Local Criteria for Triangulation of Manifolds

Authors Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh, Mathijs Wintraecken

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Jean-Daniel Boissonnat
Ramsay Dyer
Arijit Ghosh
Mathijs Wintraecken

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Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh, and Mathijs Wintraecken. Local Criteria for Triangulation of Manifolds. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex A, and a map H from the underlying space of A to M, our criteria are presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on M. No Delaunay property of A is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily verified in such a setting, they are expected to be of general use.
  • manifold
  • simplicial complex
  • homeomorphism
  • triangulation


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  1. N. Amenta and M. Bern. Surface reconstruction by Voronoi filtering. Discrete and Computational Geometry, 22(4):481-504, 1999. Google Scholar
  2. N. Amenta, S. Choi, T. K. Dey, and N. Leekha. A simple algorithm for homeomorphic surface reconstruction. Int. J. Computational Geometry and Applications, 12(2):125-141, 2002. Google Scholar
  3. J.-D. Boissonnat, R. Dyer, and A. Ghosh. The stability of Delaunay triangulations. International Journal of Computational Geometry &Applications, 23(4-5):303-333, 2013. (arXiv:1304.2947). Google Scholar
  4. J.-D. Boissonnat, R. Dyer, and A. Ghosh. Delaunay triangulation of manifolds. Foundations of Computational Mathematics, 2017. (arXiv:1311.0117). Google Scholar
  5. J.-D. Boissonnat, R. Dyer, A. Ghosh, and M. Wintraecken. Local criteria for triangulation of manifolds. Technical Report 1803.07642, arXiv, 2017. URL: http://arxiv.org/abs/1803.07642.
  6. J.-D. Boissonnat and A. Ghosh. Manifold reconstruction using tangential Delaunay complexes. Discrete and Computational Geometry, 51(1):221-267, 2014. Google Scholar
  7. J.-D. Boissonnat, A. Lieutier, and M. Wintraecken. The reach, metric distortion, geodesic convexity and the variation of tangent spaces. Technical Report hal-01661227, Inria, Sophia-Antipolis, 2017. Accepted for SoCG 2018. URL: https://hal.inria.fr/hal-01661227.
  8. J.-D. Boissonnat and S. Oudot. Provably good sampling and meshing of surfaces. Graphical Models, 67(5):405-451, 2005. Google Scholar
  9. S. S. Cairns. On the triangulation of regular loci. Annals of Mathematics. Second Series, 35(3):579-587, 1934. Google Scholar
  10. S.-W. Cheng, T. K. Dey, and E. A. Ramos. Manifold reconstruction from point samples. In SODA, pages 1018-1027, 2005. Google Scholar
  11. R. Dyer, G. Vegter, and M. Wintraecken. Riemannian simplices and triangulations. Geometriae Dedicata, 179:91-138, 2015. Google Scholar
  12. R. Dyer, H. Zhang, and T. Möller. Surface sampling and the intrinsic Voronoi diagram. Computer Graphics Forum (Special Issue of Symp. Geometry Processing), 27(5):1393-1402, 2008. Google Scholar
  13. H. Edelsbrunner and N. R. Shah. Triangulating topological spaces. Int. J. Comput. Geometry Appl., 7(4):365-378, 1997. Google Scholar
  14. J. R. Munkres. Elementary differential topology. Princton University press, second edition, 1968. Google Scholar
  15. J. H. C. Whitehead. On C¹-complexes. Ann. of Math, 41(4):809-824, 1940. Google Scholar
  16. H. Whitney. Geometric Integration Theory. Princeton University Press, 1957. Google Scholar
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