微信服务号

微信订阅号

2025年4月10日 5:44 星期四
首页 > 过刊浏览>2016年第27卷第10期 >2499-2508. DOI:10.13328/j.cnki.jos.005087
PDF HTML阅读 XML下载 导出引用 引用提醒
插值边界的四边网格离散极小曲面建模方法
作者:
基金项目:

国家自然科学基金（61472111，61272300，61602138）；浙江省杰出青年自然科学基金（LR16F020003）；浙江省自然科学基金（LQ16F020005）


Constructing Discrete Minimal Surfaces with Quadrilateral Meshes from Described Boundary
Author:
Fund Project:

National Natural Science Foundation of China (61472111, 61272300, 61602138); Natrual Science Foundation of Zhejiang Province for Distinguished Young Scholars (LR16F020003); Natural Science Foundation of Zhejiang Province (LQ16F020005)

  • 摘要
    • | |
  • 访问统计
    • |
  • 参考文献 [42]
    • |
  • 相似文献 [20]
    • | | |
  • 文章评论
    摘要:

    如何实现极小曲面的快速三维建模，是几何设计与计算领域中的难点和热点问题.给定一条封闭的边界离散折线，研究如何构造以其为边界的四边网格离散极小曲面.首先从曲面的内蕴微分几何度量出发，给出了离散四边网格极小曲面的数学定义；然后利用保长度边界投影、四边网格生成、径向基函数插值映射和非线性优化技术，提出了由给定边界离散折线快速构造离散四边网格极小曲面的一般技术框架.最后通过若干建模实例验证了所提方法的有效性.该方法可实现四边网格极小曲面的高质量建模，在建筑几何领域具有一定的应用价值.

    Abstract:

    Efficient modeling of minimal surfaces is a challenging problem and hot topic in the field of geometric design and computation. Taking boundary closed polylines, this paper proposes a general framework to construct discrete minimal surfaces with quadrilateral meshes. First, the mathematical definition of discrete minimal surface with quadrilateral mesh is given from the intrinsic differential-geometry metric of surfaces. Next, based on the length-preserving boundary projection method, quad-mesh generation approach and non-linear numerical optimization technique, a novel framework is presented to construct discrete minimal surfaces with quadrilateral meshes from a described boundary closed discrete polylines. Finally, the effectiveness of the proposed approach is illustrated by several modeling examples. The results show that the proposed method can achieve high-quality modeling of discrete minimal surfaces and provide potential usage in architecture geometry.

