Abstract
Shewchuk’s Delaunay refinement algorithm is a simple scheme to efficiently tetrahedralize a 3D domain. The original analysis provided guarantees on termination and output edge lengths. However, the guarantees are weak and the time and space complexity are not fully covered. In this paper, we present a new analysis of this algorithm. The new analysis reduces the original 90o requirement for the minimum input dihedral angle to \(\arccos\frac{1}{3} \approx 70.53^o\). The bounds on output edge lengths and vertex degrees are improved. For a set of n input points with spread Δ (the ratio between the longest and shortest pairwise distance), we prove that the number of output points is O(n logΔ). In most cases, this bound is equivalent to O(n logn). This theoretically shows that the output number of tetrahedra is small.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bern, M., Eppstein, D., Gilbert, J.R.: Provably good mesh generation. Journal of Computer and System Sciences 48(3), 384–409 (1994)
Cheng, S.-W., Dey, T., Edelsbrunner, H., Facello, M.A., Teng, S.-H.: Sliver exudation. J. Assoc. Comput. Mach. 47, 883–904 (2000)
Cheng, S.-W., Dey, T.K., Levine, J.A.: A practical Delaunay meshing algorithm for a large class of domains. In: Proc. 16th International Meshing Roundtable, pp. 477–494 (2007)
Cheng, S.-W., Dey, T.K., Ramos, E.A., Ray, T.: Quality meshing for polyhedra with small angles. International Journal on Computational Geometry and Applications 15, 421–461 (2005)
Cheng, S.-W., Poon, S.-H.: Three-dimensional Delaunay mesh generation. Discrete and Computational Geometry 36, 419–456 (2006)
Chew, P.L.: Guaranteed-quality triangular meshes. Technical Report TR 89-983, Dept. of Comp. Sci., Cornell University (1989)
Delaunay, B.N.: Sur la sphère vide. Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk 7, 793–800 (1934)
Erickson, J.: Nice point sets can have nasty Delaunay triangulations. Discrete and Computational Geometry 30(1), 109–132 (2003)
Gabriel, K.R., Sokal, R.R.: A new statistical approach to geographic analysis. Systematic Zoology 18(3), 259–278 (1969)
Hudson, B., Miller, G.L., Phillips, T.: Sparse voronoi refinement. In: Proc. 15th International Meshing Roundtable, pp. 339–356 (2006)
Hudson, B., Miller, G.L., Phillips, T., Sheehy, D.: Size complexity of volume meshes vs. surface meshes. In: 20th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1041–1047 (2009)
Miller, G.L., Phillips, T., Sheehy, D.: Linear-size meshes. In: 20th Canadian Conference on Computational Geometry (2008)
Miller, G.L., Talmor, D., Teng, S.-H., Walkington, N.J., Wang, H.: Control volume meshes using sphere packing: Generation, refinement and coarsening. In: Proc. 5th International Meshing Roundtable (1996)
Mitchell, S.A., Vavasis, S.A.: Quality mesh generation in higher dimensions. SIAM Journal on Computing 29, 1334–1370 (2000)
Oudot, S., Rineau, L., Yvinec, M.: Meshing volumes bounded by smooth surfaces. In: Proc. 14th International Meshing Roundtable, pp. 203–220 (2005)
Pav, S.E.: Delaunay Refinement Algorithm. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania (2003)
Pav, S.E., Walkington, N.J.: Robust three dimensional Delaunay refinement. In: Proc. 13th International Meshing Roundtable (2004)
Ruppert, J.: A Delaunay refinement algorithm for quality 2-dimensional mesh generation. Journal of Algorithms 18(3), 548–585 (1995)
Shewchuk, J.R.: Delaunay Refinement Mesh Generation. PhD thesis, Department of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania (1997)
Shewchuk, J.R.: Pyramid (2006) (Unpublished)
Shewchuk, J.R.: Triangle (2006), http://www.cs.cmu.edu/~quake/triangle.html
Si, H.: TetGen (2007), http://tetgen.berlios.de
Si, H.: Adaptive tetrahedral mesh generation by constrained delaunay refinement. International Journal for Numerical Methods in Engineering 75(7), 856–880 (2008)
Si, H.: Three dimensional boundary conforming Delaunay mesh generation. PhD thesis, Institute für Mathematik, Technische Universität Berlin (2008)
Talmor, D.: Well-spaced points for numerical methods. PhD thesis, Dept. of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Si, H. (2009). An Analysis of Shewchuk’s Delaunay Refinement Algorithm. In: Clark, B.W. (eds) Proceedings of the 18th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04319-2_29
Download citation
DOI: https://doi.org/10.1007/978-3-642-04319-2_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04318-5
Online ISBN: 978-3-642-04319-2
eBook Packages: EngineeringEngineering (R0)