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Proceedings of the 21st International Meshing Roundtable pp 261–277Cite as

Diamond-Kite Meshes: Adaptive Quadrilateral Meshing and Orthogonal Circle Packing

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Summary

We describe a family of quadrilateral meshes based on diamonds, rhombi with 60° and 120° angles, and kites with 60°, 90°, and 120° angles, that can be adapted to a local size function by local subdivision operations. The vertices of our meshes form the centers of the circles in a pair of dual circle packings in which each tangency between two circles is crossed orthogonally by a tangency between two dual circles.

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References

  1. Andreev, E.M.: Convex polyhedra in Lobačevskiĭ spaces. Mat. Sb (N.S.) 81(123), 445–478 (1970)

    MathSciNet  Google Scholar 

  2. Andreev, E.M.: Convex polyhedra of finite volume in Lobačevskiĭ space. Mat. Sb (N.S.) 83(125), 256–260 (1970)

    MathSciNet  Google Scholar 

  3. Benantar, M., Flaherty, J.E., Krishnamoorthy, M.S.: Coloring procedures for finite element computation on shared-memory parallel computers. In: Proc. Symp. on Adaptive, Multilevel, and Hierarchical Computation Strategies, ASME, AMD 157 (1992)

    Google Scholar 

  4. Bern, M., Eppstein, D.: Quadrilateral meshing by circle packing. In: Proc. 6th International Meshing Roundtable, pp. 7–20. Sandia National Laboratories (1997), arXiv:cs.CG/9908016, http://www.imr.sandia.gov/papers/abstracts/Be49.html

  5. Bern, M., Eppstein, D., Gilbert, J.: Provably good mesh generation. J. Computer & Systems Sciences 48(3), 384–409 (1994), doi:10.1016/S0022-0000(05)80059-5

    Article  MathSciNet  MATH  Google Scholar 

  6. Bern, M., Eppstein, D., Hutchings, B.: Algorithms for coloring quadtrees. Algorithmica 32(1), 87–94 (2002), arXiv:cs.CG/9907030, doi:10.1007/s00453-001-0054-2

    Google Scholar 

  7. Bern, M., Mitchell, S.A., Ruppert, J.: Linear-size nonobtuse triangulation of polygons. Discrete & Computational Geometry 14, 411–428 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Birkhoff, G.: Rings of sets. Duke Mathematical Journal 3(3), 443–454 (1937), doi:10.1215/S0012-7094-37-00334-X

    Article  MathSciNet  Google Scholar 

  9. Bowers, P.L., Stephenson, K.: Uniformizing dessins and Belyĭ maps via circle packing. Memoirs of the American Mathematical Society, vol. 805. American Mathematical Society (2004)

    Google Scholar 

  10. Brightwell, G.R., Scheinerman, E.R.: Representations of planar graphs. SIAM Journal on Discrete Mathematics 6(2), 214–229 (1993), doi:10.1137/0406017

    Article  MathSciNet  MATH  Google Scholar 

  11. Conway, J.H., Burgiel, H., Goodman-Strass, C.: Chapter 21: Naming Archimedean and Catalan polyhedra and tilings. In: The Symmetries of Things. A.K. Peters (2008)

    Google Scholar 

  12. Dyn, N., Levin, D., Liu, D.: Interpolatory convexity-preserving subdivision schemes for curves and surfaces. Computer-Aided Design 24(4), 211–216 (1992), doi:10.1016/0010-4485(92)90057-H

    Article  MATH  Google Scholar 

  13. Eppstein, D.: Planar Lombardi drawings for subcubic graphs. To appear in Proc. Int. Symp. Graph Drawing (GD 2012) (2012), arXiv:1206.6142

    Google Scholar 

  14. Eppstein, D.: Approximating the minimum weight Steiner triangulation. Discrete & Computational Geometry 11(2), 163–191 (1994), doi:10.1007/BF02574002

    Article  MathSciNet  MATH  Google Scholar 

  15. Eppstein, D.: Faster circle packing with application to nonobtuse triangulation. Internat. J. Comput. Geom. Appl. 7(5), 485–491 (1997), doi:10.1142/S0218195997000296

