Riemannian Simplices and Triangulations

Authors Ramsay Dyer, Gert Vegter, Mathijs Wintraecken

Thumbnail PDF


  • Filesize: 497 kB
  • 15 pages

Document Identifiers

Author Details

Ramsay Dyer
Gert Vegter
Mathijs Wintraecken

Cite AsGet BibTex

Ramsay Dyer, Gert Vegter, and Mathijs Wintraecken. Riemannian Simplices and Triangulations. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 255-269, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary finite dimension, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the weighted sum of squared distances to the vertices is the Karcher mean relative to the weights. Using barycentric coordinates as the weights, we obtain a smooth map from the standard Euclidean simplex to the manifold. A Riemannian simplex is defined as the image of the standard simplex under this barycentric coordinate map. In this work we articulate criteria that guarantee that the barycentric coordinate map is a smooth embedding. If it is not, we say the Riemannian simplex is degenerate. Quality measures for the "thickness" or "fatness" of Euclidean simplices can be adapted to apply to these Riemannian simplices. For manifolds of dimension 2, the simplex is non-degenerate if it has a positive quality measure, as in the Euclidean case. However, when the dimension is greater than two, non-degeneracy can be guaranteed only when the quality exceeds a positive bound that depends on the size of the simplex and local bounds on the absolute values of the sectional curvatures of the manifold. An analysis of the geometry of non-degenerate Riemannian simplices leads to conditions which guarantee that a simplicial complex is homeomorphic to the manifold.
  • Karcher means
  • barycentric coordinates
  • triangulation
  • Riemannian manifold
  • Riemannian simplices


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. M. Berger. A Panoramic View of Riemannian Geometry. Springer-Verlag, 2003. Google Scholar
  2. J.-D. Boissonnat, R. Dyer, and A. Ghosh. Delaunay triangulation of manifolds. Research Report RR-8389, INRIA, 2013. (also: arXiv:1311.0117). Google Scholar
  3. J.-D. Boissonnat, R. Dyer, and A. Ghosh. The stability of Delaunay triangulations. IJCGA, 23(04n05):303-333, 2013. (Preprint: arXiv:1304.2947). Google Scholar
  4. J.-D. Boissonnat and A. Ghosh. Manifold reconstruction using tangential Delaunay complexes. Discrete and Computational Geometry, 51(1):221-267, 2014. Google Scholar
  5. P. Buser and H. Karcher. Gromov’s almost flat manifolds, volume 81 of Astérique. Société mathématique de France, 1981. Google Scholar
  6. S. S. Cairns. On the triangulation of regular loci. Annals of Mathematics. Second Series, 35(3):579-587, 1934. Google Scholar
  7. S.-W. Cheng, T. K. Dey, H. Edelsbrunner, M. A. Facello, and S. H Teng. Sliver exudation. Journal of the ACM, 47(5):883-904, 2000. Google Scholar
  8. S.-W. Cheng, T. K. Dey, and E. A. Ramos. Manifold reconstruction from point samples. In SODA, pages 1018-1027, 2005. Google Scholar
  9. R. Dyer, G. Vegter, and M. Wintraecken. Riemannian simplices and triangulations. Geometriae Dedicata, 2015. To appear. (Preprint: arXiv:1406.3740). Google Scholar
  10. H. Karcher. Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics, 30:509-541, 1977. Google Scholar
  11. F. Labelle and J. R. Shewchuk. Isosurface stuffing: Fast tetrahedral meshes with good dihedral angles. ACM Trans. Graph., 26(3), 2007. Google Scholar
  12. J. R. Munkres. Elementary differential topology. Princton University press, second edition, 1968. Google Scholar
  13. R.M. Rustamov. Barycentric coordinates on surfaces. Eurographics Symposium of Geometry Processing, 29(5), 2010. Google Scholar
  14. O. Sander. Geodesic finite elements on simplicial grids. International Journal for Numerical Methods in Engineering, 92(12):999-1025, 2012. Google Scholar
  15. W. P. Thurston. Three-Dimensional Geometry and Topology. Princeton University Press, 1997. Google Scholar
  16. S. W. von Deylen. Numerische Approximation in Riemannschen Mannigfaltigkeiten mithilfe des Karcher’schen Schwerpunktes. PhD thesis, Freie Universität Berlin, 2014 (to appear). Google Scholar
  17. J. H. C. Whitehead. On C¹-complexes. Annals of Mathematics, 41(4), 1940. Google Scholar
  18. H. Whitney. Geometric Integration Theory. Princeton University Press, 1957. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail