEM include the technique and means of gathering data, designing data structure, making model, research and putting into practice arithmetic of interpolation, 3D visualization, estimating accuracy, quality control about the DEM, and interrelated computer technology, etc. So far, the above-mentioned theory and technology about the DEM are researched continually.
With the rapid development of computer and extensive application of DEM, it is very necessary to find out a modeling method with faster modeling, higher precision and better visualization quality.
DEM uses one or more mathematical functions to portray the terrain. This function is usually called interpolation function. If DEM has been built, elevation information of everypoint on the model can be easily acquired. It is similar to geometric modeling very much, so that the DEM surface models can be regarded as surface modeling of terrain. Therefore, we can use the construction methods of free-surface to DEM modeling. At the same time, we also use the construction methods of free-curve in cross-section.
The contents researched in this thesis are as follows:
1. Combine with the study of terrain surface modeling status in the world, explain the significance of the terrain surface modeling's study.
2. Introduce the means of terrain modeling base on two data format—GRID and TIN, summarize the methods and characteristics of terrain surface modeling with three dimensions.
3. Because of the data which collected from field works is discrete, it need convert into GRID. So this paper studys the network construction methods of Triangulated Irregular Network earnestly, improves the arithmetic of TIN and realizes the new method based on the VB platform.
4. Compare and analyze the interpolation arithmetic which has been put into practice from interpolation accuracy, scope of application, computational complexity and so on.
5. Learn construction methods of free-curve (surface) and theoretics of three-dimensional datafield visualization, summarize the merits and properties of B-spline basic function.
6. Put the quasi-uniform B-spline curve's construction method into sectional drawing and analysis and evaluation of tunnel cross-section's roughness. Quasi-uniform B-spline is one kind of the free-curve construction methods.
7. Using the MATLAB software to carry out the quasi-uniform B-spline, realize the three-dimensional terrain surface modeling and visualization.
引文
[1]刘锦中,马辉.基于等高线的DEM生成算法研究和实现[J].现代测绘,2004,(03)
[2]李志林,朱庆.数字高程模型[M].武汉:武汉测绘科技大学出版社,2000
[3]冯桂,林宗坚等.DEM高精度内插算法及其实现[J].遥感信息,2000,(04)
[4]唐新建,朱同林等.DEM地形表面B样条小波多尺度建模[J].中山大学学报(自然科学版),2005,(03)
[5]朱心雄等.自由曲线曲面造型技术[M].北京:科学出版社,2000
[6]周光文,黄筱蓉.一种改进的多面函数法[J].测绘通报,1996,(06)
[7]李浩,张友静.曲面移动拟合法在DEM中的插值技术[J].现代测绘,1996,(04)
[8]沈涛,李成名,苏山舞.基于水系改进的数字高程模型内插研究[J].中国图象图形学报,2006,(04)
[9]罗冀华.建立DEM快速算法分析[J].测绘科学,2003,(04)
[10]蔡学成,岳胜.数字地形模型的建立方法及其在林业中的应用[J].林业调查规划,2004,(1)
[11]杨重阳.数字地面模型及其在路线设计中的应用研究.硕士,武汉理工大学,2007
[12]韩敏,孟宇.基于ArcObjects的网络三维地形场景生成[J].计算机应用,2006,(1)
[13]V.Caselles,J.M.Morel,and C.Sbert:An axiomatic approach to image interpolation.IEEE Trans.Image Processing,Vol.7,pp376——386,Mar.1998
[14]Andrés Almansa,Frédéric Cao,Yann Gousseau,and Bernard Rougé:Interpolation of Digital Elevation Models Using AMLE and Related Methods.IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING,2002,(2)
[15]J.Meinguet.Surface spline interpolation:Basic theory and computational Aspects.Approximation Theory and Spline Functions,S.P.Singh et al,Eds.Dordrecht,Holland:D.Reidel,1984,pp.124-142
[16]R.Franke.Scattered data interpolation:Test of some methods.Math.Comput.1982,(157)pp.181-200
[17]L.Mitas and H.Mitasova:Spatial Interpolation.New York:Wiley,1999,(1)pp.481-492
[18]杨德麟.数字地面模型[J].测绘通报,1998,(03)
[19]吴焕萍,潘懋等.基于空间索引的规则格网DTM内插算法研究[J].地理与地理信息科学,2004,(01)
[20]沈涛,李成名,苏山舞.基于水系改进的数字高程模型内插研究[J].中国图象图形学报,2006,(04)
[21]郭清.DEM内插方法比较及ARC/INFO内插方法分析[J].铁路航测,1999,(01).
[22]赵建三.基于格网DEM的自适应等高线内插方法[J].中南工业大学学报(自然科学版)2003,(3)
[23]黎细波,郑建.内插法在建立数字高程模型中的应用[J].大坝与安全,2004,(S1)
[24]涂震飚,刘永才等.一种面向大场景3维显示的地形建模方法[J].测绘学报,2004,(01)
[25]张会平,杨农等.数字高程模型(DEM)在构造地貌研究中的应用新进展[J].地质通报,2006,(06)
[26]刘少华,程朋根,史文中.约束Detaunay三角网生成算法研究[J].测绘通报,2004,(03)
[27]王建雄.CAD环境下基于不规则三角网的DEM算法及实现[J].云南农业大学学报,2005,(04)
[28]谭仁春,杜清运等.地形建模中不规则三角网构建的优化算法研究[J].武汉大学学报(信息科学版),2006,(05)
[29]梁海勇,潘燕芳,刘少华,程朋根.DEM的建立及其在线路勘测设计中的应用[J].东华理工学院学报(自然科学版),2005,(01)
[30]施法中.计算机辅助几何设计与非均匀有理B样条[M].北京:高等教育出版社,2001
[31]朱心雄等.自由曲线曲面造型技术[M].北京:科学出版社,2000
[32]朱心雄,吴瑞祥.B样条曲线曲面的原理和应用.北京航空学院讲义,1980
[33]徐叔贤.多项式样条函数.北京航空学院,1979
[34]Zhu Xinxiong.The Numerical Evaluation of B-Spline Curves and Surfaces,Lecture Notes,University of Minnesota,1981
[35]朱心雄,雷毅,吴瑞祥.复杂曲线曲面造型方法—B样条方法的原理和应用.北京航空学院,1988
[36]周宏,陈吉森,吴继敏.基于粗糙度的连拱隧道开挖断面线分析[J].地下空间与工程学报,2006,(2)
[37]秦之富.多功能激光断面检测仪在隧道拱墙衬砌前横断面检测中的应用[J].公路隧道,2006,(3)
[38]冯洁,王国辉等.计算机辅助测量在铁路隧道断面测量中的应用[J].铁道建筑,2005,(02)
[39]肖清华,张继春.隧道断面自动成图的数学模型研究[J].铁道建筑,2006,(9)
[40]王沫然.MATLAB 6.0与科学计算[M].北京:电子工业出版社,2001
[41]尚涛.工程计算可视化与MATLAB实现[M].武汉:武汉大学出版社,2002
[42]阮沈勇,王永利,桑群芳.MATLAB程序设计[M].北京:电子工业出版社,2004
[43]苏金明,张莲花,刘波等编.MATLAB工具箱应用[M].电子工业出版社,2004
[44]唐泽圣等著.三维数据场可视化[M].清华大学出版社,2000
[45]苏金明,阮沈勇.MATLAB 6.1实用指南[M].北京:电子工业出版社,2002
[46]刘卫国.MATLAB程序设计教程[M].北京:中国水利水电出版社,2005
[47]刘卫国.MATLAB程序设计与应用[M].北京:高等教育出版社,2006
[48]李清泉,杨必胜等著.三维空间数据的实时获取,建模与可视化[M].武汉:武汉大学出版社,2003
[49]徐青著.地形三维可视化技术[M].测绘出版社,北京国家科学技术学术著作出版基金资助出版2000
[50]Halstead Mark A.,Barsky Brian A.,Klein Stanley A.etal.Reconstructing Curved Surfaces From Specular Reflection Patterns Using Spline Surface Fitting of Normals Special.In:Computer Graphics,Annual Conference Series(ACM SIGGRAPH).New York:ACM Press,1996,163-170
[51]曹毓秀,李勇军等.荫罩式CRT校正透镜曲面设计与实现[J].计算机辅助设计与图形学学报,2000,(04)
[52]Lee Yuan-Chung,Jen Chein-Wei,Improved Quadratic Normal Vector Interpolation for Realistic Shading,Visual Computer,2001.6.17,337-352
[53]Jorg Peters,Local cubic and bicubic C~1 surface interpolation with linearly varying boundary normal.Computer Aided Geometric Design,(1990.6.7)499-516
