The Reach, Metric Distortion, Geodesic Convexity and the Variation of Tangent Spaces

Authors Jean-Daniel Boissonnat, André Lieutier, Mathijs Wintraecken

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Jean-Daniel Boissonnat
André Lieutier
Mathijs Wintraecken

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Jean-Daniel Boissonnat, André Lieutier, and Mathijs Wintraecken. The Reach, Metric Distortion, Geodesic Convexity and the Variation of Tangent Spaces. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


In this paper we discuss three results. The first two concern general sets of positive reach: We first characterize the reach by means of a bound on the metric distortion between the distance in the ambient Euclidean space and the set of positive reach. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the distance between the points and the reach.
  • Reach
  • Metric distortion
  • Manifolds
  • Convexity


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  1. N. Amenta and M. W. Bern. Surface reconstruction by Voronoi filtering. In SoCG, pages 39-48, 1998. URL:
  2. D. Attali, H. Edelsbrunner, and Yu. Mileyko. Weak Witnesses for Delaunay triangulations of Submanifold. In ACM Symposium on Solid and Physical Modeling, pages 143-150, Beijing, China, 2007. URL:
  3. D. Attali and A. Lieutier. Geometry-driven collapses for converting a Čech complex into a triangulation of a nicely triangulable shape. Discrete &Computational Geometry, 54(4):798-825, 2015. Google Scholar
  4. M. Belkin, J. Sun, and Y. Wang. Constructing laplace operator from point clouds in ℝ^d. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1031-1040, 2009. URL:
  5. J.-D. Boissonnat and F. Cazals. Natural neighbor coordinates of points on a surface. Computational Geometry Theory &Applications, 19(2-3):155-173, Jul 2001. URL:
  6. J.-D. Boissonnat, R. Dyer, and A. Ghosh. Constructing intrinsic Delaunay triangulations of submanifolds. Research Report RR-8273, INRIA, 2013. arXiv:1303.6493. URL:
  7. J.-D. Boissonnat, R. Dyer, A. Ghosh, and M.H.M.J. Wintraecken. Local criteria for triangulation of manifolds. Accepted for SoCG 2018, 2018. URL:
  8. J.-D. Boissonnat and A. Ghosh. Triangulating smooth submanifolds with light scaffolding. Mathematics in Computer Science, 4(4):431-461, 2010. Google Scholar
  9. J.-D. Boissonnat and S. Oudot. Provably good surface sampling and approximation. In Symp. Geometry Processing, pages 9-18, 2003. Google Scholar
  10. Jean-Daniel Boissonnat, André Lieutier, and Mathijs Wintraecken. The reach, metric distortion, geodesic convexity and the variation of tangent spaces. full version, 2017. URL:
  11. S.-W. Cheng, T. K. Dey, and E. A. Ramos. Manifold reconstruction from point samples. In SODA, pages 1018-1027, 2005. Google Scholar
  12. T. K. Dey. Curve and Surface Reconstruction: Algorithms with Mathematical Analysis (Cambridge Monographs on Applied and Computational Mathematics). Cambridge University Press, New York, NY, USA, 2006. Google Scholar
  13. T.K. Dey, J. Giesen, E.A. Ramos, and B. Sadri. Critical points of distance to an ε-sampling of a surface and flow-complex-based surface reconstruction. International Journal of Computational Geometry &Applications, 18(01n02):29-61, 2008. URL:
  14. M. P. do Carmo. Riemannian Geometry. Birkhäuser, 1992. Google Scholar
  15. H. Federer. Curvature measures. Trans. Amer. Math. Soc., 93(3):418-491, 1959. Google Scholar
  16. M. Gromov, M. Katz, P. Pansu, and S. Semmes. Metric structures for Riemannian and non-Riemannian spaces. Birkhauser, 2007. Google Scholar
  17. K. Menger. Untersuchungen uber allgemeine metrik, vierte untersuchungen zur metrik kurven. Mathematische Annalen, 103:466-501, 1930. Google Scholar
  18. J. Milnor. Morse Theory. Cambridge, 2006. Google Scholar
  19. P. Niyogi, S. Smale, and S. Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete &Comp. Geom., 39(1-3):419-441, 2008. Google Scholar
  20. S. Scholtes. On hypersurfaces of positive reach, alternating Steiner formulae and Hadwiger’s Problem. ArXiv e-prints, 2013. URL:
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