过渡曲面的B-样条曲线构造法研究
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摘要
在计算机辅助几何设计中,过渡曲面的构造是一个重要的研究课题,所谓的过渡曲面是指曲线或曲面间的具有一定连续性的过渡面。也就是说对于给定的两片或多片曲面以及原曲面上的边界线,怎样来构造一个曲面,让这个曲面与原曲面之间在边界线处光滑连接。因为它在汽车,船舶,飞机等工业制造领域的强实用性,所以就成为计算机辅助几何设计(CAGD)的一个重要学术分支。
本文总结了一些学者构造过渡曲面的方法,并对这些方法经行了分析与研究,在分析过程中发现,这些方法可分为参数形式研究和隐式形式的研究,对比这两种方法,很容易发现参数形式更容易画图,细分,约束或者是说更容易去调整曲面的形状,而隐式形式则很容易确定过渡曲面与原曲面之间的连续性,使其能很容易达到想要达到的连续性。如果我们选择其中一种形式而另一种形式的优点则减弱,因此能提出一种两种形式优点相结合的方法是一个很重要的研究课题,这就要求我们在研究构造方法中既要满足一定的连续性又要满足灵活的调整性。
通过分析对比本文提出了一种新的方法。因为所要构造的过渡曲面是由一束空间曲线构造而成,而空间曲线是由原始曲面上的点与调控线上的点插值构成的B-样条参数曲线,所以称它为B样条曲线构造法,并且用这个方法构造的过渡曲面能和原始曲面达到G 1连续。又因为这个过渡曲面具有参数形式所以它的形状很容易用参数因子和调控线去控制。文中最后通过一些例子展现了这种方法的灵活性与有效性。
In engineering design and geometric modeling process, how to construct a blending surface among several curves or surfaces is an important problem in the CAGD, which is to construct a blending surface that smoothly joins two or more given curves or surfaces. That is, for given two or several surfaces and those blending borders, how to construct a blending surface which contacts original surfaces as high order continuity as possible. Because it has strong practicability in automobile, airplane, shipping and other industrial manufacturing field, so it become an important academic branch of computer aided geometric design (CAGD).
This paper summary, analysis, research some methods which other researchers construct blending surface,and find as many construction of blending surface methods, surface can be classified into two types: one of the implicit surfaces, another is parametric surfaces .In comparing the two types of surfaces one notes that parametric surfaces are generally easier to draw, subdivided and bound, or easier to adjust the shape of blending surface whereas an implicit surface is easily determine the continuity between the blending surface and the original surface, make it can easily achieve satisfactory continuity. If we choose one kind of form and another form advantages is abate. So it is an important research fields to combine with the two forms advantage. This requires that we research the construction methods to satisfy some continuity but also to satisfy the flexible adjustment.
Through the comparative analysis, this paper put forward a new method. Because the blending surface is constructed with a collection of space curves, which are defined by points on the blending boundaries of primary surfaces and the points on the given adjustment curves. The iso-parametric curves are B-Spline curves, which interpolate the blending boundaries. So we call it B-spline curves constructing method. And the constructing method that smoothly joining two or more surfaces with G 1 continuity. Because the blending surface has a parametric form, so its shape is easy to be controlled by adjusting the parameters and adjustment curves. In the end of this paper several numerical examples are presented to show the feasibility and validity of our method.
本文总结了一些学者构造过渡曲面的方法,并对这些方法经行了分析与研究,在分析过程中发现,这些方法可分为参数形式研究和隐式形式的研究,对比这两种方法,很容易发现参数形式更容易画图,细分,约束或者是说更容易去调整曲面的形状,而隐式形式则很容易确定过渡曲面与原曲面之间的连续性,使其能很容易达到想要达到的连续性。如果我们选择其中一种形式而另一种形式的优点则减弱,因此能提出一种两种形式优点相结合的方法是一个很重要的研究课题,这就要求我们在研究构造方法中既要满足一定的连续性又要满足灵活的调整性。
通过分析对比本文提出了一种新的方法。因为所要构造的过渡曲面是由一束空间曲线构造而成,而空间曲线是由原始曲面上的点与调控线上的点插值构成的B-样条参数曲线,所以称它为B样条曲线构造法,并且用这个方法构造的过渡曲面能和原始曲面达到G 1连续。又因为这个过渡曲面具有参数形式所以它的形状很容易用参数因子和调控线去控制。文中最后通过一些例子展现了这种方法的灵活性与有效性。
In engineering design and geometric modeling process, how to construct a blending surface among several curves or surfaces is an important problem in the CAGD, which is to construct a blending surface that smoothly joins two or more given curves or surfaces. That is, for given two or several surfaces and those blending borders, how to construct a blending surface which contacts original surfaces as high order continuity as possible. Because it has strong practicability in automobile, airplane, shipping and other industrial manufacturing field, so it become an important academic branch of computer aided geometric design (CAGD).