[54]Farin Gerald,Smooth Interpolation to Scattered 3D-Data,In:R E Barnhill,W Boehm eds.Surfaces in CAGD,Am sterdam:North-Holland,1983,43-63
[55]Jones A K.,Nonrectangular surface patches with curvature continuity,Computer Aided Design,(1988.6.20)325-335
[56]Biermann Henning,Levin Adi,Zorin Denis,Piecewise smooth subdivision surfaces with normal control,In:Computer Graphics,Annual Conference Series(ACM SIGGRAPH),New York:ACM Press,2000,113-120
[57]Halstead Mark,Kass Michael,DeRose Tony,Efficient,fair interpolation using CatmulI-Clark surfaces,In: Computer Graphics,Annual Conference Series(ACM SIGGRAPH),New York:ACM Press,1993,35-44
[58]Nasri A.,Surface interpolation on irregular networks with normal conditions,Computer Aided Geometric Design,(1991.1.8)89-96
[59]李桂清,李现民,李华.二次B样条曲面顶点和法向插值[J].计算机辅助设计与图形学学报,2001
[60]李桂清,李华.基于混合子分方法的曲面网格顶点与法向插值[J].计算机辅助设计与图形学学报,2001,(6)
[61]朱长青,张真等.基于非均匀B样条曲面的DTM内插模型[J].华中科技大学学报(自然科学版)2006,(4)
[62]刘浩,廖文和.非均匀B样条曲面顶点及法向插值[J].数值计算与计算机应用,2006,(04).
[63]孙永辉,姜昱明.准均匀B样条曲面的多分辨率表示及应用[J].计算机工程与设计,2003,(9)
[64]荣辉,吴国庆,王格芳.逆向工程中测量数据曲面重构技术研究[J].微计算机信息,2006,(34)
[65]唐建新,任伟中等.多尺度数字高程模型[J].岩土力学,2006,(12)
[66]曹利新,游洪等.面向截面测量数据B样条曲面光顺拟合及其精度计算[J].大连理工大学学报,2003,(3)
[67]张丽艳,周来水,蔡炜斌等.基于截面测量数据的B样条曲面重建[J].应用科学学报,2002,(2)
[68]方逵.双三次B样条插值曲面[J].数学理论与应用,2001,(3)
[2]李志林,朱庆.数字高程模型[M].武汉:武汉测绘科技大学出版社,2000
[3]冯桂,林宗坚等.DEM高精度内插算法及其实现[J].遥感信息,2000,(04)
[4]唐新建,朱同林等.DEM地形表面B样条小波多尺度建模[J].中山大学学报(自然科学版),2005,(03)
[5]朱心雄等.自由曲线曲面造型技术[M].北京:科学出版社,2000
[6]周光文,黄筱蓉.一种改进的多面函数法[J].测绘通报,1996,(06)
[7]李浩,张友静.曲面移动拟合法在DEM中的插值技术[J].现代测绘,1996,(04)
[8]沈涛,李成名,苏山舞.基于水系改进的数字高程模型内插研究[J].中国图象图形学报,2006,(04)
[9]罗冀华.建立DEM快速算法分析[J].测绘科学,2003,(04)
[10]蔡学成,岳胜.数字地形模型的建立方法及其在林业中的应用[J].林业调查规划,2004,(1)
[11]杨重阳.数字地面模型及其在路线设计中的应用研究.硕士,武汉理工大学,2007
[12]韩敏,孟宇.基于ArcObjects的网络三维地形场景生成[J].计算机应用,2006,(1)
[13]V.Caselles,J.M.Morel,and C.Sbert:An axiomatic approach to image interpolation.IEEE Trans.Image Processing,Vol.7,pp376——386,Mar.1998
