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Decomposition Algorithms in Geometry

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Algebraic Geometry and its Applications

Abstract

Decomposing complex shapes into simpler components has always been a focus of attention in computational geometry. The reason is obvious: most geometric algorithms perform more efficiently and are easier to implement and debug if the objects have simple shapes. For example, mesh-generation is a standard staple of the finite-element method; partitioning polygons or polyhedra into convex pieces or simplices is a typical preprocessing step in automated design, robotics, and pattern recognition. In computer graphics, decompositions of two-dimensional scenes are used in contour filling, hit detection, clipping and windowing; polyhedra are decomposed into smaller parts to perform hidden surface removal and ray-tracing.

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References

  1. T. Asano, and T. Asano, Partitioning Polygonal Regions into Trapezoids, Proc. 24th Annual IEEE Symposium on Foundations of Computer Science (1983), 233–241.

    Google Scholar 

  2. T. Asano, T. Asano, and H. Imai, Partitioning a Polygonal Region into Trapezoids, Journal of the ACM 33 (1986), 290–312.

    Article  MathSciNet  Google Scholar 

  3. C. L. Bajaj, and T. K. Dey, Convex Decompositions of Polyhedra and Robustness,SIAM Journal on Computing 21 (1992), 339–364.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. L. Bentley, and T. Ottmann, Algorithms for Reporting and Counting Geometric Intersections, IEEE Transactions on Computing C-28 (1979), 643–647.

    Article  MATH  Google Scholar 

  5. K. Q. Brown, Comments on “Algorithms for Reporting and Counting Geometric Intersections,”IEEE Transactions on Computing C30 (1981), 147–148.

    MATH  Google Scholar 

  6. B. Chazelle, An Optimal Algorithm for Intersecting Three-Dimensional Convex Polyhedra, Proc. 30th Annual IEEE Symposium on Foundations of Computer Science (1989), 586–591.

    Google Scholar 

  7. B. Chazelle, Triangulating a Simple Polygon in Linear Time, Proc. 31st Annual Symposium on Foundations of Computer Science (1990), to appear in Discrete and Computational Geometry.

    Google Scholar 

  8. B. Chazelle, and D. P. Dobkin, Optimal Convex Decompositions, Computational Geometry, North Holland, (1985), 63–133.

    Google Scholar 

  9. B. Chazelle, and H. Edelsbrunner, An Optimal Algorithm for Intersecting Line Segments in the Plane,Proc. 29th Annual IEEE Symposium on Foundations of Computer Science (1988), 590–600.

    Google Scholar 

  10. B. Chazelle, H. Edelsbrunner, L. J. Guibas, and M. Sharir, A Singly Exponential Stratification Scheme for real Semi-Algebraic Varieties and its Applications, Lecture Notes in Computer Science 372 (1989), 179–193.

    Article  MathSciNet  Google Scholar 

  11. B. Chazelle, and J. Incerpi, Triangulation and Shape-Complexity, ACM Transactions on Graphics 3 (1984), 135–152.

    Article  MATH  Google Scholar 

  12. B. Chazelle, and L. Palios, Triangulating a Nonconvex Polytope, Discrete and Computational Geometry 5 (1990), 505–526.

    Article  MathSciNet  MATH  Google Scholar 

  13. L. P. Chew, Guaranteed-Quality Triangulation Meshes, Tech. Report, Dept. of Computer Science, Cornell University.

    Google Scholar 

  14. K. L. Clarkson, and R. Cole, in preparation.

    Google Scholar 

  15. K. L. Clarkson, and P. W. Shor, Applications of Random Sampling in Computational Geometry, II, Discrete and Computational Geometry 4 (1989), 387–421.

    Article  MathSciNet  MATH  Google Scholar 

  16. K. L. Clarkson, R. E. Tarjan, and C. J. Van Wyk, A Fast Las Vegas Algorithm for Triangulating a Simple Polygon, Discrete and Computational Geometry 4 (1989), 423–432.

    Article  MathSciNet  MATH  Google Scholar 

  17. G. E. Collins, Quantifier Elimination for Real Closed Fields by Cylindric Algebraic Decomposition, Lecture Notes in Computer Science 33 (1975), 134–183.

    Google Scholar 

  18. J. Davenport, and J. Heintz, Real Quantifier Elimination is Doubly Exponential, Journal of Symbolic Computation 5 (1988), 29–35.

    Article  MathSciNet  MATH  Google Scholar 

  19. T. K. Dey, C. L. Bajaj, and K. Sugihara, On Good Triangulations in Three Dimensions, International Journal of Computational Geometry and Applications 2 (1992), 75–95.

    Article  MathSciNet  MATH  Google Scholar 

  20. S. J. Fortune, A Sweepline Algorithm for Voronoi Diagrams, Algorithmica 2 (1987), 153–174.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Fournier, and D. Y. Montuno, Triangulating Simple Polygons and Equivalent Problems, ACM Transactions on Graphics 3 (1984), 153–174.

    Article  MATH  Google Scholar 

  22. M. R. Garey, and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, 1979.

    MATH  Google Scholar 

  23. M. R. Garey, D. S. Johnson, F. P. Preparata, and R. E. Tarjan, Triangulating a Simple Polygon, Information Processing Letters 7 (1978), 175–179.

