An acyclicity theorem for cell complexes ind dimension

Abstract

LetC be a cell complex ind-dimensional Euclidean space whose faces are obtained by orthogonal projection of the faces of a convex polytope ind+ 1 dimensions. For example, the Delaunay triangulation of a finite point set is such a cell complex. This paper shows that the in_front/behind relation defined for the faces ofC with respect to any fixed viewpointx is acyclic. This result has applications to hidden line/surface removal and other problems in computational geometry.

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References

  1. [1]

    F. Aurenhammer: Power diagrams: properties, algorithms, and applications,SIAM J. Comput.,16 (1987), 78–96.

    Google Scholar 

  2. [2]

    B. Chazelle: How to search in history,Inform. Control,64 (1985), 77–99.

    Google Scholar 

  3. [3]

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

    Google Scholar 

  4. [4]

    H. Edelsbrunner:Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, Germany,1987.

    Google Scholar 

  5. [5]

    H. Edelsbrunner, D. G. Kirkpatrick, andR. Seidel: On the shape of a set of points in the plane,IEEE Trans. Inform. Theory,IT-29 (1983), 551–559.

    Google Scholar 

  6. [6]

    L. De Floriani, B. Falcidieno, C. Pienovi, andG. Nagy:On sorting triangles in a Delaunay tessellation, Techn. Rept., Istituto per la Matematica Applicata, Consiglio Nazionale delle Richerche, Genove, Italy,1988.

    Google Scholar 

  7. [7]

    J. D. Foley, andA. van Dam:Fundamentals of Interactive Computer Graphics, Addison-Wesley, Reading, Massachusetts,1982.

    Google Scholar 

  8. [8]

    H. Fuchs, Z. M. Kedem, andB. Naylor: On visible surface generation by a priori structures,Comput. Graphics,14 (1980), 124–133.

    Google Scholar 

  9. [9]

    G. Voronoi: Sur quelques propriétés des formes quadratiques parfaites,J. Reine Angew. Math.,133 (1907), 212–287.

    Google Scholar 

Download references

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Research reported in this paper was supported by the National Science Foundation under grant CCR-8714565

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Edelsbrunner, H. An acyclicity theorem for cell complexes ind dimension. Combinatorica 10, 251–260 (1990). https://doi.org/10.1007/BF02122779

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AMS subject classification (1980)

  • 52 A 45
  • 05 B 45
  • 05 B 30