带多调节参数曲线的研究
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摘要
在样条函数的逼近领域及应用领域中,在原来设计样条函数的基础上做必要的修正以求达到更好的逼近效果是非常重要的的课题,在这方面已经有很多的方法,而运用调节参数就是一个主要的方法.本文在对[15, 16, 17, 18, 19, 20, 21, 22, 23]的工作的充分分析的基础上,重新给出了两类带多调节参数的参数曲线的构造方法,包括:
1.用变换矩阵给出带多调节参数的Be′zier曲线扩展,并具体分析了其性质,根据几何性质将其进行了分类.
本文在第二章给出了带两个调节参数的二阶Be′zier曲线的扩展:
在第三章给出了带n个调节参数的n阶Be′zier曲线的扩展:
2.运用前面的思路给出带多个调节参数的均匀B样条曲线的构造方法,并体分析了其性质.
在第四章首先给出了带多个调节参数的三阶B样条曲线扩展,其表示为:继带多个调节参数的三阶B样条曲线后,本文在第四章构造了带多个调节参数的n阶B样条曲线,其表示为:
在本文中给出的这两类曲线引入了多个调节参数,这使得参数成为局部参数,它既能整体调控曲线的形状,又能局部调控.而且有别于其它参数曲线,它是在基函数不变即仍然使用Be′zier和B样条基的前提下,通过通过使用参数调节控制点来调节曲线形状.由于函数基是不变的,因此减弱了对参数的范围限制,使参数的取值有任意性,增加增加了带参数的曲线的自由度,因此能很好的调节曲线.
In the field of B-spline function approximation theory and application,in orderto arriving at better effect ,modulating basis functions is an important way,and therehave been many methods abort that,using shape modification parameters is one ofthese.In this paper,at the base of the job of [15, 16, 17, 18, 19, 20, 21, 22, 23], we constructand research two kinds of curves with multiple modification parameters are listed asfollows:
1. Given an extension of Be′zier curve with multiple modification parameters,andprofoundly research the properties of it.Further more ,we class it with its Geo-metric characters.
In the second chapter,we are given a kind of an extension of order two Be′zier
In the third chapter,we are given a kind of an extension of n order Be′zier curvewith n modification parameters:
2. Given an extension of uniform B-spline curve with multiple modification param-eters,and profoundly research the properties of it.
In the forth chapter,we are given a kind of an extension of three order B-splinecurve with multiple modification parameters:curve with two modification parameters:
After the extension of three order B-spline curve with multiple modification pa-rameters,we give n order uniform B-spline curve with multiple modification pa-rameters:
In the paper we Take different values of modification parameters make the pa-rameters become of local parameters,so we can globally or locally adjust the curveshape.Not liking other kinds of parameters curves,the two kinds of curves given inthis paper are given modification parameters at the base of without changing basisfunctions of the original Be′zier and uniform B-spline basis functions,therefore reduc-ing limits of parameters value. Because of values of these parameters can been chosenat random,we can adjust Be′zier curve with parameters much more liberty,can adjustthe shape of curves effectively.
1.用变换矩阵给出带多调节参数的Be′zier曲线扩展,并具体分析了其性质,根据几何性质将其进行了分类.
本文在第二章给出了带两个调节参数的二阶Be′zier曲线的扩展:
在第三章给出了带n个调节参数的n阶Be′zier曲线的扩展:
2.运用前面的思路给出带多个调节参数的均匀B样条曲线的构造方法,并体分析了其性质.
在第四章首先给出了带多个调节参数的三阶B样条曲线扩展,其表示为:继带多个调节参数的三阶B样条曲线后,本文在第四章构造了带多个调节参数的n阶B样条曲线,其表示为:
在本文中给出的这两类曲线引入了多个调节参数,这使得参数成为局部参数,它既能整体调控曲线的形状,又能局部调控.而且有别于其它参数曲线,它是在基函数不变即仍然使用Be′zier和B样条基的前提下,通过通过使用参数调节控制点来调节曲线形状.由于函数基是不变的,因此减弱了对参数的范围限制,使参数的取值有任意性,增加增加了带参数的曲线的自由度,因此能很好的调节曲线.