This paper summary, analysis, research some methods which other researchers construct blending surface,and find as many construction of blending surface methods, surface can be classified into two types: one of the implicit surfaces, another is parametric surfaces .In comparing the two types of surfaces one notes that parametric surfaces are generally easier to draw, subdivided and bound, or easier to adjust the shape of blending surface whereas an implicit surface is easily determine the continuity between the blending surface and the original surface, make it can easily achieve satisfactory continuity. If we choose one kind of form and another form advantages is abate. So it is an important research fields to combine with the two forms advantage. This requires that we research the construction methods to satisfy some continuity but also to satisfy the flexible adjustment.
Through the comparative analysis, this paper put forward a new method. Because the blending surface is constructed with a collection of space curves, which are defined by points on the blending boundaries of primary surfaces and the points on the given adjustment curves. The iso-parametric curves are B-Spline curves, which interpolate the blending boundaries. So we call it B-spline curves constructing method. And the constructing method that smoothly joining two or more surfaces with G 1 continuity. Because the blending surface has a parametric form, so its shape is easy to be controlled by adjusting the parameters and adjustment curves. In the end of this paper several numerical examples are presented to show the feasibility and validity of our method.
引文
[1] S.Allen,D.Dutta.Supercyclides and blending.Computer Aided Geometric Design.1995, 14(7):637–651.
[2] Bloor MIG,Wilson MJ.Generation blend surfaes using partial differential Equations. Computer-Aided Design.1989,21(3):165–171.
[3] H.J.Zhang,L.You.Pde based surface representation-vase design. Computers and Graphics -Uk.2002,26(1):89–98.
[4] L.H.You,P.Comninos,J.J.Zhang.Pde blending surfaces with C 2continuity.Computers and Graphics-Uk.2004,28(6):895–906.
[5] M.I.G.Bloor, M.J.Wilson.Spectral approximations to pde surfaces.Computer-Aided Design.1996,28(2):145–152.
[6] M.I.G.Bloor,M.J.Wilson.Method for effcient shape parametrization of fluid membranes and vesicles.Physical Review E.2000, 61(4):4218–4229.
[7] E. Hartmann. Blending an implicit with a parametric surface. Computer Aided Geometric Design.1995,12:825-835.
[8] D.S.Meek,D.J.Walton.Blending two parametric curves.Computer-Aided Design.2009, 41(6):423–431.
[10] B.Belkhatir, A.Zidna. Construction of flexible blending parametric surfaces via curves. Math-matics and Computers in Simulation.2009,79:3599-3608.
[11] Qunlin,J.G.Rokne.Blendingimplicit interval curves and surfaces.Applied Mathematics and Computation.2009,10:1-12.
[12] JR Rossignac,AAG Requicha.Constant radius blending in solid modeling.Computer in Mechanical Engineering,1984.
[13] Alan E.Middledithch and Kenneth H.Sears.Blend surfaces for set theoretic volume modelling systems.Proceedings of the 12th annual conference on Computer graphics and interactive techniques.1985:161–170.
[14] Rockwood A P,Owen J C.Blending surfaces in solid geometric modeling.Geometric dodeling and robotics.1985.
[15]陈发来,陈长松,邓建松.用分片代数曲面构造管道曲面的过渡曲.计算学报.2000,23(9):911-916
[16] Christioph M.Hoffmann,John E.Hopcroft.The potential method for blending surfaces and corners.Geometric modeling.1985:301–307.
[17]J.Warren.Blending algebraic-surfaces.Acm Transactions on Graphics.1989,8(4):263-278.
[18] W.T.Wu,D.K.Wang.On surface-ftting problem in CAGD.Mathematics in Practice and Theory.1994:26–31.
[19]T.R.Wu,Y.S.Zhou.On blending of several quadratic algebraic surfaces.Computer Aided Geometric Design.2000,17(8):759–766.
[20] M.I.G. Bloor,J. Wilson.Generating blending surface using partial differential equation.Computer Aided Design.1990,22(4):202-212.
[21]宋遒志.融合过渡曲面的构造理论与方法研究:(博士学位论文).合肥:华中科技大学,2001.
[22]石雄辉.高阶几何连续过渡曲面的造型研究与应用:(硕士学位论文).西安:西北工业大学,2003.
[23]朱汉东,金东光.过渡曲面的母线构造法.工程图学学报.1998,3:45-48.
[24]马岭,朱心雄.用偏微分方程构造过渡曲面.工程图学学报.1995,4(01):1-8.
[25] J.Cheng, X.S. Gao, Constructing surfaces for two arbitrary surfaces, Research MMR, AMMSS,vol.21, Academia, Sinica, Beijing, 2002:15-22.
[26]陈浩乾.过渡曲面生成算法及其在隐形牙套模型中应用的研究:(硕士学位论文).西安:西安科技大学,2008.