    Article  MathSciNet  MATH  Google Scholar 

  16. Eppstein, D.: The graphs of planar soap bubbles (submitted, 2012), arXiv:1207.3761

    Google Scholar 

  17. Eppstein, D., Miller, G.L., Teng, S.-H.: A deterministic linear time algorithm for geometric separators and its applications. Fundamenta Informaticae 22(4), 309–331 (1995), doi:10.3233/FI-1995-2241

    MathSciNet  MATH  Google Scholar 

  18. Frey, P.J., Maréchal, L.: Fast adaptive quadtree mesh generation. In: Proc. 7th International Meshing Roundtable, pp. 211–224. Sandia National Laboratories (1998), http://www.imr.sandia.gov/papers/abstracts/Fr104.html

  19. He, Z.-X.: Solving Beltrami equations by circle packing. Transactions of the American Mathematical Society 322(2), 657–670 (1990), doi:10.2307/2001719

    Article  MathSciNet  MATH  Google Scholar 

  20. Hurdal, M.K., Bowers, P.L., Stephenson, K., Sumners, D.W.L., Rehm, K., Schaper, K., Rottenberg, D.A.: Quasi-Conformally Flat Mapping the Human Cerebellum. In: Taylor, C., Colchester, A. (eds.) MICCAI 1999. LNCS, vol. 1679, pp. 279–286. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  21. Keszegh, B., Pach, J., Pálvölgyi, D.: Drawing Planar Graphs of Bounded Degree with Few Slopes. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 293–304. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  22. Kobbelt, L.: \(\sqrt3\)-subdivision. In: Proc. 27th Conf. Computer Graphics and Interactive Techniques (SIGGRAPH 2000), pp. 103–112 (2000), doi:10.1145/344779.344835

    Google Scholar 

  23. Koebe, P.: Kontaktprobleme der Konformen Abbildung. Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88, 141–164 (1936)

    Google Scholar 

  24. Li, X.-Y., Teng, S.-H., Üngör, A.: Biting spheres in 3D. In: Proc. 8th International Meshing Roundtable, pp. 85–95. Sandia National Laboratories (1999), http://www.imr.sandia.gov/papers/abstracts/Li133.html

  25. Li, X.-Y., Teng, S.-H., Üngör, A.: Biting: advancing front meets sphere packing. International Journal for Numerical Methods in Engineering 49(1-2), 61–81 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liang, X., Zhang, Y.: Hexagon-based all-quadrilateral mesh generation with guaranteed angle bounds. Computer Methods in Applied Mechanics and Engineering 200(23-24), 2005–2020 (2011), doi:10.1016/j.cma.2011.03.002

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, J., Li, S., Chen, Y.: A fast and practical method to pack spheres for mesh generation. Acta Mechanica Sinica 24(4), 439–447 (2008), doi:10.1007/s10409-008-0165-y

    Article  MathSciNet  Google Scholar 

  28. Lo, S.H., Wang, W.X.: Generation of tetrahedral mesh of variable element size by sphere packing over an unbounded 3D domain. Computer Methods in Applied Mechanics and Engineering 194(48-49), 5002–5018 (2005), doi:10.1016/j.cma.2004.11.022

    Article  MATH  Google Scholar 

  29. Malitz, S., Papakostas, A.: On the angular resolution of planar graphs. SIAM Journal on Discrete Mathematics 7(2), 172–183 (1994), doi:10.1137/S0895480193242931

    Article  MathSciNet  MATH  Google Scholar 

  30. Miller, G.L., Talmor, D., Teng, S.-H., Walkington, N.: A Delaunay based numerical method for three dimensions: generation, formulation, and partition. In: Proce. 27th ACM Symp. on Theory of Computing (STOC 1995), pp. 683–692 (1995), doi:10.1145/225058.225286

    Google Scholar 

  31. Miller, G.L., Talmor, D., Teng, S.-H., Walkington, N.: On the radius-edge condition in the control volume method. SIAM Journal on Numerical Analysis 36(6), 1690–1708 (1999), doi:10.1137/S0036142996311854

