Quasiisothermic Mesh Layout

  • Stefan Sechelmann
  • Thilo Rörig
  • Alexander I. Bobenko
Conference paper

Abstract

The quality of a quad-mesh depends on the shape of the individual quadrilaterals. The ideal shape from an architectural point of view is the planar square or rectangles with fixed aspect ratio. A parameterization that divides a surface into such shapes is called isothermic, i.e., angle-preserving and curvature-aligned. Such a parameterization exists only for the special class of isothermic surfaces. We extend this notion and introduce quasiisothermic parameterizations for arbitrary triangulated surfaces.

We describe an algorithm that creates quasiisothermic meshes. Interestingly many surfaces appearing in architecture are close to isothermic surfaces, namely those coming from form finding methods and physical simulation. For those surfaces our method works particularly well and gives a high quality and robust mesh layout. We show how to optimize such meshes further to obtain disk packing representations. The quadrilaterals of these meshes are planar and possess touching incircles.

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References

  1. ALLIEZ, P., COHEN-STEINER, D., DEVILLERS, O., LÉVY, B., AND DESBRUN, M. 2003. Anisotropic polygonal remeshing. ACM Trans. Graph. 22, 3, 485–493.CrossRefGoogle Scholar
  2. BALAY, S., BROWN, J., BUSCHELMAN, K., GROPP, W. D., KAUSHIK, D., KNEPLEY, M. G., MCINNES, L. C., SMITH, B. F., AND ZHANG, H., 2011. PETSc Web page. http://www.mcs.anl.gov/petsc.Google Scholar
  3. BENSON, S., MCINNES, L. C., MORÉ, J., MUNSON, T., AND SARICH, J., 2007. TAO user manual (revision 1.9). http://www.mcs.anl.gov/tao.Google Scholar
  4. BOBENKO, A. I., AND PINKALL, U. 1999. Discretization of surfaces and integrable systems. In Discrete integrable geometry and physics (Vienna, 1996), vol. 16 of Oxford Lecture Ser. Math. Appl. Oxford Univ. Press, New York, 3–58.Google Scholar
  5. BOBENKO, A. I., AND SURIS, Y. B. 2008. Discrete differential geometry — Integrable structure, vol. 98 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI.Google Scholar
  6. BOBENKO, A. I., HOFFMANN, T., AND SPRINGBORN, B. 2006. Minimal surfaces from circle patterns: geometry from combinatorics. Ann. of Math. (2) 164, 1, 231–264.CrossRefGoogle Scholar
  7. BOBENKO, A. I., PINKALL, U., AND SPRINGBORN, B. 2010. Discrete conformal maps and ideal hyperbolic polyhedra. Preprint; http://arxiv.org/abs/1005.2698.Google Scholar
  8. COHEN-STEINER, D., AND MORVAN, J.-M. 2003. Restricted Delaunay triangulations and normal cycle. In Symposium on Computational Geometry, 312–321.Google Scholar
  9. FLOATER, M. S., AND HORMANN, K. 2005. Surface parameterization: a tutorial and survey. In Advances in Multiresolution for Geometric Modelling, N. A. Dodgson, M. S. Floater, and M. A. Sabin, Eds., Mathematics and Visualization. Springer, Berlin, Heidelberg, 157–186.CrossRefGoogle Scholar
  10. KÄLBERER, F., NIESER, M., AND POLTHIER, K. 2007. Quadcover — surface parameterization using branched coverings. Comput. Graph. Forum 26, 3, 375–384.CrossRefGoogle Scholar
  11. LIU, Y., POTTMANN, H., WALLNER, J., YANG, Y.-L., AND WANG, W. 2006. Geometric modeling with conical meshes and developable surfaces. ACM Trans. Graph. 25, 3, 681–689.CrossRefGoogle Scholar
  12. POTTMANN, H., LIU, Y., WALLNER, J., BOBENKO, A., AND WANG, W. 2007. Geometry of multi-layer freeform structures for architecture. ACM Trans. Graphics 26, 3, #65, 1–11. Proc. SIGGRAPH.CrossRefGoogle Scholar
  13. POTTMANN, H., SCHIFTNER, A., BO, P., SCHMIEDHOFER, H., WANG, W., BALDASSINI, N., AND WALLNER, J. 2008. Freeform surfaces from single curved panels. ACM Trans. Graph. 27, 3, #76, 1–10.CrossRefGoogle Scholar
  14. SCHIFTNER, A., HÖBINGER, M., WALLNER, J., AND POTTMANN, H. 2009. Packing circles and spheres on surfaces. ACM Trans. Graphics 28, 5, #139, 1–8. Proc. SIGGRAPH Asia.CrossRefGoogle Scholar
  15. SHEFFER, A., AND DE STURLER, E. 2001. Parameterization of Faceted Surfaces for Meshing using Angle-Based Flattening. Engineering with Computers 17, 326–337.CrossRefGoogle Scholar
  16. SHEFFER, A., PRAUN, E., AND ROSE, K. 2006. Mesh parameterization methods and their applications. Foundations and Trends in Computer Graphics and Vision 2, 2, 105–171.CrossRefGoogle Scholar
  17. SOMMER, H., 2010. jPETScTao JNI library web page. http://jpetsctao.zwoggel.net/.Google Scholar
  18. SPRINGBORN, B., SCHRÖDER, P., AND PINKALL, U. 2008. Conformal equivalence of triangle meshes. ACM Trans. Graph. 27, 3, #77, 1–11.CrossRefGoogle Scholar
  19. ZADRAVEC, M., SCHIFTNER, A., AND WALLNER, J. 2010. Designing quad-dominant meshes with planar faces. Computer Graphics Forum 29, 5, 1671–1679. Proc. Symp. Geometry Processing.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag/Wien 2013

Authors and Affiliations

  • Stefan Sechelmann
    • 1
  • Thilo Rörig
    • 1
  • Alexander I. Bobenko
    • 1
  1. 1.Institut für MathematikTechnische Universität BerlinGermany

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