Skip to main content
SpringerLink
Log in

Quadrangulations of planar sets

  • Invited Presentation
  • Conference paper
  • First Online:
Algorithms and Data Structures (WADS 1995)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 955))

Included in the following conference series:

Abstract

Given a set S such as a polygon or a set of points, a quadrangulation of S is a partition of the interior of S, if S is a polygon, or the interior of the convex hull of S, if S is a set of points, into quadrangles (quadrilaterals) obtained by inserting edges between pairs of points (diagonals between vertices of the polygon) such that the edges intersect each other only at their end points. Not all polygons or sets of points admit quadrangulations, even when the quadrangles are not required to be convex (convex quadrangulations). In this paper we briefly survey some recent results concerning the characterization of those planar sets that always admit quadrangulations (convex and non-convex) as well as some related computational problems.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Takao Asano, Tetsuo Asano and H. Imai, “Partitioning a polygonal region into trapezoids,” Journal of the A.C.M., vol. 33, No. 2, April 1986, pp. 290–312.

    Google Scholar 

  2. Arkin, E., M. Held, J. Mitchell, and S. Skiena, “Hamiltonian triangulations for fast rendering,” Algorithms-ESA'94, J. van Leeuwen, ed., Utrecht, NL, LNCS 855, pp. 36–47, September 1994.

    Google Scholar 

  3. Bern, M. and Eppstein, D., “Mesh generation and optimal triangulation,” in Computing in Euclidean Geometry, F. K. Hwang and D.-Z. Du, eds., World Scientific 1992.

    Google Scholar 

  4. Baumgarten, H., Jung, H. and Mehlhorn, K., “Dynamic point location in general subdivisions,” Journal of Algorithms, vol. 17, 1994, pp. 342–380.

    Article  Google Scholar 

  5. Barequet, G., and M. Sharir, “Piecewise-linear interpolation between polygonal slices,” Proceedings of the 10th Annual Symposium on Computational Geometry, pp. 93–102, 1994.

    Google Scholar 

  6. Bose, P., and G. Toussaint, “No Quadrangulation is Extremely Odd,” technical report #95-03, Dept. of Computer Science, University of British Columbia, January 1995.

    Google Scholar 

  7. P. Bose and G. T. Toussaint, “On computing quadrangulations of planar point sets,” 10th Colloquium on Graph Theory, Combinatorics and Applications, Feb. 27–March 3, 1995, Xalapa, Mexico.

    Google Scholar 

  8. P. Bose and G. T. Toussaint, “Generating quadrangulations of planar point sets,” accepted for publication in Computer Aided Geometric Design.

    Google Scholar 

  9. Chazelle, B., “On the convex layers of a convex set,” IEEE Transactions on Information Theory, vol. IT-31, 1985, pp. 509–517.

    Article  Google Scholar 

  10. B. Chazelle, “Triangulating a simple polygon in linear time,” Discrete Comput. Geom., vol. 6, 1991, pp. 485–524.

    Google Scholar 

  11. V. Chvátal, “A combinatorial theorem in plane geometry,” J. Combin. Theory Ser. B, vol. 18, 1975, pp. 39–41.

    Article  Google Scholar 

  12. Cheng, S. W. and Janardan, R., “New results on dynamic planar point location,” SIAM Journal on Computing, vol. 21, 1992, pp. 972–999.

    Article  Google Scholar 

  13. H. E. Conn and J. O'Rourke, “Minimum weight quadrilateralization in O(n3 log n) time,” Proc. of the 28th Allerton Conference on Comm. Control and Computing, October 1990, pp. 788–797.

    Google Scholar 

  14. Everett, H., W. Lenhart, M. Overmars, T. Shermer, and J. Urrutia, “Strictly convex quadrilateralizations of polygons,” in Proceedings of the 4th Canadian Conference on Computational Geometry, pp. 77–83, 1992.

    Google Scholar 

  15. S. Fisk, “A short proof of Chvátal's watchman theorem,” J. Combin. Theory Ser. B, vol. 24, 1978, pp. 374.

    Article  Google Scholar 

  16. A. Fournier and D. Y. Montuno, “Triangulating simple polygons and equivalent problems,” ACM Transactions on Graphics, vol. 3, No. 2, 1984, pp. 153–175.

    Article  Google Scholar 

  17. Heighway, E., “A mesh generator for automatically subdividing irregular polygons into quadrilaterals,” IEEE Transactions on Magnetics, 19, 6, pp. 2535–2538, 1983.

    Article  Google Scholar 

  18. Ho-Le, K., “Finite element mesh generation methods: A review and classification,” Computer Aided Design, 20, pp. 27–38, 1988.

    Article  Google Scholar 

  19. Hershberger, J., and S. Suri, “Applications of a semi-dynamic convex hull algorithm,” Proceedings of the second S.W.A.T., Lecture Notes in Computer Science 447, Bergen, Sweden, pp. 380–392, 1990.

    Google Scholar 

  20. Johnston, B. P., Sullivan, J. M. and Kwasnik, A., “Automatic conversion of triangular finite meshes to quadrilateral elements,” International Journal of Numerical Methods in Engineering, vol. 31, No. 1, 1991, pp. 67–84.

    Google Scholar 

  21. Kahn, J., M. Klawe, D. Kleitman, “Traditional galleries require fewer watchmen,” SIAM J. Algebraic Discrete Methods, 4, pp. 194–206, 1983.

    Google Scholar 

  22. A. A. Kooshesh and B. M. E. Moret, “Three-coloring the vertices of a triangulated simple polygon,” Pattern Recognition, vol. 25, 1992.

    Google Scholar 

  23. J. M. Keil and J.-R. Sack, “Minimum decompositions of polygonal objects,” in Computational Geometry, Ed., G. T. Toussaint, North-Holland, Amsterdam, pp. 197–216.

    Google Scholar 

  24. Lai, M. J., “Scattered data interpolation and approximation by using C1 piecewise cubic polynomials,” submitted for publication.

    Google Scholar 

  25. W. Lipski, Jr., E. Lodr, F. Luccio, C. Mugnal and L. Pagli, “On two dimensional data organization II, Fundanmenta Informaticae, vol. 2, 1979, pp. 245–260.

    Google Scholar 

  26. Lai, M. J. and Schumaker, L. L., “Scattered data interpolation using C2 piecewise polynomials of degree six,” Third Workshop on Proximity Graphs, Mississippi State University, Starkville, Mississippi, December 1–3, 1994.

    Google Scholar 

  27. Lubiw, A., “Decomposing polygonal regions into convex quadrilaterals,” Proc. Symposium on Computational Geometry, 1985, pp. 97–106.

    Google Scholar 

  28. Okabe, A., B. Boots, and K. Sugihara, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, John Wiley & Sons, Chichester, England, 1992.

    Google Scholar 

  29. T. Ohtsuki, “Minimum dissection of rectilinear regions,” in Proc. IEEE International Symposium on Circuits and Systems, Rome, 1982, pp. 1210–1213.

    Google Scholar 

  30. O'Rourke, J., Computational Geometry in C., Cambridge University Press, 1994.

    Google Scholar 

  31. Preparata, F. P. and Shamos, M. I., Computational Geometry: An Introduction, Springer-Verlag, New York, 1985.

    Google Scholar 

  32. Preparata, F. P. and Tamassia, R., “Fully dynamic point location in a monotone subdivision,” SIAM Journal on Computing, vol. 18, 1989, pp. 811–830.

    Article  Google Scholar 

  33. Quak, E. and Schumaker, L. L., “Cubic spline fitting using data dependent triangulations.” Computer-Aided Geometric Design, vol. 7, 1990, pp. 293–301.

