The Topological Correctness of PL-Approximations of Isomanifolds

Authors Jean-Daniel Boissonnat, Mathijs Wintraecken

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Jean-Daniel Boissonnat
  • Université Côte d'Azur, INRIA, Sophia-Antipolis, France
Mathijs Wintraecken
  • IST Austria, Klosterneuburg, Austria


First and foremost, we acknowledge Siargey Kachanovich for discussions. We thank Herbert Edelsbrunner and all members of his group, all former and current members of the Datashape team (formerly known as Geometrica), and André Lieutier for encouragement. We thank the reviewers for their comments which improved the exposition.

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Jean-Daniel Boissonnat and Mathijs Wintraecken. The Topological Correctness of PL-Approximations of Isomanifolds. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f: ℝ^d → ℝ^(d-n). A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation 𝒯. This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • PL-approximations
  • isomanifolds
  • solution manifolds
  • topological correctness


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  1. Eugene L. Allgower and Kurt Georg. Simplicial and continuation methods for approximating fixed points and solutions to systems of equations. Siam review, 22(1):28-85, 1980. Google Scholar
  2. Eugene L. Allgower and Kurt Georg. Estimates for piecewise linear approximations of implicitly defined manifolds. Applied Mathematics Letters, 2(2):111-115, 1989. Google Scholar
  3. Eugene L. Allgower and Kurt Georg. Numerical continuation methods: an introduction, volume 13. Springer Science & Business Media, 1990. Google Scholar
  4. Dominique Attali and André Lieutier. Geometry-driven collapses for converting a Čech complex into a triangulation of a nicely triangulable shape. Discrete Comput. Geom., 54(4):798-825, December 2015. URL:
  5. P. Bendich, S. Mukherjee, and B. Wang. Stratification learning through homology inference. In 2010 AAAI Fall Symposium Series, 2010. Google Scholar
  6. Paul Bendich, David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Dmitriy Morozov. Inferring local homology from sampled stratified spaces. In Proc. of the IEEE Symp. on Foundations of Computer Science, pages 536-546, 2007. Google Scholar
  7. J.-D. Boissonnat, R. Dyer, and A. Ghosh. The Stability of Delaunay Triangulations. International Journal of Computional Geometry & Applications, 23(4-5):303-334, 2013. URL:
  8. J-D. Boissonnat, M. Rouxel-Labbé, and M. Wintraecken. Anisotropic triangulations via discrete Riemannian Voronoi diagrams. SIAM Journal on Computing, 48(3):1046-1097, 2019. URL:
  9. Jean-Daniel Boissonnat, Frédéric Chazal, and Mariette Yvinec. Geometric and Topological Inference. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2018. URL:
  10. Jean-Daniel Boissonnat, David Cohen-Steiner, and Gert Vegter. Isotopic implicit surface meshing. Discrete & Computational Geometry, 39(1):138-157, March 2008. URL:
  11. Jean-Daniel Boissonnat, Ramsay Dyer, and Arijit Ghosh. Delaunay stability via perturbations. International Journal of Computational Geometry & Applications, 24(02):125-152, 2014. Google Scholar
  12. Jean-Daniel Boissonnat, Ramsay Dyer, and Arijit Ghosh. Delaunay Triangulation of Manifolds. Foundations of Computational Mathematics, 45:38, 2017. URL:
  13. Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh, André Lieutier, and Mathijs Wintraecken. Local conditions for triangulating submanifolds of Euclidean space. Preprint, July 2019. URL:
  14. Jean-Daniel Boissonnat and Arijit Ghosh. Manifold reconstruction using tangential Delaunay complexes. Discrete & Computational Geometry, 51(1):221-267, 2014. Google Scholar
  15. Jean-Daniel Boissonnat, Siargey Kachanovich, and Mathijs Wintraecken. Triangulating submanifolds: An elementary and quantified version of Whitney’s method. Preprint, December 2018. URL:
  16. Jean-Daniel Boissonnat, Siargey Kachanovich, and Mathijs Wintraecken. Sampling and Meshing Submanifolds in High Dimension. Preprint, November 2019. URL:
  17. Jean-Daniel Boissonnat and Mathijs Wintraecken. The topological correctness of PL-approximations of isomanifolds. In 36th International Symposium on Computational Geometry, SoCG 2020, Zurich, Switzerland, 2020. Google Scholar
  18. L. E. J. Brouwer. Über Abbildung von Mannigfaltigkeiten. Mathematische Annalen, 71(4):598-598, December 1912. URL:
  19. Yen-Chi Chen. Solution manifold and its statistical applications, 2020. URL:
  20. S-W. Cheng, T. K. Dey, and E. A. Ramos. Manifold reconstruction from point samples. In Proc. 16th ACM-SIAM Symp. Discrete Algorithms, pages 1018-1027, 2005. Google Scholar
  21. S.-W. Cheng, T. K. Dey, and J. R. Shewchuk. Delaunay Mesh Generation. Computer and information science series. CRC Press, 2013. Google Scholar
  22. Aruni Choudhary, Siargey Kachanovich, and Mathijs Wintraecken. Coxeter triangulations have good quality. Preprint, December 2017. URL:
  23. Frank H. Clarke. Optimization and Nonsmooth Analysis, volume 5 of Classics in applied mathematics. SIAM, 1990. Google Scholar
  24. Harold S. M. Coxeter. Discrete groups generated by reflections. Annals of Mathematics, pages 588-621, 1934. Google Scholar
  25. T. K. Dey. Curve and Surface Reconstruction; Algorithms with Mathematical Analysis. Cambridge University Press, 2007. Google Scholar
  26. Tamal K. Dey and Joshua A. Levine. Delaunay meshing of piecewise smooth complexes without expensive predicates. Algorithms, 2(4):1327-1349, 2009. URL:
  27. Tamal K. Dey and Andrew G. Slatton. Localized Delaunay refinement for volumes. Computer Graphics Forum, 30(5):1417-1426, 2011. URL:
  28. Tamal Krishna Dey and Andrew G. Slatton. Localized Delaunay refinement for piecewise-smooth complexes. In Guilherme Dias da Fonseca, Thomas Lewiner, Luis Mariano Pe~naranda, Timothy M. Chan, and Rolf Klein, editors, Symposium on Computational Geometry 2013, SoCG '13, Rio de Janeiro, Brazil, June 17-20, 2013, pages 47-56. ACM, 2013. URL:
  29. Akio Doi and Akio Koide. An efficient method of triangulating equi-valued surfaces by using tetrahedral cells. IEICE TRANSACTIONS on Information and Systems, E74-D, 1991. Google Scholar
  30. J. J. Duistermaat and J. A. C. Kolk. Multidimensional Real Analysis I: Differentiation. Number 86 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2004. Google Scholar
  31. 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
  32. B. Curtis Eaves. A course in triangulations for solving equations with deformations, volume 234. Lecture Notes in Economics and Mathematical Systems, 1984. Google Scholar
  33. Herbert Edelsbrunner and Nimish R. Shah. Triangulating topological spaces. International Journal of Computational Geometry & Applications, 7(04):365-378, 1997. Google Scholar
  34. Hans Freudenthal. Simplizialzerlegungen von beschrankter flachheit. Annals of Mathematics, pages 580-582, 1942. Google Scholar
  35. M. W. Hirsch. Differential Topology. Springer-Verlag, 1976. Google Scholar
  36. Harold W. Kuhn. Some combinatorial lemmas in topology. IBM Journal of research and development, 4(5):518-524, 1960. Google Scholar
  37. William E. Lorensen and Harvey E. Cline. Marching cubes: A high resolution 3d surface construction algorithm. In ACM siggraph computer graphics, volume 21, pages 163-169. ACM, 1987. Google Scholar
  38. J. Milnor. Morse Theory. Princeton University Press, 1969. Google Scholar
  39. John Milnor. Lectures on the H-Cobordism Theorem. Princeton University Press, 1965. URL:
  40. Chohong Min. Simplicial isosurfacing in arbitrary dimension and codimension. Journal of Computational Physics, 190(1):295-310, 2003. Google Scholar
  41. Timothy S. Newman and Hong Yi. A survey of the marching cubes algorithm. Computers & Graphics, 30(5):854-879, 2006. URL:
  42. Timothy S Newman and Hong Yi. A survey of the marching cubes algorithm. Computers & Graphics, 30(5):854-879, 2006. Google Scholar
  43. Steve Oudot, Laurent Rineau, and Mariette Yvinec. Meshing Volumes Bounded by Smooth Surfaces. Computational Geometry, Theory and Applications, 38:100-110, 2007. URL:
  44. Simon Plantinga and Gert Vegter. Isotopic approximation of implicit curves and surfaces. In Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing, pages 245-254. ACM, 2004. Google Scholar
  45. Laurent Rineau. Meshing Volumes bounded by Piecewise Smooth Surfaces. Theses, Université Paris-Diderot - Paris VII, November 2007. URL:
  46. Mael Rouxel-Labbé, Mathijs Wintraecken, and Jean-Daniel Boissonnat. Discretized Riemannian Delaunay Triangulations. Research Report RR-9103, INRIA Sophia Antipolis - Méditerranée, October 2017. URL:
  47. Jennifer Schultens. Introduction to 3-manifolds, volume 151. American Mathematical Soc., 2014. Google Scholar
  48. Jonathan Richard Shewchuk. Lecture notes on Delaunay mesh generation, 1999. Google Scholar
  49. Michael J. Todd. The computation of fixed points and applications, volume 124. Lecture Notes in Economics and Mathematical Systems, 1976. Google Scholar
  50. H. Whitney. Geometric Integration Theory. Princeton University Press, 1957. Google Scholar
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