[14]Andrés Almansa,Frédéric Cao,Yann Gousseau,and Bernard Rougé:Interpolation of Digital Elevation Models Using AMLE and Related Methods.IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING,2002,(2)
[15]J.Meinguet.Surface spline interpolation:Basic theory and computational Aspects.Approximation Theory and Spline Functions,S.P.Singh et al,Eds.Dordrecht,Holland:D.Reidel,1984,pp.124-142
[16]R.Franke.Scattered data interpolation:Test of some methods.Math.Comput.1982,(157)pp.181-200
[17]L.Mitas and H.Mitasova:Spatial Interpolation.New York:Wiley,1999,(1)pp.481-492
[18]杨德麟.数字地面模型[J].测绘通报,1998,(03)
[19]吴焕萍,潘懋等.基于空间索引的规则格网DTM内插算法研究[J].地理与地理信息科学,2004,(01)
[20]沈涛,李成名,苏山舞.基于水系改进的数字高程模型内插研究[J].中国图象图形学报,2006,(04)
[21]郭清.DEM内插方法比较及ARC/INFO内插方法分析[J].铁路航测,1999,(01).
[22]赵建三.基于格网DEM的自适应等高线内插方法[J].中南工业大学学报(自然科学版)2003,(3)
[23]黎细波,郑建.内插法在建立数字高程模型中的应用[J].大坝与安全,2004,(S1)
[24]涂震飚,刘永才等.一种面向大场景3维显示的地形建模方法[J].测绘学报,2004,(01)
[25]张会平,杨农等.数字高程模型(DEM)在构造地貌研究中的应用新进展[J].地质通报,2006,(06)
[26]刘少华,程朋根,史文中.约束Detaunay三角网生成算法研究[J].测绘通报,2004,(03)
[27]王建雄.CAD环境下基于不规则三角网的DEM算法及实现[J].云南农业大学学报,2005,(04)
[28]谭仁春,杜清运等.地形建模中不规则三角网构建的优化算法研究[J].武汉大学学报(信息科学版),2006,(05)
[29]梁海勇,潘燕芳,刘少华,程朋根.DEM的建立及其在线路勘测设计中的应用[J].东华理工学院学报(自然科学版),2005,(01)
[30]施法中.计算机辅助几何设计与非均匀有理B样条[M].北京:高等教育出版社,2001
[31]朱心雄等.自由曲线曲面造型技术[M].北京:科学出版社,2000
[32]朱心雄,吴瑞祥.B样条曲线曲面的原理和应用.北京航空学院讲义,1980
[33]徐叔贤.多项式样条函数.北京航空学院,1979
[34]Zhu Xinxiong.The Numerical Evaluation of B-Spline Curves and Surfaces,Lecture Notes,University of Minnesota,1981
[35]朱心雄,雷毅,吴瑞祥.复杂曲线曲面造型方法—B样条方法的原理和应用.北京航空学院,1988
[36]周宏,陈吉森,吴继敏.基于粗糙度的连拱隧道开挖断面线分析[J].地下空间与工程学报,2006,(2)
[37]秦之富.多功能激光断面检测仪在隧道拱墙衬砌前横断面检测中的应用[J].公路隧道,2006,(3)
[38]冯洁,王国辉等.计算机辅助测量在铁路隧道断面测量中的应用[J].铁道建筑,2005,(02)
[39]肖清华,张继春.隧道断面自动成图的数学模型研究[J].铁道建筑,2006,(9)
[40]王沫然.MATLAB 6.0与科学计算[M].北京:电子工业出版社,2001
[41]尚涛.工程计算可视化与MATLAB实现[M].武汉:武汉大学出版社,2002
[42]阮沈勇,王永利,桑群芳.MATLAB程序设计[M].北京:电子工业出版社,2004
[43]苏金明,张莲花,刘波等编.MATLAB工具箱应用[M].电子工业出版社,2004
[44]唐泽圣等著.三维数据场可视化[M].清华大学出版社,2000
[45]苏金明,阮沈勇.MATLAB 6.1实用指南[M].北京:电子工业出版社,2002
[46]刘卫国.MATLAB程序设计教程[M].北京:中国水利水电出版社,2005
[47]刘卫国.MATLAB程序设计与应用[M].北京:高等教育出版社,2006
[48]李清泉,杨必胜等著.三维空间数据的实时获取,建模与可视化[M].武汉:武汉大学出版社,2003
