Skip to main content
Log in

Edge Flips in Surface Meshes

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

Little theoretical work has been done on edge flips in surface meshes despite their popular usage in graphics and solid modeling to improve mesh equality. We propose the class of \((\varepsilon ,\alpha )\)-meshes of a surface that satisfy several properties: the vertex set is an \(\varepsilon \)-sample of the surface, the triangle angles are no smaller than a constant \(\alpha \), some triangle has a good normal, and the mesh is homeomorphic to the surface. We believe that many surface meshes encountered in practice are \((\varepsilon ,\alpha )\)-meshes or close to being one. We prove that flipping the appropriate edges can smooth a dense \((\varepsilon ,\alpha )\)-mesh by making the triangle normals better approximations of the surface normals and the dihedral angles closer to \(\pi \). Moreover, the edge flips can be performed in time linear in the number of vertices. This helps to explain the effectiveness of edge flips as observed in practice and in our experiments. A corollary of our techniques is that, in \(\mathbb {R}^2\), every triangulation with a constant lower bound on the angles can be flipped in linear time to the Delaunay triangulation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30

Similar content being viewed by others

Notes

  1. In \(\mathbb {R}^2\), if \(pq\) is flippable, condition (iv) of Definition 3 forces \(pqrs\) to be a convex quadrilateral and conditions (i, ii) of Definition 3 force \(r\) to be inside the circumcircle of \(pqs\). That is, Definition 3 and the empty circumcircle criterion are equivalent in \(\mathbb {R}^2\).

  2. If \(\sigma \) and \(\tau \) are coplanar, then \(H\) must be perpendicular to the support plane of \(\sigma \) and \(\tau \).

References

  1. Alliez, P., Meyer, M., Desbrun, M.: Interactive geometry remeshing. ACM Trans. Graph. 21, 347–354 (2002)

    Article  Google Scholar 

  2. Amenta, N., Bern, M.: Surface reconstruction by Voronoi filtering. Discrete Comput. Geom. 22(4), 481–504 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Amenta, N., Choi, S., Dey, T.K., Leekha, N.: A simple algorithm for homeomorphic surface reconstruction. Int. J. Comput. Geom. Appl. 12, 125–141 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Aurenhammer, F., Klein, R.: Voronoi diagrams. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 201–290. Elsevier, Amsterdam (2000)

    Chapter  Google Scholar 

  5. Boissonnat, J.-D., Cazals, F.: Natural neighbor coordinates of points on a surface. Comput. Geom. 19, 87–120 (2001)

    Article  MathSciNet  Google Scholar 

  6. Cheng, S.-W., Dey, T.K.: Maintaining deforming surface meshes. In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 112–121. Society for Industrial and Applied Mathematics, Philadelphia (2008)

  7. Cheng, S.-W., Jin, J.: Edge flips and deforming surface meshes. In: Proceedings of the 27th Annual Symposium on Computational Geometry, pp. 331–340. ACM Press, New York (2011)

  8. Cheng, S.-W., Jin, J., Lau, M.-K.: A fast and simple surface reconstruction algorithm. Proceedings of the 28th Annual Symposium on Computational Geometry, pp. 69–78. University of North Carolina, Chapel Hill (2012)

  9. Dey, T.K.: Curve and Surface Reconstruction: Algorithms with Mathematical Analysis. Cambridge University Press, New York (2006)

    Book  Google Scholar 

  10. Dey, T.K., Funke, S., Ramos, E.A.: Surface reconstruction in almost linear time under locally uniform sampling. In: European Workshop on Computational Geometry. Springer, Berlin (2001)

  11. Dyn, N., Hormann, K., Kim, S.J., Levin, D.: Optimizing 3D Triangulations Using Discrete Curvature Analysis, pp. 135–146. Vanderbilt University, Nashville (2001)

    Google Scholar 

  12. Enright, D., Fedkiw, R., Ferziger, J., Mitchell, I.: A hybrid particle level set method for improved interface capturing. J. Comput. Phys. 183, 83–116 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Funke, S., Ramos, E.A.: Smooth-surface reconstruction in near-linear time. In: Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 781–790. Society for Industrial and Applied Mathematics, New York (2002)

  14. Galaktionov, O.S., Anderson, P.D., Peters, G.W.M., Van de Vosse, F.N.: An adaptive front tracking technique for three-dimensional transient flows. Int. J. Numer. Methods Fluids 32, 201–217 (2000)

    Article  MATH  Google Scholar 

  15. Giesen, J., Wagner, U.: Shape dimension and intrinsic metric from samples of manifolds. Discrete Comput. Geom. 32(2), 245–267 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hurtado, F., Noy, M., Urrutia, J.: Flipping edges in triangulations. Discrete Comput. Geom. 22(3), 333–346 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. LeVeque, R.J.: High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 33, 627–665 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Surazhsky, V., Gotsman, C.: Explicit surface remeshing. In: Proceedings of the Eurogrpahics Symposium on Geometry Processing, pp. 20–30. Eurographics Association, Aire-la-Ville (2005)

Download references

Acknowledgments

We would like to thank the anonymous reviewers for their helpful comments. We also thank Man-Kit Lau for helping us to generate Fig. 3. This research was supported by Research Grant Council, Hong Kong, China (612109). The work was done while Jin was at the Department of Computer Science and Engineering, HKUST.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Siu-Wing Cheng.

Additional information

Editor in Charge: Herbert Edelsbrunner

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, SW., Jin, J. Edge Flips in Surface Meshes. Discrete Comput Geom 54, 110–151 (2015). https://doi.org/10.1007/s00454-015-9693-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-015-9693-y

Keywords

Mathematics Subject Classification

Navigation