Abstract
This paper studies applications of envelopes of piecewise linear functions to problems in computational geometry. Among these applications we find problems involving hidden line/surface elimination, motion planning, transversals of polytopes, and a new type of Voronoi diagram for clusters of points. All results are either combinatorial or computational in nature. They are based on the combinatorial analysis in two companion papers [PS] and [E2] and a divide-and-conquer algorithm for computing envelopes described in this paper.
Article PDF
Similar content being viewed by others
References
Aronov, B. and Sharir, M., Triangles in space, or: Building (and analyzing) castles in the air,Proc. 4th Ann. ACM Sympos. Comput. Geom., 1988, pp. 381–391.
Avis, D. and Doskas, M., Algorithms for high dimensional stabbing problems, Report SOCS-87.2, School of Computer Science, McGill University, Montreal, Quebec, 1987.
Chazelle, B., Convex partitions of polyhedra: a lower bound and worst-case optimal algorithm,SIAM J. Comput. 13 (1984), 488–507.
Defays, D., An efficient algorithm for a complete link method,Comput. J. 20 (1977), 364–366.
Devai, F., Quadratic bounds for hidden line elimination,Proc. 2nd Ann. ACM Sympos. Comput. Geom., 1986, pp. 269–275.
Edelsbrunner, H.,Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.
Edelsbrunner, H., The upper envelope of piecewise linear functions: tight bounds on the number of faces, Report UIUCDCS-R-87-1396, Department of Computer Science, University of Illinois, 1987.
Edelsbrunner, H., Maurer, H. A., Preparata, F. P., Rosenberg, A. L., Welzl, E., and Wood, D., Stabbing line segments,BIT 22 (1982), 274–281.
Edelsbrunner, H. and Mücke, E. P., Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms,Proc. 4th Ann. ACM Sympos. Comput. Geom., 1988, pp. 118–133.
Guibas, L. J., Ramshaw, L., and Stolfi, J., A kinematic framework for computational geometry,Proc. 24th Ann. IEEE Sympos. Found. Comput. Sci., 1983, pp. 100–111.
Guibas, L. J. and Seidel, R., Computing convolutions by reciprocal search,Discrete Comput. Geom. 2 (1987), 175–193.
Guibas, L. J., Sharir, M., and Sifrony, S., On the general motion planning problem with two degrees of freedom,Proc. 4th Ann. ACM Sympos. Comput. Geom., 1988, pp. 289–298.
Hart, S. and Sharir, M., Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes,Combinatorica 6 (1986), 151–177.
Hartigan, J. A.,Clustering Algorithms, Wiley, New York, 1975.
Hopcroft, J., Schwartz, J., and Sharir, M. (eds.),Planning, Geometry and Complexity of Robot Motion, Ablex, Norwood, NJ, 1987.
Kedem, K., Livne, R., Pach, J., and Sharir, M., On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles,Discrete Comput. Geom. 1 (1986), 59–71.
Leven, D. and Sharir, M., Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams,Discrete Comput. Geom. 2 (1987), 9–31.
Lozano-Pérez, T. and Wesley, M. A., An algorithm for planning collision-free paths among polyhedral obstacles,Comm. ACM 22 (1979), 560–570.
McKenna, M., Worst-case optimal hidden-surface removal,ACM Trans. Graphics 6 (1987), 19–28.
Pach, J. and Sharir, M., The upper envelope of piecewise linear functions and the boundary of a region enclosed by convex plates: combinatorial analysis,Discrete Comput. Geom., to appear.
Pollack, R., Sharir, M., and Sifrony, S., Separating two simple polygons by a sequence of translations,Discrete Comput. Geom. 3 (1988), 123–136.
Preparata, F. P. and Shamos, M. I.,Computational Geometry—An Introduction, Springer-Verlag, New York, 1985.
Schwartz, J. T. and Sharir, M., On the two-dimensional Davenport-Schinzel problem, Report 193 (revised), Computer Science Department, Courant Institute, New York, 1987.
Shor, P., Private communication.
Sutherland, I. E., Sproull, R. F. and Shumacker, R. A., A characterization of ten hidden surface algorithms,Comput. Surveys 6 (1974), 1–55.
Tamir, A., Improved complexity bounds for center location problems on networks by using dynamic data structures, Manuscript.
Tarjan, R. E., Depth-first search and linear graph algorithms,SIAM J. Comput. 2 (1972), 146–160.
Toussaint, G., Movable separability of sets, inComputational Geometry, G. T. Toussaint, ed., North-Holland, Amsterdam, 1985, pp. 335–375.
Wiernik, A. and Sharir, M., Planar realization of nonlinear Davenport-Schinzel sequences by segments,Discrete Comput. Geom. 3 (1988), 15–47.
Author information
Authors and Affiliations
Additional information
Work on this paper by the first author has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862. Work by the third author has been supported by the Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, by grants from the Digital Equipment Corporation and the IBM Corporation, and by a research grant from NCRD, the Israeli National Council for Research and Development.
Rights and permissions
About this article
Cite this article
Edelsbrunner, H., Guibas, L.J. & Sharir, M. The upper envelope of piecewise linear functions: Algorithms and applications. Discrete Comput Geom 4, 311–336 (1989). https://doi.org/10.1007/BF02187733
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02187733