[49]徐青著.地形三维可视化技术[M].测绘出版社,北京国家科学技术学术著作出版基金资助出版2000
[50]Halstead Mark A.,Barsky Brian A.,Klein Stanley A.etal.Reconstructing Curved Surfaces From Specular Reflection Patterns Using Spline Surface Fitting of Normals Special.In:Computer Graphics,Annual Conference Series(ACM SIGGRAPH).New York:ACM Press,1996,163-170
[51]曹毓秀,李勇军等.荫罩式CRT校正透镜曲面设计与实现[J].计算机辅助设计与图形学学报,2000,(04)
[52]Lee Yuan-Chung,Jen Chein-Wei,Improved Quadratic Normal Vector Interpolation for Realistic Shading,Visual Computer,2001.6.17,337-352
[53]Jorg Peters,Local cubic and bicubic C~1 surface interpolation with linearly varying boundary normal.Computer Aided Geometric Design,(1990.6.7)499-516
[54]Farin Gerald,Smooth Interpolation to Scattered 3D-Data,In:R E Barnhill,W Boehm eds.Surfaces in CAGD,Am sterdam:North-Holland,1983,43-63
[55]Jones A K.,Nonrectangular surface patches with curvature continuity,Computer Aided Design,(1988.6.20)325-335
[56]Biermann Henning,Levin Adi,Zorin Denis,Piecewise smooth subdivision surfaces with normal control,In:Computer Graphics,Annual Conference Series(ACM SIGGRAPH),New York:ACM Press,2000,113-120
[57]Halstead Mark,Kass Michael,DeRose Tony,Efficient,fair interpolation using CatmulI-Clark surfaces,In: Computer Graphics,Annual Conference Series(ACM SIGGRAPH),New York:ACM Press,1993,35-44
[58]Nasri A.,Surface interpolation on irregular networks with normal conditions,Computer Aided Geometric Design,(1991.1.8)89-96
[59]李桂清,李现民,李华.二次B样条曲面顶点和法向插值[J].计算机辅助设计与图形学学报,2001
[60]李桂清,李华.基于混合子分方法的曲面网格顶点与法向插值[J].计算机辅助设计与图形学学报,2001,(6)
[61]朱长青,张真等.基于非均匀B样条曲面的DTM内插模型[J].华中科技大学学报(自然科学版)2006,(4)
[62]刘浩,廖文和.非均匀B样条曲面顶点及法向插值[J].数值计算与计算机应用,2006,(04).
[63]孙永辉,姜昱明.准均匀B样条曲面的多分辨率表示及应用[J].计算机工程与设计,2003,(9)
[64]荣辉,吴国庆,王格芳.逆向工程中测量数据曲面重构技术研究[J].微计算机信息,2006,(34)
[65]唐建新,任伟中等.多尺度数字高程模型[J].岩土力学,2006,(12)
[66]曹利新,游洪等.面向截面测量数据B样条曲面光顺拟合及其精度计算[J].大连理工大学学报,2003,(3)
[67]张丽艳,周来水,蔡炜斌等.基于截面测量数据的B样条曲面重建[J].应用科学学报,2002,(2)
[68]方逵.双三次B样条插值曲面[J].数学理论与应用,2001,(3)
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