Quality Surface Meshing Using Discrete Parametrizations

  • Emilie Marchandise
  • Jean-François Remacle
  • Christophe Geuzaine
Conference paper


We present 3 mapping/flattening techniques for triangulations of poor quality triangles. The implementation of those mappings as well as the boundary conditions are presented in a very comprehensive manner such that it becomes accessible to a wider community than the one of computer graphics. The resulting parameterizations are used to generate new triangulations or quadrilateral meshes for the model that are of high quality.


Discrete Parametrization Quadrilateral Mesh Fiedler Vector Initial Triangulation Remeshing Procedure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alliez, P., Meyer, M., Desbrun, M.: Interactive geometry remeshing. In: Computer graphics (Proceedings of the SIGGRAPH 2002), pp. 347–354 (2002)Google Scholar
  2. 2.
    Batdorf, M., Freitag, L.A., Ollivier-Gooch, C.: A computational study of the effect of unstructured mesh quality on solution efficiency. In: Proc. 13th AIAA Computational Fluid Dynamics Conf. (1997)Google Scholar
  3. 3.
    Bechet, E., Cuilliere, J.-C., Trochu, F.: Generation of a finite element mesh from stereolithography (stl) files. Computer-Aided Design 34(1), 1–17 (2002)CrossRefGoogle Scholar
  4. 4.
    Ben-Chen, M., Gotsman, C., Bunin, G.: Conformal flattening by curvature prescription and metric scaling. Computer Graphics Forum 27(2) (2008)Google Scholar
  5. 5.
    Bennis, C., Vézien, J.-M., Iglésias, G.: Piecewise surface flattening for non-distorted texture mapping. In: ACM SIGGRAPH Computer Graphics, pp. 237–246 (1991)Google Scholar
  6. 6.
    Borouchaki, H., Laug, P., George, P.L.: Parametric surface meshing using a combined advancing-front generalized delaunay approach. International Journal for Numerical Methods in Engineering 49, 223–259 (2000)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Choquet, C.: Sur un type de représentation analytique généralisant la représentation conforme et défininie au moyen de fonctions harmoniques. Bull. Sci. Math. 69(156-165) (1945)Google Scholar
  8. 8.
    Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M., Stuetzle, W.: Multiresolution analysis of arbitrary meshes. In: SIGGRAPH 1995: Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, pp. 173–182 (1995)Google Scholar
  9. 9.
    Floater, M.S.: Parametrization and smooth approximation of surface triangulations. Computer Aided Geometric Design 14(231-250) (1997)Google Scholar
  10. 10.
    Floater, M.S.: Parametric tilings and scattered data approximation. International Journal of Shape Modeling 4, 165–182 (1998)CrossRefGoogle Scholar
  11. 11.
    Floater, M.S.: One-to-one piecewise linear mappings over trinagulations. Math. Comp. 72(685-696) (2003)Google Scholar
  12. 12.
    Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Advances in Multiresolution for Geometric Modelling (2005)Google Scholar
  13. 13.
    George, P.-L., Frey, P.: Mesh Generation. Hermes (2000)Google Scholar
  14. 14.
    Geuzaine, C., Remacle, J.-F.: Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering 79(11), 1309–1331 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Greiner, G., Hormann, K.: Interpolating and approximating scattered 3d data with hierarchical tensor product splines. In: Surface Fitting and Multiresolution Methods, pp. 163–172 (1996)Google Scholar
  16. 16.
    Hernandez, V., Roman, J.E., Vidal, V.: Slepc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Soft. 31(3), 351–362 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Hormann, K., Greiner, G.: Mips: An efficient global. parametrization method. In: Curve and Surface Design, Vanderbilt University Press (2000)Google Scholar
  18. 18.
    Ito, Y., Nakahashi, K.: Direct surface triangulation using stereolithography data. AIAA Journal 40(3), 490–496 (2002)CrossRefGoogle Scholar
  19. 19.
    Laug, P., Boruchaki, H.: Interpolating and meshing 3d surface grids. International Journal for Numerical Methods in Engineering 58, 209–225 (2003)CrossRefzbMATHGoogle Scholar
  20. 20.
    Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK Users Guide: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. Society for Industrial and Applied Mathematics (1997)Google Scholar
  21. 21.
    Lévy, B., Petitjean, S., Ray, N., Maillot, J.: Least squares conformal maps for automatic texture atlas generation. In: Computer Graphics (Proceedings of SIGGRAPH 2002), pp. 362–371 (2002)Google Scholar
  22. 22.