In the field of B-spline function approximation theory and application,in orderto arriving at better effect ,modulating basis functions is an important way,and therehave been many methods abort that,using shape modification parameters is one ofthese.In this paper,at the base of the job of [15, 16, 17, 18, 19, 20, 21, 22, 23], we constructand research two kinds of curves with multiple modification parameters are listed asfollows:
1. Given an extension of Be′zier curve with multiple modification parameters,andprofoundly research the properties of it.Further more ,we class it with its Geo-metric characters.
In the second chapter,we are given a kind of an extension of order two Be′zier
In the third chapter,we are given a kind of an extension of n order Be′zier curvewith n modification parameters:
2. Given an extension of uniform B-spline curve with multiple modification param-eters,and profoundly research the properties of it.
In the forth chapter,we are given a kind of an extension of three order B-splinecurve with multiple modification parameters:curve with two modification parameters:
After the extension of three order B-spline curve with multiple modification pa-rameters,we give n order uniform B-spline curve with multiple modification pa-rameters:
In the paper we Take different values of modification parameters make the pa-rameters become of local parameters,so we can globally or locally adjust the curveshape.Not liking other kinds of parameters curves,the two kinds of curves given inthis paper are given modification parameters at the base of without changing basisfunctions of the original Be′zier and uniform B-spline basis functions,therefore reduc-ing limits of parameters value. Because of values of these parameters can been chosenat random,we can adjust Be′zier curve with parameters much more liberty,can adjustthe shape of curves effectively.
引文
[1]韩旭里,刘圣军.二次Bézier曲线的扩展[J].中南工业大学学报(自然科学版), 2003,34(02):214-256.
[2]刘植.Bézier曲线的扩展[J].合肥工业大学学报(自然科学版), 2004,27(08):976-979.
[3]齐从谦,邬弘毅.一类可调控Bézier曲线及其逼近性[J].湖南大学学报, 1996,23(06):15-19.
[4]卢跃奇.具有可调自由参数的广义Bézier曲线[D].浙江大学,2006.
[5]吴晓勤,韩旭里,罗善明.四次Bézier曲线的两种不同扩展[J].工程图学学报,2006,27(05):59-64.
[6]谢进,邬弘毅.一类带双参数的二次三角Be′zier曲线[J].合肥学院学报(自然科学版),2006,16(1):20-24.
[7]王刘强,刘旭敏.带形状参数的二次TC一Be′zier曲线[J].计算机工程与设计,2007,5(28):1095-1098.
[8]杨联强,邬弘毅.带形状参数的三次三角Be′zier曲线[J].合肥工业大学学报(自然科学版),2005,11(25):1472-1477.
[9]陶淑一.构造带形状参数的二次均匀B样条曲线[A].全国第18届计算机技术与应用(CACIS)学术会议论文集(下册)[C], 2007:1066-1070.
[10]王文涛,汪国昭.带形状参数的均匀B样条[J].计算机辅助设计与图形学学报, 2004,16(06):783-788.
[11]施晓燕.低阶的带形状参数的均匀B样条[D].浙江大学, 2003.
[12]韩旭里,刘圣军.三次均匀B样条曲线的扩展[J].计算机辅助设计与图形学学报, 2003,15(05):576-578.
[13]刘长明. B样条的扩展及其应用[D].合肥工业大学, 2004 .
[14]施晓燕.带形状参数的七阶均匀B样条[J].浙江工业大学学报,2004,32(04):477-481.
[15]夏成林.带多个形状参数的Bézier曲线面的扩展[D].合肥工业大学,2005 .
[16]邬弘毅,夏成林.带多个形状参数的Bézier曲线与曲面的扩展[J].计算机辅助设计与图形学学报,2005,17(12):2607-2612.
[17]邬弘毅,左华.多形状参数的二次非均匀双曲B-样条曲线[J].计算机辅助设计与图形学学报,2007,19(07):876-883.
[18]左传桂,王国昭.多形状参数的四阶均匀B样条曲线设计[J].浙江大学学报(理学版),2001.30(2):252-256.
[19]左传桂.一种多形状参数四阶均匀B样条的研究[D].浙江大学, 2006
[20]张莉,邬弘毅.多形状参数的均匀B样条曲线[J].合肥工业大学,2007.30(2):252-256.
[21]熊建.带多形状参数的样条曲线曲面及其应用研究[D].合肥工业大学, 2008
[22]谢进,洪素珍.一类带两个形状参数的三次Bézier曲线[J].计算机工程与设计, 2007,28(06):1361-1363.
[23]严兰兰,宋来忠.带两个形状参数的Bézier曲线[J].工程图学学报, 2008,29(03):88-92.
[24] Farin G.Curves and surfaces for computer aided geometric design[M].New York:AcademicPress, 1988:50-150.
[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:高等教育出版社,2001.
[26] Piegl,L. Modifying the shape of rational B-spline , part 1 :surfaces [J]. Computer Aided Design, 1989 , 21(9) : 538 -546.
[27]王文涛.CAD中带形状参数曲线曲面研究[D].浙江大学数学系(浙江大学计算机图像图形研究所),2005.
[2]刘植.Bézier曲线的扩展[J].合肥工业大学学报(自然科学版), 2004,27(08):976-979.
[3]齐从谦,邬弘毅.一类可调控Bézier曲线及其逼近性[J].湖南大学学报, 1996,23(06):15-19.
[4]卢跃奇.具有可调自由参数的广义Bézier曲线[D].浙江大学,2006.
[5]吴晓勤,韩旭里,罗善明.四次Bézier曲线的两种不同扩展[J].工程图学学报,2006,27(05):59-64.
[6]谢进,邬弘毅.一类带双参数的二次三角Be′zier曲线[J].合肥学院学报(自然科学版),2006,16(1):20-24.
[7]王刘强,刘旭敏.带形状参数的二次TC一Be′zier曲线[J].计算机工程与设计,2007,5(28):1095-1098.
[8]杨联强,邬弘毅.带形状参数的三次三角Be′zier曲线[J].合肥工业大学学报(自然科学版),2005,11(25):1472-1477.
[9]陶淑一.构造带形状参数的二次均匀B样条曲线[A].全国第18届计算机技术与应用(CACIS)学术会议论文集(下册)[C], 2007:1066-1070.
[10]王文涛,汪国昭.带形状参数的均匀B样条[J].计算机辅助设计与图形学学报, 2004,16(06):783-788.
[11]施晓燕.低阶的带形状参数的均匀B样条[D].浙江大学, 2003.
[12]韩旭里,刘圣军.三次均匀B样条曲线的扩展[J].计算机辅助设计与图形学学报, 2003,15(05):576-578.
[13]刘长明. B样条的扩展及其应用[D].合肥工业大学, 2004 .
[14]施晓燕.带形状参数的七阶均匀B样条[J].浙江工业大学学报,2004,32(04):477-481.
[15]夏成林.带多个形状参数的Bézier曲线面的扩展[D].合肥工业大学,2005 .
[16]邬弘毅,夏成林.带多个形状参数的Bézier曲线与曲面的扩展[J].计算机辅助设计与图形学学报,2005,17(12):2607-2612.
[17]邬弘毅,左华.多形状参数的二次非均匀双曲B-样条曲线[J].计算机辅助设计与图形学学报,2007,19(07):876-883.
[18]左传桂,王国昭.多形状参数的四阶均匀B样条曲线设计[J].浙江大学学报(理学版),2001.30(2):252-256.
[19]左传桂.一种多形状参数四阶均匀B样条的研究[D].浙江大学, 2006
[20]张莉,邬弘毅.多形状参数的均匀B样条曲线[J].合肥工业大学,2007.30(2):252-256.
[21]熊建.带多形状参数的样条曲线曲面及其应用研究[D].合肥工业大学, 2008
[22]谢进,洪素珍.一类带两个形状参数的三次Bézier曲线[J].计算机工程与设计, 2007,28(06):1361-1363.
[23]严兰兰,宋来忠.带两个形状参数的Bézier曲线[J].工程图学学报, 2008,29(03):88-92.
[24] Farin G.Curves and surfaces for computer aided geometric design[M].New York:AcademicPress, 1988:50-150.
[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:高等教育出版社,2001.
[26] Piegl,L. Modifying the shape of rational B-spline , part 1 :surfaces [J]. Computer Aided Design, 1989 , 21(9) : 538 -546.
[27]王文涛.CAD中带形状参数曲线曲面研究[D].浙江大学数学系(浙江大学计算机图像图形研究所),2005.