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Computational Geometry and Visualization: Problems at the Interface

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Scientific Visualization of Physical Phenomena

Abstract

In this paper we survey certain geometric problems that arise in volume visualization and discuss how techniques from computational geometry can be applied to them. Such problems include depth-sorting of polyhedral complexes, point-location, ray shooting and tracing, and others. We give a few worked-out illustrative examples, as well as references to the extant literature.

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References

  1. P. Agarwal, A deterministic algorithm for partitioning arrangements of lines and its applications, Proc. 5th ACM Geometry Symp., 1989, 11–22.

    Google Scholar 

  2. P. Agarwal, Ray shooting and other applications of spanning trees with low stabbing number, Proc. 5th ACM Geometry Symp., 1989, 315–325.

    Google Scholar 

  3. E. Brisson, Representation of d-dimensional geometric objects, Technical Report 90-08-03, Department of Comp. Sc. and Eng., University of Washington, 1990.

    Google Scholar 

  4. B. Chazelle, How to search in history, Inf. and Control, 64 (1985), 77–99.

    Article  MATH  MathSciNet  Google Scholar 

  5. B. Chazelle and H. Edelsbrunner, An optimal algorithm for intersecting line segments in the plane, To appear in J. Assoc. Comp. Mach..

    Google Scholar 

  6. B. Chazelle, H. Edelsbrunner, L. Guibas, M. Sharir, and J. Stolfi, Lines in space: combinatorics and algorithms, Rep. UIUCDCS-R-90-1569, Dept. Comp. Sc., Univ. of Illinois, Urbana, 1990; also to appear in Algorithmica.

    Google Scholar 

  7. B. Chazelle, H. Edelsbrunner, L. Guibas, R. Pollack, R. Seidel, M. Sharir, and J. Snoeyink, Counting and cutting cycles of lines and rods in space, 31st Annual FOCS Conference, (1990), 242–251.

    Google Scholar 

  8. D. Cohen and A. Kaufman, Scan-conversion algorithms for linear and quadratic objects, Volume Visualization, IEEE Press, 1991, 280–301.

    Google Scholar 

  9. B. Delaunay, Sur la sphere vide, Izv. Akad. Nauk SSSR. Otdelenie Matematicheskii i Estestvennyka Nauk, 7 (1934), 793–800.

    Google Scholar 

  10. D. Dobkin and M. Laszlo, Primitives for the manipulation of three-dimensional subdivisions, Proc. 3rd ACM Comp. Geom. Symp., 1987, 86–99.

    Google Scholar 

  11. D. Dobkin, S. Levy, W. Thurston, and A. Wilks, Contour tracing by piecewise linear approx-imations, ACM Trans, on Graphics, 9 (1990), 389–423.

    Article  Google Scholar 

  12. J. Driscoll, N. Sarnak, D. Sleator, and R. Tarjan, Making data structures persistent, J. Comp. and System Sc., 38 (1989), 86–124.

    Article  MATH  MathSciNet  Google Scholar 

  13. H. Edelsbrunner, An acyclicity theorem for cell complexes in n dimensions, Proc. 4th Annual ACM Geom. Conf., 1989, 145–151.

    Google Scholar 

  14. H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer-Verlag, 1987.

    Google Scholar 

  15. H. Edelsbrunner, L. Guibas, and J. Stolfi, Optimal point-location in a monotone subdivision, SIAM J. Comp., vol. 15, 2, 1986, 317–340.

    Article  MATH  MathSciNet  Google Scholar 

  16. H. Edelsbrunner and E. Welzl, Halfplanar range search in linear space and O(n 0.695 ) query time, Inf. Proc. Letters, 23 (1986), 289–293.

    Article  MATH  Google Scholar 

  17. R. Forrest, Computational geometry, Proc. Royal Soc. London, 321 series A (1971), 187–195.

    Google Scholar 

  18. H. Fuchs, Z. Kedem, and B. Naylor, On visible surface generation by a priori tree structures, Comp. Graphics, 1980, 124–133 (SIGGRAPH ’80).

    Google Scholar 

  19. Raytracing irregular volume data, Proc. 1990 San Diego Workshop on Vol. Vis., 1990, 35–40.

    Google Scholar 

  20. A. Glassner, An Introduction to Ray-Tracing, Academic Press, 1989.

    Google Scholar 

  21. L. Guibas and M. Sharir, Triangulations with low crossing number, in preparation.

    Google Scholar 

  22. M. Goodrich and R. Tamassia, Dynamic trees and dynamic point location, to appear in STOC ’91.

    Google Scholar 

  23. L. Guibas and J. Stolfi, Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams, ACM Trans, on Graphics, 4 (1985), 74–123.

    Article  MATH  Google Scholar 

  24. J. Helman and L. Hesselink, Surface representations of two- and three-dimensional flow topology, IEEE Visualization ’90, 1990, 6–13.