[27]陆晓辉.裁剪B样条曲面间过渡曲面的构造:(硕士学位论文).长春:吉林大学,2005.
[28]施法中.计算机辅助几何设计与非均匀有理B样条,北京:高等教育出版社,2002.
[29]周裕庚.基于PHTSpline的双三次曲面间过渡曲面的构造:(硕士学位论文).安徽:中国科技大学,2010.
[30]雷娜,崔丽,伍铁如.两个轴异面的管道拼接问题.吉林大学学报.2002,4:23-28.
[31]宋遒志,陈立平等.基于物理的能量最小法构造融合过渡曲面.工程图学学报.2002,3:122-132.
[2] Bloor MIG,Wilson MJ.Generation blend surfaes using partial differential Equations. Computer-Aided Design.1989,21(3):165–171.
[3] H.J.Zhang,L.You.Pde based surface representation-vase design. Computers and Graphics -Uk.2002,26(1):89–98.
[4] L.H.You,P.Comninos,J.J.Zhang.Pde blending surfaces with C 2continuity.Computers and Graphics-Uk.2004,28(6):895–906.
[5] M.I.G.Bloor, M.J.Wilson.Spectral approximations to pde surfaces.Computer-Aided Design.1996,28(2):145–152.
[6] M.I.G.Bloor,M.J.Wilson.Method for effcient shape parametrization of fluid membranes and vesicles.Physical Review E.2000, 61(4):4218–4229.
[7] E. Hartmann. Blending an implicit with a parametric surface. Computer Aided Geometric Design.1995,12:825-835.
[8] D.S.Meek,D.J.Walton.Blending two parametric curves.Computer-Aided Design.2009, 41(6):423–431.
[10] B.Belkhatir, A.Zidna. Construction of flexible blending parametric surfaces via curves. Math-matics and Computers in Simulation.2009,79:3599-3608.
[11] Qunlin,J.G.Rokne.Blendingimplicit interval curves and surfaces.Applied Mathematics and Computation.2009,10:1-12.
[12] JR Rossignac,AAG Requicha.Constant radius blending in solid modeling.Computer in Mechanical Engineering,1984.
[13] Alan E.Middledithch and Kenneth H.Sears.Blend surfaces for set theoretic volume modelling systems.Proceedings of the 12th annual conference on Computer graphics and interactive techniques.1985:161–170.
[14] Rockwood A P,Owen J C.Blending surfaces in solid geometric modeling.Geometric dodeling and robotics.1985.
[15]陈发来,陈长松,邓建松.用分片代数曲面构造管道曲面的过渡曲.计算学报.2000,23(9):911-916
[16] Christioph M.Hoffmann,John E.Hopcroft.The potential method for blending surfaces and corners.Geometric modeling.1985:301–307.
[17]J.Warren.Blending algebraic-surfaces.Acm Transactions on Graphics.1989,8(4):263-278.
[18] W.T.Wu,D.K.Wang.On surface-ftting problem in CAGD.Mathematics in Practice and Theory.1994:26–31.
[19]T.R.Wu,Y.S.Zhou.On blending of several quadratic algebraic surfaces.Computer Aided Geometric Design.2000,17(8):759–766.
[20] M.I.G. Bloor,J. Wilson.Generating blending surface using partial differential equation.Computer Aided Design.1990,22(4):202-212.
[21]宋遒志.融合过渡曲面的构造理论与方法研究:(博士学位论文).合肥:华中科技大学,2001.
[22]石雄辉.高阶几何连续过渡曲面的造型研究与应用:(硕士学位论文).西安:西北工业大学,2003.
[23]朱汉东,金东光.过渡曲面的母线构造法.工程图学学报.1998,3:45-48.
[24]马岭,朱心雄.用偏微分方程构造过渡曲面.工程图学学报.1995,4(01):1-8.
[25] J.Cheng, X.S. Gao, Constructing surfaces for two arbitrary surfaces, Research MMR, AMMSS,vol.21, Academia, Sinica, Beijing, 2002:15-22.
[26]陈浩乾.过渡曲面生成算法及其在隐形牙套模型中应用的研究:(硕士学位论文).西安:西安科技大学,2008.
[27]陆晓辉.裁剪B样条曲面间过渡曲面的构造:(硕士学位论文).长春:吉林大学,2005.
[28]施法中.计算机辅助几何设计与非均匀有理B样条,北京:高等教育出版社,2002.
[29]周裕庚.基于PHTSpline的双三次曲面间过渡曲面的构造:(硕士学位论文).安徽:中国科技大学,2010.
[30]雷娜,崔丽,伍铁如.两个轴异面的管道拼接问题.吉林大学学报.2002,4:23-28.
[31]宋遒志,陈立平等.基于物理的能量最小法构造融合过渡曲面.工程图学学报.2002,3:122-132.