    MathSciNet  Google Scholar 

  34. Ramaswami, S., Ramos, P. and G. T. Toussaint, “Converting triangulations to quadrangulations,” manuscript in preparation.

    Google Scholar 

  35. Sack, J. R., “An O(n log n) algorithm for decomposing simple rectilinear polygons into convex quadrilaterals,” Proc. 20th Annual Conf. on Communications, Control and Computing, Allerton, 1982, pp. 64–74.

    Google Scholar 

  36. Srinivasan, V., Nackman, L. R., Tang, J.-M. and Meshkat, S. N., “Automatic mesh generation using the symmetric axis transformation of polygonal domains,” Proceedings of the IEEE, vol. 80, No. 9, September 1992, pp. 1485–1501.

    Article  Google Scholar 

  37. Sapidis, N. and Perucchio, R., “Advanced techniques for automatic finite element meshing from solid models,” Computer Aided Design, vol. 21, No. 4, May 1989, pp. 248–253.

    Article  Google Scholar 

  38. N. Sugiyama and K. Saitoh, “Electron-beam exposure system AMDES,” Computer Aided Design, vol. 11, 1979, pp. 59–65.

    Article  Google Scholar 

  39. Schroeder, W., and M. Shephard, “Geometry-based fully automatic mesh generation and the Delaunay triangulation,” International Journal for Numerical Methods in Engineering, 24, pp. 2503–2515, 1988.

    Google Scholar 

  40. Sack, J.-R. and Toussaint, G. T., “A linear-time algorithm for decomposing rectilinear star-shaped polygons into convex quadrilaterals,” Proc. 19th Annual Conf. on Communications, Control and Computing, Allerton, 1981, pp. 21–30.

    Google Scholar 

  41. Sack, J.-R. and Toussaint, G. T., “Guard placement in rectilinear polygons,” in Computational Morphology, Ed., G. T. Toussaint, North-Holland, 1988, pp. 153–175.

    Google Scholar 

  42. Toussaint, G. T., “Solving geometric problems with the rotating calipers,” Proc. IEEE MELECON 83, Athens, Greece, 1983, pp. A10002/1–4.

    Google Scholar 

  43. Toussaint, G. T., “New results in computational geometry relevant to pattern recognition in practice,” in Pattern Recognition in Practice II, E. S. Gelsema and L. N. Kanal, Eds., North-Holland, 1986, pp. 135–146.

    Google Scholar 

  44. Wang, T., “A C 2-quintic spline interpolation scheme on triangulation,” Computer Aided Geometric Design, vol. 9, 1992, pp. 379–386.

    Article  Google Scholar 

  45. Wang, Y., and J. Aggarwal, “Surface reconstruction and representation of 3-d scenes,” Pattern Recognition, 19, pp. 197–207, 1986.

    Article  Google Scholar 

  46. Wismath, S. K., “Triangulations: An algorithmic study,” Tech. Report 80-106, Queens University, Kingston, Canada, July 1980.

    Google Scholar 

  47. Yoeli, P., “Compilation of data for computer-assisted relief cartography,” in Display and Analysis of Spatial Data, J. Davis, and M. McCullagh, eds., John Wiley & Sons, New York, 1975.

    Google Scholar 

  48. Zienkiewicz, O. C. and Taylor, R. L., The Finite Element Method, vol. I, McGraw-Hill, London, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, McGill University, Montreal, Canada

    Godfried Toussaint

Authors
  1. Godfried Toussaint
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Selim G. Akl Frank Dehne Jörg-Rüdiger Sack Nicola Santoro

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Toussaint, G. (1995). Quadrangulations of planar sets. In: Akl, S.G., Dehne, F., Sack, JR., Santoro, N. (eds) Algorithms and Data Structures. WADS 1995. Lecture Notes in Computer Science, vol 955. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60220-8_64

Download citation

  • DOI: https://doi.org/10.1007/3-540-60220-8_64

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60220-0

  • Online ISBN: 978-3-540-44747-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation