Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris

Computational geometry

Reference work entry
DOI: https://doi.org/10.1007/1-4020-0611-X_142


Computational geometry is the discipline of exploring algorithms and data structures for computing geometric objects and their — often extremal — attributes. The objects are predominantly finite collections of points, flats, hyperplanes — “arrangements” — or polyhedra, all in finite dimensions. The algorithms are typically finite, their complexity playing a central role. Emphasis is on problems in low dimensions, exploiting special properties of the plane and 3-space.

A young field — its name coined in the early 1970s — it has since witnessed explosive growth, stimulated in part by the largely parallel development of computer graphics, pattern recognition, cluster analysis, and modern industry's reliance on computer-aided design (CAD) and robotics (Forrest, 1971; Graham and Yao, 1990; Lee and Preparata, 1984). It plays a key role in the emerging fields of automated cartography and computational metrology.

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  1. [1]
    Alfeld, P. and Barnhill, R. E. (1984). “A Transfinite C 2 Interpolant over Triangles,” Rocky Mountain Jl. Mathematics, 14, 17–39.Google Scholar
  2. [2]
    Asano, T., Edahiro, M., Imai, H., and Iri, M. (1985). “Practical Use of Bucketing Techniques in Computational Geometry,” in Computational Geometry, G. T. Toussaint, ed., North Holland, New York.Google Scholar
  3. [3]
    Atherton, P., Weiler, K., and Greenberg, D. P. (1978). “Polygon Shadow Generation,” Comput. Graph., 12, 275–281.Google Scholar
  4. [4]
    Barnhill, R. E. (1977). “Representation and Approximation of Surfaces” in Mathematical Software III, J. R. Rice, ed., Academic Press, New York.Google Scholar
  5. [5]
    Bartels, R. H., Beatty, J. C., Barski, B. A. (1987). An Introduction to Splines for Use in Computer Graphics, Morgan Kaufmann, Los Altos, California.Google Scholar
  6. [6]
    Beichl, I. and Sullivan, F. (1990). “A Robust Parallel Triangulation and Shelling Algorithm,” Proc. 2nd Canad. Conf. Comput. Geom., 107–111. Google Scholar
  7. [7]
    Bentley, J. L. and Carruthers, W. (1980). “Algorithms for Testing the Inclusion of Points in Polygons,” Proc. 18th Allerton Conf. Commun. Control Comput., 11–19. Google Scholar
  8. [8]
    Bentley, J. L., Weide, B. W., and Yao, A. C. (1980). “Optimal Expected-Time Algorithms for Closest Point Problems,” ACM Trans. Math. Software, 6, 563–580.Google Scholar
  9. [9]
    de Berg, M., van Kreveld, M., Overmars, M. (1997). Computational Geometry: Algorithms and Applications, Springer Verlag, New York.Google Scholar
  10. [10]
    Chazelle, B. (1990). “Triangulating the Simple Polygon in Linear Time,” Proc. 31st Annu. IEEE Sympos. Found. Comput. Sci., 220–230. Google Scholar
  11. [11]
    Chazelle, B. (1991). “An Optimal Convex Hull Algorithm and New Results on Cuttings,” Proc. 32nd Annu. IEEE Sympos. Found. Comput. Sci., 29–38. Google Scholar
  12. [12]
    Chin, F., Snoeyink, J., and Wang,C. A. (1995). “Finding the medial axis of a simple polygon in linear time,” Proceedings 6th Annu. Internat. Sympos. Algorithms Comput., vol. 1004 of Lecture Notes in Computer Science, 382–391, Springer Verlag, New York.Google Scholar
  13. [13]
    Clarkson, K. L. (1985). “A Probabilistic Algorithm for the Post Office Problem,” Proc. 17th Annu. ACM Sympos. Theory Comput., 175–184. Google Scholar
  14. [14]
    Clarkson, K. L. (1986). “Linear Programming in0(n3 d2)Time,” Inform. Process. Lett., 22, 21–24.Google Scholar
  15. [15]
    Devroye, L. (1986). Lecture Notes on Bucket Algorithms, Birkhäuser Verlag, Boston.Google Scholar
  16. [16]
    Dobkin, D. P. and Edelsbrunner, H. (1984). “Ham-sandwich Theorems Applied to Intersection Problems,” Proc. 10th Internat. Workshop Graph-Theoret. Concepts Comput. Sci. (WG 84), 88–99. Google Scholar
  17. [17]
    Dobkin, D. P. and Souvaine, D. L. (1990). “Computational Geometry in a Curved World,” Algorithmica, 5, 421–457.Google Scholar
  18. [18]
    Dobkin, D., Guibas, L., Hershberger, J., and Snoeyink, J. (1988). “An Efficient Algorithm for Finding the CSG Representation of a Simple Polygon,” Computer Graphics, 22, 31–40.Google Scholar
  19. [19]
    Edelsbrunner, H. (1983). “A New Approach to Rectangle Intersections, Parts I and II,” Internat. Jl. Comput. Math., 13, 209–219, 221–229.Google Scholar
  20. [20]
    Edelsbrunner, H. (1987). Algorithms in Combinatorial Geometry, Springer Verlag, New York.Google Scholar
  21. [21]
    Edelsbrunner, H. and Maurer, H. A. (1985). “Finding Extreme Points in Three Dimensions and Solving the Post-Office Problem in the Plane,” Inform. Process. Lett., 21, 39–47.Google Scholar
  22. [22]
    Edelsbrunner, H. and Mücke, E. P. (1988). “Simulation of Simplicity: a Technique to Cope with Degenerate Algorithms,” Proc. 4th Annu. ACM Sympos. Com-put. Geom., 118–133. Google Scholar
  23. [23]
    Farin, G. (1988). Curves and Surfaces for Computer Aided Geometric Design, Academic Press, New York.Google Scholar
  24. [24]
    Forrest, A. R. (1971). “Computational Geometry,” Proc. Roy. Soc. Lond. Ser. A, 321, 187–195.Google Scholar
  25. [25]
    Forrest, A. R. (1972). “Interactive Interpolation and Approximation by Bézier Polynomials,” The Computer Jl., 15, 71–79.Google Scholar
  26. [26]
    Franklin, W. R. (1980). “A Linear Time Exact Hidden Surface Algorithm,” Proc. SIGGRAPH ‘80, Comput. Graph., 14, 117–123.Google Scholar
  27. [27]
    Fortune, S. and Milenkovic, V. (1991). “Numerical Stability of Algorithms for Line Arrangements,” Proc. 7th Annu. ACM Sympos. Comput. Geom., 3342–341. Google Scholar
  28. [28]
    Fortune, S. and Van Wyck, C. (1993). “Efficient Exact Arithmetic for Computational Geometry,” ACM Symposium on Computational Geometry, 9, 163–172.Google Scholar
  29. [29]
    Garey, M. R., Johnson, D. S., Preparata, F. P., and Tarjan, R. E. (1978). “Triangulating a Simple Polygon,” Inform. Process. Lett., 7, 175–179.Google Scholar
  30. [30]
    Graham, R. L. and Yao, F. F. (1983). “Finding the Convex Hull of a Simple Polygon,” Jl. Algorithms, 4, 324–331.Google Scholar
  31. [31]
    Graham, R. and Yao, F. (1990). “A Whirlwind Tour of Computational Geometry,” Amer. Math. Monthly, 97, 687–701.Google Scholar
  32. [32]
    Grünbaum, B. (1967). Convex Polytopes, Wiley Interscience, New York.Google Scholar
  33. [33]
    Guibas, L. J., Salesin, D., and Stolfi, J. (1989). “Epsilon Geometry: Building Robust Algorithms from Imprecise Computations,” Proc. 5th Annu. ACM Sympos. Comput. Geom., 208–217. Google Scholar
  34. [34]
    Heller, M. (1990). “Triangulation Algorithms for Adaptive Terrain Modeling,” 4th Symposium on Spatial Data Handling, 163–174. Google Scholar
  35. [35]
    Kirkpatrick, D. (1983). “Optimal Search in Planar Subdivisions,” SIAM Jl. Comput., 12, 28–35.Google Scholar
  36. [36]
    Kirkpatrick, D. G. and Seidel, R. (1986). “The Ultimate Planar Convex Hull Algorithm?,” SIAM Jl. Comput., 15, 287–299.Google Scholar
  37. [37]
    Lawson, C. L. (1977). “Software for C 1 Surface Interpolation,” in Mathematical Software III, J. R. Rice, ed., Academic Press, New York.Google Scholar
  38. [38]
    Lee, D. T. (1982). “Medial axis transformation of a planar shape,” IEEE Trans. Pattern Anal. Mach. Intell., 4, 363–369. Google Scholar
  39. [39]
    Lee, D. T. and Preparata, F. P. (1984). “Computational Geometry–A Survey,” IEEE Transactions on Computers, c-33, 1072–1101.Google Scholar