    参考文献
    [1] Johannes CCN.Lectures on Minimal Surfaces,Vol.1.Cambridge:Cambridge University Press,1989.
    [2] Robert O.A survey of Minimal Surfaces.2nd ed.,New York:Dover Publications,1986.
    [7] Xu G,Wang GZ.AHT Bézier surfaces and NUAHT B-spline curves.Journal of Computer Science and Technology,2007,22(4):597-607.[doi:10.1007/s11390-007-9073-z]
    [8] Xu G,Wang GZ.Control mesh representation of a class of minimal surfaces.Journal of Computer-Aided Design and Computer Graphics,2007,19(2):240-244(in Chinese with English abstract).[doi:10.3321/j.issn:1003-9775.2007.02.019]
    [9] Man JJ,Wang GZ.Polynomial minimal surface in Isothermal parameter.Chinese Journal of Computers,2002,25(2):197-201(in Chinese with English abstract).[doi:10.3321/j.issn:0254-4164.2002.02.012]
    [10] Xu G,Wang GZ.Quintic parametric polynomial minimal surfaces and their properties.Differential Geometry and Its Applications,2010,28(6):697-704.[doi:10.1016/j.difgeo.2010.07.003]
    [11] Xu G,Zhu YG,Wang GZ,Galligo A,Zhang L,Hui KC.Explicit form of parametric polynomial minimal surfaces with arbitrary degree.Applied Mathematics and Computation,2015,259:124-131.[doi:10.1016/j.amc.2015.02.065]
    [12] Monterde J,Ugail H.On harmonic and biharmonic Bézier surfaces.Computer Aided Geometric Design,2004,21(7):697-715.[doi:10.1016/j.cagd.2004.07.003]
    [13] Xu G,Wang GZ.Harmonic B-B surfaces over the triangular domain.Chinese Journal of Computers,2006,29(12):2180-2185(in Chinese with English abstract).[doi:10.3321/j.issn:0254-4164.2006.12.015]
    [14] Monterde J.Bézier surfaces of minimal area:The Dirichlet approach.Computer Aided Geometric Design,2004,21(2):117-136.[doi:10.1016/j.cagd.2003.07.009]
    [15] Arnal A,Lluch A,Monterde J.Triangular Bézier surfaces of minimal area.In:Proc.of the Int'l Workshop on Computer Graphics and Geometric Modeling.Montreal,2003.366-375.[doi:10.1007/3-540-44842-X_38]
    [16] Wu H,Wang GJ.Extending Dirichlet approach to design B-spline surface of minimal area.Ruan Jian Xue Bao/Journal of Software,2011,22(12):3015-3022(in Chinese with English abstract).http://www.jos.org.cn/1000-9825/4009.htm[doi:10.3724/SP.J.1001.2011.04009]
    [17] Wu H,Wang GJ.Designing approximate minimal NURBS surfaces with boundaries interpolation.Ruan Jian Xue Bao/Journal of Software,2010,21(zk):67-74(in Chinese with English abstract).http://www.jos.org.cn/1000-9825/10008.htm
    [18] Xu G,Wang GZ.A new method to solve Plateau-Bézier problems over the triangular domain.Chinese Journal of Computers,2011,34(3):548-555(in Chinese with English abstract).[doi:10.3724/SP.J.1016.2011.00548]
    [19] Hao YX,Wang RH,Li CJ.Minimal quasi-Bézier surface.Applied Mathematical Modelling,2012,36(2):5751-5757.[doi:10.1016/j.apm.2012.01.040]
    [20] Hao YX,Li CJ,Wang RH.An approximation method based on MRA for the quasi-Plateau problem.BIT Numerical Mathematics,2013,53(2):411-442.[doi:10.1007/s10543-012-0412-2]
    [21] Man JJ,Wang GZ.Approximating to nonparameterzied minimal surface with B-spline surface.Ruan Jian Xue Bao/Journal of Software,2003,14(4):824-829(in Chinese with English abstract).http://www.jos.org.cn/ch/reader/view_abstract.aspx?flag=1&file_no=20030416&journal_id=jos
    [22] Qin L,Guan LT.Minimal surface modeling based on NURBS.Numerical Mathematics:A Journal of Chinese Universities,2005,16(S1):175-181(in Chinese with English abstract).
    [23] Pan Q,Xu GL.Construction of minimal subdivision surface with a given boundary.Computer-Aided Design,2011,43(4):374-380.[doi:10.1016/j.cad.2010.12.013]
    [24] Li CY,Wang RH,Zhu CG.Designing approximation minimal parametric surfaces with geodesics.Applied Mathematical Modelling,2013,37(9):6415-6424.[doi:10.1016/j.apm.2013.01.035]
