以离散曲线为曲率线构造拟可展离散曲面
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摘要
提出一种构造插值给定离散曲率线的近似可展的离散曲面的算法.首先,计算离散Frenet标架.其次将其旋转得到一族旋转最小标架,通过指定已知曲线某点处的曲面法向来选择其一.最后,用它来实时地构建近似可展的离散曲面.该算法具有很好的交互性,而且所得曲面拥有插值性、良好的可展性和光顺性.实验结果表明,该算法简单、有效,富有潜在应用价值.
This paper proposes an algorithm to design almost developable meshes interpolating a given discrete line of curvature. First of all, compute the discrete Frenet frame of the given curve. Then rotate this frame to get the rotation minimizing frames. Select one of them by assigning the normal vector of the resulting surface at some vertex of the given discrete line. Finally, use it to construct an almost developable mesh in real time. This algorithm has very good interactivity, and the obtained interpolatory meshes possess good developability and smoothness. The experiment results show that this algorithm is very simple and efficient, and it has big potential value in practical application.
This paper proposes an algorithm to design almost developable meshes interpolating a given discrete line of curvature. First of all, compute the discrete Frenet frame of the given curve. Then rotate this frame to get the rotation minimizing frames. Select one of them by assigning the normal vector of the resulting surface at some vertex of the given discrete line. Finally, use it to construct an almost developable mesh in real time. This algorithm has very good interactivity, and the obtained interpolatory meshes possess good developability and smoothness. The experiment results show that this algorithm is very simple and efficient, and it has big potential value in practical application.
引文
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[16]Liu Y J,Lai Y K,Hu S M.Stripification of free-form surfaces with global error bounds for developable approximation[J].IEEE Transactions on Automation Science and Engineering,2009,6(4):700-709
[17]Tang K,Wang C C L.Modeling developable folds on a strip[J].Journal of Computing and Information Science in Engineering,2005,5(1):35-47
[18]Liu Y J,Tang K,Joneja A.Modeling dynamic developable meshes by the hamilton principle[J].Computer-Aided Design,2007,39(9):719-731
[19]Liu Y J,Tang K,Gong W Y,et al.Industrial design using interpolatory discrete developable surfaces[J].Computer-Aided Design,2011,43(9):1089-1098
[20]Julius D,Kraevoy V,Sheffer A.D-charts:quasi-developable mesh segmentation[J].Computer Graphics Forum,2005,24(3):581-590
[21]Tang K,Chen M.Quasi-developable mesh surface interpolation via mesh deformation[J].IEEE Transactions on Visualization and Computer Graphics,2009,15(3):518-528
[22]Zeng L,Liu Y J,Chen M,et al.Least squares quasi-developable mesh approximation[J].Computer Aided Geometric Design,2012,29(7):565-578
[23]Guggenheimer H.Computing frames along a trajectory[J].Computer Aided Geometric Design,1989,6(1):77-78
[24]Klok F.Two moving coordinate frames for sweeping along a3D trajectory[J].Computer Aided Geometric Design,1986,3(3):217-229
[25]Willmore T J.An introduction to differential geometry[M].Oxford:Oxford University Press,1959
[2]Weiss G,Furtner P.Computer-aided treatment of developable surfaces[J].Computers&Graphics,1988,12(1):39-51
[3]Wang G J,Tang K,Tai C L.Parametric representation of a surface pencil with a common spatial geodesic[J].ComputerAided Design,2004,36(5):447-459
[4]Zhao H Y,Wang G J.A new method for designing a developable surface utilizing the surface pencil through a given curve[J].Progress in Natural Science,2008,18(1):105-110
[5]Bayram E,Güler F,Kasap E.Parametric representation of a surface pencil with a common asymptotic curve[J].ComputerAided Design,2012,44(7):637-643
[6]Chen M,Tang K.Quasi-developable surface modeling of contours with curved triangular patches[J].Computers&Graphics,2013,37(7):851-861
[7]Chen M,Tang K.2G quasi-developable bezier surface interpolation of two space curves[J].Computer-Aided Design,2013,45(11):1365-1377
[8]Martin R R.Principal patches-a new class of surface patch based on differential geometry[C]//Proceedings of Eurographics'83.Amsterdam:North-Holland,1983:47-55
[9]Li C Y,Wang R H,Zhu C G.Parametric representation of a surface pencil with a common line of curvature[J].ComputerAided Design,2011,43(9):1110-1117
[10]Li C Y,Wang R H,Zhu C G.An approach for designing a developable surface through a given line of curvature[J].Computer-Aided Design,2013,45(3):621-627
[11]Kalogerakis E,Nowrouzezahrai D,Simari P,et al.Extracting lines of curvature from noisy point clouds[J].Computer-Aided Design,2009,41(4):282-292
[12]Pang X F,Song Z,Pang M Y.Extraction of the lines of curvature from raw point cloud[C]//Computer Graphics Proceedings,Annual Conference Series,ACM SIGGRAPH.New York:ACM Press,2010:225-228
[13]Rose K,Sheffer A,Wither J,et al.Developable surfaces from arbitrary sketched boundaries[C]//Proceedings of the 5th Eurographics Symposium on Geometry Processing.Aire-la-Ville:Eurographics Association Press,2007:163-172
[14]Frey W H.Boundary triangulations approximating developable surfaces that interpolate a closed space curve[J].International Journal of Foundations of Computer Science,2002,13(2):285-302
[15]Frey W H.Modeling buckled developable surfaces by triangulation[J].Computer-Aided Design,2004,36(4):299-313
[16]Liu Y J,Lai Y K,Hu S M.Stripification of free-form surfaces with global error bounds for developable approximation[J].IEEE Transactions on Automation Science and Engineering,2009,6(4):700-709
[17]Tang K,Wang C C L.Modeling developable folds on a strip[J].Journal of Computing and Information Science in Engineering,2005,5(1):35-47
[18]Liu Y J,Tang K,Joneja A.Modeling dynamic developable meshes by the hamilton principle[J].Computer-Aided Design,2007,39(9):719-731
[19]Liu Y J,Tang K,Gong W Y,et al.Industrial design using interpolatory discrete developable surfaces[J].Computer-Aided Design,2011,43(9):1089-1098
[20]Julius D,Kraevoy V,Sheffer A.D-charts:quasi-developable mesh segmentation[J].Computer Graphics Forum,2005,24(3):581-590
[21]Tang K,Chen M.Quasi-developable mesh surface interpolation via mesh deformation[J].IEEE Transactions on Visualization and Computer Graphics,2009,15(3):518-528
[22]Zeng L,Liu Y J,Chen M,et al.Least squares quasi-developable mesh approximation[J].Computer Aided Geometric Design,2012,29(7):565-578
[23]Guggenheimer H.Computing frames along a trajectory[J].Computer Aided Geometric Design,1989,6(1):77-78
[24]Klok F.Two moving coordinate frames for sweeping along a3D trajectory[J].Computer Aided Geometric Design,1986,3(3):217-229
[25]Willmore T J.An introduction to differential geometry[M].Oxford:Oxford University Press,1959