    Article  MathSciNet  MATH  Google Scholar 

  32. Miller, G.L., Teng, S.-H., Thurston, W.P., Vavasis, S.A.: Separators for sphere-packings and nearest neighbor graphs. Journal of the ACM 44(1), 1–29 (1997), doi:10.1145/256292.256294

    Article  MathSciNet  MATH  Google Scholar 

  33. Miller, G.L., Teng, S.-H., Thurston, W.P., Vavasis, S.A.: Geometric separators for finite-element meshes. SIAM Journal on Scientific Computing 19(2), 364–386 (1998), doi:10.1137/S1064827594262613

    Article  MathSciNet  MATH  Google Scholar 

  34. Rodin, B., Sullivan, D.: The convergence of circle packings to the Riemann mapping. Journal of Differential Geometry 26(2), 349–360 (1987), http://projecteuclid.org/getRecord?id=euclid.jdg/1214441375

    MathSciNet  MATH  Google Scholar 

  35. Ruppert, J.: A Delaunay refinement algorithm for quality 2-dimensional mesh generation. Journal of Algorithms 18(3), 548–585 (1995), doi:10.1006/jagm.1995.1021

    Article  MathSciNet  MATH  Google Scholar 

  36. Shimada, K., Gossard, D.C.: Bubble mesh: automated triangular meshing of non-manifold geometry by sphere packing. In: Proc. 3rd ACM Symp. Solid Modeling and Applications (SMA 1995), pp. 409–419 (1995), doi:10.1145/218013.218095

    Google Scholar 

  37. Stephenson, K.: The approximation of conformal structures via circle packing. In: Computational Methods and Function Theory 1997, Nicosia. Approx. Decompos., vol. 11, pp. 551–582. World Scientific Publishing (1999)

    Google Scholar 

  38. Sußner, G., Greiner, G.: Hexagonal Delaunay triangulation. In: Proc. 18th International Meshing Roundtable, pp. 519–538. Springer (2009), doi:10.1007/978-3-642-04319-2_30

    Google Scholar 

  39. Szalay, A.S., Kunszt, P.Z., Thakar, A., Gray, J., Slutz, D., Brunner, R.J.: Designing and mining multi-terabyte astronomy archives: the Sloan Digital Sky Survey. In: Proc. ACM International Conf. on Management of Data (SIGMOD 2000), pp. 451–462 (2000), doi:10.1145/342009.335439

    Google Scholar 

  40. Thurston, W.P.: The Geometry and Topology of Three-Manifolds. Mathematical Sciences Research Inst. (2002), http://library.msri.org/books/gt3m/ ; See especially Section 13.6, Andreev’s theorem and generalizations

  41. Wang, W.X., Ming, C.Y., Lo, S.H.: Generation of triangular mesh with specified size by circle packing. Advances in Engineering Software 38(2), 133–142 (2007), doi:10.1016/j.advengsoft.2006.04.006

    Article  Google Scholar 

  42. Yerry, M.A., Shephard, M.S.: A modified quadtree approach to finite element mesh generation. IEEE Computer Graphics and Applications 3(1), 39–46 (1983), doi:10.1109/MCG.1983.262997

    Article  Google Scholar 

  43. Zorin, D.: Subdivision zoo. Subdivision for Modeling and Animation. SIGGRAPH Course Notes, 65–102 (1999)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of California, Irvine, USA

    David Eppstein

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  1. David Eppstein
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Correspondence to David Eppstein .

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Editors and Affiliations

  1. , Department of Applied Mathematics &, Stony Brook University, Stony Brook, 11794-3600, New York, USA

    Xiangmin Jiao

  2. , Bruyeres le Chatel, CEA, Arpajon Cedex, 91297, France

    Jean-Christophe Weill

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Eppstein, D. (2013). Diamond-Kite Meshes: Adaptive Quadrilateral Meshing and Orthogonal Circle Packing. In: Jiao, X., Weill, JC. (eds) Proceedings of the 21st International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33573-0_16

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  • DOI: https://doi.org/10.1007/978-3-642-33573-0_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33572-3

  • Online ISBN: 978-3-642-33573-0

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