    [25] Ulrich P,Konrad P.Computing discrete minimal surface and their conjugates.Experimental Mathematics,1993,2(1):15-36.[doi:10.1080/10586458.1993.10504266]
    [26] Gerhard D,Hutchinson JE.The discrete Plateau problem.Mathematics of Computation,1999,68(225):1-23.[doi:10.1090/S0025-5718-99-01025-X]
    [27] Zhang Q,Xu GL.Weighted minimal surfaces and discrete weighted minimal surfaces.Journal of Information&Computational Science,2005,2(2):395-408.
    [28] Bobenko AI,Hoffmann T,Springborn BA,Minimal surfaces from circle patterns:Geometry from combinatorics.Annals of Mathematics,2006,164(1):231-264.[doi:10.4007/annals.2006.164.231]
    [29] Wallner J,Pottmann H.Infinitesimally flexible meshes and discrete minimal surfaces.Monatshefte für Mathematik,2008,153:347-365.[doi:10.1007/s00605-007-0502-4]
    [30] Hao P,Choi YK,Liu Y,Hu WC,Du Q,Polthier K,Zhang CM,Wang WP.Robust modeling of constant mean curvature surfaces.In:Proc.of the SIGGRAPH ACM Trans.Graph.2012.[doi:10.1145/2185520.2185581]
    [31] Liu D,Xu GL,Zhang Q.A discrete scheme of Laplace-Beltrami operator and its convergence over quadrilateral meshes.Computers&Mathematics with Applications,2008,55(6):1081-1093.[doi:10.1016/j.camwa.2007.04.047]
    [32] Wang CCL.Computing length-preserved free boundary for quasi-developable mesh segmentation.IEEE Trans.on Visualization and Computer Graphics,2008,14(1):25-36.[doi:10.1109/TVCG.2007.1067]
    [33] Olga S.Differential representations for mesh processing.Computer Graphics Forum,2006,25(4):789-807.[doi:10.1111/j.1467-8659.2006.00999.x]
    [34] Jorge N,Stephen JW.Numerical Optimization.Springer-Verlag,1999.[doi:10.1007/b98874]
    [35] Xu G,Mourrain B,Duvigneau R,Galligo A.Parameterization of computational domain in isogeometric analysis:Methods and comparison.Computer Methods in Applied Mechanics and Engineering,2011,(200):2021-2031.[doi:10.1016/j.cma.2011.03.005]
    [4] 金文标,汪国昭.一类极小曲面的几何设计.计算机学报,1999,23(12):1277-1279.[doi:10.3321/j.issn:0254-4164.1999.12.008]
    [5] 徐岗,汪国昭.计算机辅助几何设计中极小曲面造型的研究进展.系统科学与数学,2008,28(8):984-992.
    [6] 满家巨,汪国昭.正螺面与悬链面的表示与构造.计算机辅助设计与图形学学报,2005,17(3):431-436.[doi:10.3321/j.issn:1003-9775.2005.03.009]
    [8] 徐岗,汪国昭.一类极小曲面的控制网格表示.计算机辅助设计与图形学学报,2007,19(2):240-244.[doi:10.3321/j.issn:1003-9775.2007.02.019]
    [9] 满家巨,汪国昭.等温参数多项式极小曲面.计算机学报,2002,25(2):197-201.[doi:10.3321/j.issn:0254-4164.2002.02.012]
    [13] 徐岗,汪国昭.三角域上的调和B-B曲面.计算机学报,2006,29(12):2180-2185.[doi:10.3321/j.issn:0254-4164.2006.12.015]
    [16] 吴花精灵,王国瑾.推广Dirichlet方法用于B样条极小曲面设计.软件学报,2011,22(12):3015-3022.http://www.jos.org.cn/1000-9825/4009.htm[doi:10.3724/SP.J.1001.2011.04009]
    [17] 吴花精灵,王国瑾.插值边界曲线的NURBS近似极小曲面设计.软件学报,2010,21(增刊):67-74.http://www.jos.org.cn/1000-9825/10008.htm
    [18] 徐岗,汪国昭,陈小雕.三角域上的Plateau-Bézier问题求解新方法.计算机学报,2011,34(3):548-555.[doi:10.3724/SP.J.1016.2011.00548]
    [21] 满家巨,汪国昭.B-样条函数极小曲面造型.软件学报,2003,14(4):824-829.http://www.jos.org.cn/ch/reader/view_abstract.aspx?flag=1&file_no=20030416&journal_id=jos
    [22] 覃廉,关履泰.基于NURBS的极小曲面造型.高等学校计算数学学报,2005,16(S1):175-181.
    引证文献
    网友评论
    网友评论
    分享到微博
    发 布
引用本文

徐岗,朱亚光,李鑫,许金兰,汪国昭,许健泉.插值边界的四边网格离散极小曲面建模方法.软件学报,2016,27(10):2499-2508

复制
分享
文章指标
  • 点击次数:4457
  • 下载次数: 7073
  • HTML阅读次数: 3135
  • 引用次数: 0
历史
  • 收稿日期:2016-01-21
  • 最后修改日期:2016-03-25
  • 在线发布日期: 2016-08-11
文章二维码
友情链接:中国科学院软件研究所中国计算机学会中国科学院国家自然科学基金委员会CCF数字图书馆Int'l J of Softw Info计算机系统应用
您是第19808887位访问者
版权所有：中国科学院软件研究所 京ICP备05046678号-3
地址：北京市海淀区中关村南四街4号,邮政编码：100190
电话：010-62562563 传真：010-62562533 Email：jos@iscas.ac.cn
技术支持：北京勤云科技发展有限公司

京公网安备 11040202500063号