    Article  MathSciNet  MATH  Google Scholar 

  24. F. Gavril, Algorithms for a Maximum Clique and a Maximum Independent Set of a Circle Graph, Networks 3 (1973), 261–273.

    Article  MathSciNet  MATH  Google Scholar 

  25. D. H. Greene, The Decomposition of Polygons into Convex Parts, manuscript, Xerox PARC, 1982.

    Google Scholar 

  26. L. J. Guibas, and J. Stolfi, Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams, A CM Transactions on Graphics 4 (1985), 74–123.

    Article  MATH  Google Scholar 

  27. S. Hertel, and K. Mehlhorn, Fast Triangulation of a Simple Polygon, Lecture Notes in Computer Science 158 (1985), 207–218.

    MathSciNet  Google Scholar 

  28. K. Hoffman, K. Mehlhorn, P. Rosenstiehl, and R. E. Tarjan, Sorting Jordan Sequences in Linear Time Using Level-Linked Search Trees, Information and Control 68 (1986), 170–184.

    Article  MathSciNet  Google Scholar 

  29. J. M. Keil, Decomposing a Polygon into Simpler Components, SIAM Journal on Computing 14 (1985), 799–811.

    Article  MathSciNet  MATH  Google Scholar 

  30. D. G. Kirkpatrick, Efficient Computation of Continuous Skeletons, Proc. 20th Annual IEEE Symposium on Foundations of Computer Science (1979), 18–27.

    Google Scholar 

  31. D. G. Kirkpatrick, M. M. Klawe, and R. E. Tarjan, Polygon Triangulation in 0(n log log n) Time with Simple Data Structures, Proc. 6th Annual ACM Symposium on Computational Geometry (1990), 34–43.

    Google Scholar 

  32. A. Knight, J. May, J. McAffer, T. Nguyen, and J.-R. Sack, A Computational Geometry Workbench, Proc. 6th Annual ACM Symposium on Computational Geometry (1990), 370.

    Google Scholar 

  33. D. T. Lee, and F. P. Preparata, Location of a Point in a Planar Subdivision and its Applications,SIAM Journal on Computing 6 (1977), 594–606.

    Article  MathSciNet  MATH  Google Scholar 

  34. D. Lichtenstein, Planar Formulae and their Uses, SIAM Journal on Computing 11 (1982), 329–343.

    Article  MathSciNet  MATH  Google Scholar 

  35. A. Lingas, The Power of Non-Rectilinear Holes, Lecture Notes in Computer Science 140 (1982), 369–383.

    Article  MathSciNet  Google Scholar 

  36. A. Lingas, R. Pinter, R. Rivest, and A. Shamir, Minimum Edge Length Partitioning of Rectilinear Polygons, Proc. 20th Annual Allerton Conference on Communication, Control, and Computing (1982), 53–63.

    Google Scholar 

  37. K. Mulmuley, A Fast Planar Partition Algorithm, II, Proc. 5th Annual ACM Symposium on Computational Geometry (1989), 33–43.

    Google Scholar 

  38. J. O’Rourke, and K. J. Supowit, Some NP-Hard Polygon Decomposition Problems, IEEE Transactions on Information Theory IT-29 (1983), 181–190.

    Article  MathSciNet  MATH  Google Scholar 

  39. J. Ruppert, and R. Seidel, On the Difficulty of Tetrahedralizing 3- Dimensional Non-Convex Polyhedra, Proc. 5th Annual ACM Symposium on Computational Geometry (1989), 380–392.

    Google Scholar 

  40. R. Seidel, in preparation.

    Google Scholar 

  41. R. E. Tarjan, and C. J. Van Wyk, An O(nloglogn)-time Algorithm for Triangulating a Simple Polygon, SIAM Journal on Computing 17 (1988), 143–178.

    Article  MathSciNet  MATH  Google Scholar 

  42. G. T. Toussaint, Pattern Recodnition and Geometrical Complexity, Proc. 5th International Conference on Pattern Recognition (1980), 1324–1347.

    Google Scholar 

  43. G. T. Toussaint, and D. Avis, On a Convex Hull Algorithm for Polygons and its Application to Triangulation Problems,Pattern Recognition 15 (1982), 23–29.

    Article  MathSciNet  Google Scholar 

  44. C. K. Yap, An O(nlogn) Algorithm for the Voronoi Diagram of a Set of Simple Curve Segments, Discrete and Computational Geometry 2 (1987), 365–393.

    Article  MathSciNet  MATH  Google Scholar 

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Authors
  1. Bernard Chazelle
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  2. Leonidas Palios
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Purdue University, 47907, West Lafayette, IN, USA

    Chandrajit L. Bajaj

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© 1994 Springer-Verlag New York, Inc.

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Chazelle, B., Palios, L. (1994). Decomposition Algorithms in Geometry. In: Bajaj, C.L. (eds) Algebraic Geometry and its Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2628-4_27

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  • DOI: https://doi.org/10.1007/978-1-4612-2628-4_27

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7614-2

  • Online ISBN: 978-1-4612-2628-4

  • eBook Packages: Springer Book Archive

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