    Spagnuolo, M., Attene, M., Falcidieno, B., Wyvill, G.: A mapping-independent primitive for the triangulation of parametric surfaces. Graphical Models 65(260-273) (2003)Google Scholar
  23. 23.
    Maillot, J., Yahia, H., Verroust, A.: Interactive texture mapping. In: Proceedings of ACM SIGGRAPH 1993, pp. 27–34 (1993)Google Scholar
  24. 24.
    Marchandise, E., Carton de Wiart, C., Vos, W.G., Geuzaine, C., Remacle, J.-F.: High quality surface remeshing using harmonic maps. Part II: Surfaces with high genus and of large aspect ratio. International Journal for Numerical Methods in Engineering 86(11), 1303–1321 (2011)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Marchandise, E., Compère, G., Willemet, M., Bricteux, G., Geuzaine, C., Remacle, J.-F.: Quality meshing based on stl triangulations for biomedical simulations. International Journal for Numerical Methods in Biomedical Engineering 83, 876–889 (2010)Google Scholar
  26. 26.
    Marcum, D.L.: Efficient generation of high-quality unstructured surface and volume grids. Engrg. Comput. 17, 211–233 (2001)CrossRefzbMATHGoogle Scholar
  27. 27.
    Marcum, D.L., Gaither, A.: Unstructured surface grid generation using global mapping and physical space approximation. In: Proceedings, 8th International Meshing Roundtable, pp. 397–406 (1999)Google Scholar
  28. 28.
    Mullen, P., Tong, Y., Alliez, P., Desbrun, M.: Spectral conformal parameterization. In: In ACM/EG Symposium of Geometry Processing (2008)Google Scholar
  29. 29.
    Rado, T.: Aufgabe 41. Math-Verien, p. 49 (1926)Google Scholar
  30. 30.
    Rebay, S.: Efficient unstructured mesh generation by means of delaunay triangulation and bowyer-watson algorithm. Journal of Computational Physics 106, 25–138 (1993)CrossRefGoogle Scholar
  31. 31.
    Remacle, J.-F., Henrotte, F., Carrier-Baudouin, T., Bechet, E., Marchandise, E., Geuzaine, C., Mouton, T.: A frontal delaunay quad mesh generator using the l  ∞  norm. International Journal for Numerical Methods in Engineering (2011)Google Scholar
  32. 32.
    Remacle, J.-F., Lambrechts, J., Seny, B., Marchandise, E., Johnen, A., Geuzaine, C.: Blossom-quad: a non-uniform quadrilateral mesh generator using a minimum cost perfect matching algorithm. International Journal for Numerical Methods in Engineering (submitted 2011)Google Scholar
  33. 33.
    Sheffer, A., Praun, E., Rose, K.: Mesh parameterization methods and their applications. Found. Trends. Comput. Graph. Vis. 2(2), 105–171 (2006)CrossRefGoogle Scholar
  34. 34.
    Sheffer, A., Lévy, B., Lorraine, I., Mogilnitsky, M., Bogomyakov, E.: Abf++: fast and robust angle based flattening. ACM Transactions on Graphics 24(311-330) (2005)Google Scholar
  35. 35.
    Szczerba, D., McGregor, R.H.P., Székely, G.: High quality surface mesh generation for multi-physics bio-medical simulations. In: Shi, Y., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds.) ICCS 2007. LNCS, vol. 4487, pp. 906–913. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  36. 36.
    Tristano, J.R., Owen, S.J., Canann, S.A.: Advancing front surface mesh generation in parametric space using riemannian surface definition. In: Proceedings of 7th International Meshing Roundtable, Sandia National Laboratory, pp. 429–455 (1998)Google Scholar
  37. 37.
    Tutte, W.T.: How to draw a graph. In: Proceedings of the London Mathematical Society, vol. 13, pp. 743–768 (1963)Google Scholar
  38. 38.
    Wang, D., Hassan, O., Morgan, K., Weatheril, N.: Enhanced remeshing from stl files with applications to surface grid generation. Commun. Numer. Meth. Engng 23, 227–239 (2007)CrossRefzbMATHGoogle Scholar
  39. 39.
    Zayer, R., Lévy, B., Seidel, H.-P.: Linear angle based parameterization. In: ACM/EG Symposium on Geometry Processing Conference Proceedings (2007)Google Scholar
  40. 40.
    Zheng, Y., Weatherill, N.P., Hassan, O.: Topology abstraction of surface models for three-dimensional grid generation. Engrg. Comput. 17(28-38) (2001)Google Scholar
  41. 41.
    Zigelman, G., Kimmel, R., Kiryati, N.: Texture mapping using surface flattening via multi-dimensional scaling. IEEE Trans. on Visualisation and Computer Graphics 8(198-207) (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Emilie Marchandise
    • 1
  • Jean-François Remacle
    • 1
  • Christophe Geuzaine
    • 2
  1. 1.Institute of Mechanics, Materials and Civil Engineering (iMMC)Université catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Department of Electrical Engineering and Computer Science, Montefiore Institute B28Université de LiègeLiègeBelgium

Personalised recommendations