    Google Scholar 

  25. G. Herman and H.K. Liu, Three-dimensional display of human organs from computed tomograms, Comp. Graphics and Image Proc., 1979, 1–21.

    Google Scholar 

  26. A. Kaufman, Efficient algorithms for 3D scan-conversion of parametric curves, surfaces, and volumes, Computer Graphics, 21 (1987), 171–179 (SIGGRAPH ’87).

    Google Scholar 

  27. D. Kirkpatrick, Optimal search in a planar subdivision, SIAM J. Comp., vol. 12, 1, 1983, 28–35.

    MATH  MathSciNet  Google Scholar 

  28. D. Knuth, The Art of Computer Programming, Vol. I: Fundamental Algorithms,Addison-Wesley (1973).

    Google Scholar 

  29. W. Lorensen and H. Cline, Marching cubes: a high resolution 3D surface reconstruction algorithm, Comp. Graphics, 21 (1987), 163–169 (SIGGRAPH ’87).

    Google Scholar 

  30. M. Levoy, Display of surfaces from volume data, IEEE CGAnd;A, 8 (1988), 29–37.

    Google Scholar 

  31. N. Max, P. Hanrahan, and R. Crawfils, Area and volume coherence for efficient visualization of 3D scalar functions, Proc. 1990 San Diego Workshop on Vol. Vis., 1990, 27–33.

    Google Scholar 

  32. H. Neeman, A decomposition algorithm for visualizing irregular grids, Computer Graphics, vol. 24, 5, 1990, 49–56.

    Article  Google Scholar 

  33. M. Overmars and M. Sharir, Output-sensitive hidden surface removal, Proc. 30th IEEE FOCS Symp., 1989.

    Google Scholar 

  34. T. Porter and T. Duff, Compositing digital images, Computer Graphics, vol. 18, 3, 1984, 253–259.

    Article  Google Scholar 

  35. F. Preparata and I. Shamos, Computational Geometry: An Introduction, Springer-Verlag, 1985.

    Google Scholar 

  36. F. Preparata and R. Tamassia, Fully dynamic point location in a monotone subdivision, SIAM J. on Comp., 18 (1989), 811–830.

    Article  MATH  MathSciNet  Google Scholar 

  37. F. Preparata and R. Tamassia, Efficient spatial point location, Proc. WADS ’89., Lect. Notes in Comp. Sc. 382, 3–11.

    Google Scholar 

  38. A. Rosenfeld, Three-dimensional digital topology, Inform, and Control, 50 (1981), 119–127.

    Article  MATH  MathSciNet  Google Scholar 

  39. K. Schumacher, R. Brand,, A. Gilliland, and A. Sharp, Study for applying computer generated images for visual simulation, U.S. Air Force Human Resources Laboratory, Tech. Rep. AFHRL-TR-69-14 (1969).

    Google Scholar 

  40. A. Schmitt, H. Müller, and W. Leister, Ray-tracing algorithms — theory and practice, Theoretical Foundations of Computer Graphics and CAD, Springer-Verlag, 1988, 997–1030.

    Google Scholar 

  41. D. Speray and S. Kennon, Volume probes: interactive data exploration of arbitrary grids, Proc. 1990 San Diego Workshop on Vol. Vis., 1990, 5–12.

    Google Scholar 

  42. J. Stolfi, Primitives for computational geometry, DEC/SRC Research Report 36, 1989.

    Google Scholar 

  43. L. Wilhams, Pyramidal parametrics, Comp. Graphics, 18 (1984), 213–222 (SIGGRAPH ’84).

    Article  Google Scholar 

  44. E. Welzl, Partition trees for triangle counting and other range searching problems, Proc. 4th ACM Geom. Conf., (1988), 23–33.

    Google Scholar 

  45. J. Wilhelms and A. Van Gelder, Topological consideration in isosurface generation Proc. 1990 San Diego Workshop on Vol. Vis., 1990, 79–86;

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Room NE43-307, 545 Technology Square, Cambridge, MA, USA, 02139

    Leonidas J. Guibas

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  1. Leonidas J. Guibas
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Editor information

Editors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

    Nicholas M. Patrikalakis (Associate Professor of Ocean Engineering) (Associate Professor of Ocean Engineering)

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© 1991 Springer-Verlag Tokyo

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Guibas, L.J. (1991). Computational Geometry and Visualization: Problems at the Interface. In: Patrikalakis, N.M. (eds) Scientific Visualization of Physical Phenomena. Springer, Tokyo. https://doi.org/10.1007/978-4-431-68159-5_4

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  • DOI: https://doi.org/10.1007/978-4-431-68159-5_4

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-68161-8

  • Online ISBN: 978-4-431-68159-5

  • eBook Packages: Springer Book Archive

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