  40. [40]
    Lozano-Pérez, T. and Wesley, M. A. (1979). “An Algorithm for Planning Collision-Free Paths Among Polyhedral Obstacles,” Commun. ACM, 22, 560–570.Google Scholar
  41. [41]
    Megiddo, N. (1982). “Linear-Time Algorithms for Linear Programming in R 3 and Related Problems,” Proc. 23rd Annu. IEEE Sympos. Found. Comput. Sci., 329–338. Google Scholar
  42. [42]
    Megiddo, N. (1984). “Linear Programming in Linear Time When the Dimension is Fixed,” Jl. ACM, 31, 114–127.Google Scholar
  43. [43]
    Muhidinov, N. and Nazirov, S. (1978). “Computerized Recognition of Closed Plane Domains,” Voprosy Vychisl. i Prikl. Mat. (Tashkent), 53, 96–107, 182.Google Scholar
  44. [44]
    Reingold, E. M. and Supowit, K. J. (1983). “Probabilistic Analysis of Divide-and-Conquer Heuristics for Minimum Weighted Euclidean Matching,” Networks, 13, 49–66.Google Scholar
  45. [45]
    O'Rourke, J. (1987). Art Gallery Theorems and Algorithms, Oxford University Press, New York.Google Scholar
  46. [46]
    Papadimitriou, C. H. (1977). “The Euclidean Traveling Salesman Problem is NP-Complete,” Theoret. Comput. Sci., 4, 237–244.Google Scholar
  47. [47]
    Preparata, F. P. (1990). “Planar Point Location Revisited,” Internat. Jl. Found. Comput. Science 24(1), 71–86.Google Scholar
  48. [48]
    Preparata, F. P. and Hong, S. J. (1977). “Convex Hulls of Finite Sets of Points in Two and Three Dimensions,” Commun. ACM, 20, 87–93.Google Scholar
  49. [49]
    Preparata, F. P. and Shamos, M. I. (1985). Computational Geometry: An Introduction, Springer Verlag, New York.Google Scholar
  50. [50]
    Requicha, A. A.G. (1980). “Representations for Rigid Solids: Theory, Methods, and Systems,” ACM Comput. Surveys, 12, 437–464.Google Scholar
  51. [51]
    Ruppert, J. and Seidel, R. (1989). “On the Difficulty of Tetrahedralizing 3-dimensional Non-convex Polyhedra,” Proc. 5-th Annu. ACM Sympos. Comput. Geom., 380–392. Google Scholar
  52. [52]
    Schwartz, J. T. and Sharir, M. (1983). “On the ‘Piano Movers' Problem, I: The Case of a Two-dimensional Rigid Polygonal Body Moving Amidst Polygonal Barriers,” Commun. Pure Appl. Math., 36, 345–398.Google Scholar
  53. [53]
    Schwartz, J. T. and Sharir, M. (1989). “A Survey of Motion Planning and Related Geometric Algorithms,” in Geometric Reasoning, D. Kapur and J. Mundy, eds., 157–169, MIT Press, Cambridge, Massachusetts.Google Scholar
  54. [54]
    Samet, H. (1990). The Design and Analysis of Spatial Data Structures, Addison Wesley, Reading, Pennsylvania.Google Scholar
  55. [55]
    Samet, H. (1990). Applications of Spatial Data Structures: Computer Graphics, Image Processing and GIS, Addison Wesley, Reading, Pennsylvania.Google Scholar
  56. [56]
    Shamos, M. I. and Hoey, D. (1975). “Closest-Point Problems,” Proc. 16th Annu. IEEE Sympos. Found. Comput. Sci., 151–162. Google Scholar
  57. [57]
    Shamos, M. I. and Hoey, D. (1976). “Geometric Intersection Problems,” Proc. 17th Annu. IEEE Sympos. Found. Comput. Sci., 208–215. Google Scholar
  58. [58]
    Sutherland, I. E., Sproull, R. F., and Shumacker, R. A. (1974). “A Characterization of Ten Hidden Surface Algorithms,” ACM Comput. Surveys, 6, 1–55.Google Scholar
  59. [59]
    Tiller, W. (1983). “Rational B–splines for Curve and Surface Representation,” IEEE Computer Graphics and Applications, 3(6), 61–69.Google Scholar
  60. [60]
    Yap, C. (1993). “Towards Exact Geometric Computation,” Proc. 5th Canadian Conference on Computational Geometry, 405–419. Google Scholar

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© Kluwer Academic Publishers 2001

Authors and Affiliations

  1. 1.National Institute of Standards & TechnologyGaithersburgUSA
  2. 2.Supercomputing Research CenterBowieUSA