Lenses in arrangements of pseudo-circles and their applications

Journal of the ACM

Volumn 51, Issue 2, 2004, Pages 139-186

  Agarwal, Pankaj K ^a,f   Nevo, Eran ^b   Pach, János ^c   Pinchasi, Rom ^d   Sharir, Micha ^c,e   Smorodinsky, Shakhar ^e  

^a DUKE UNIVERSITY   (United States)

^b HEBREW UNIVERSITY OF JERUSALEM   (Israel)

^c NEW YORK UNIVERSITY   (United States)

^d MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^e TEL AVIV UNIVERSITY   (Israel)

^f Duke University   (United States)

Author keywords

Arrangements; Incidence problems; Pseudo circles

Indexed keywords

ARRANGEMENTS; GEOMETRIC CURVES; INCIDENCE PROBLEMS; PSEUDO-CIRCLES;

ALGEBRA; ALGORITHMS; COMPUTATIONAL COMPLEXITY; COMPUTER GRAPHICS; FINITE DIFFERENCE METHOD; FORMAL LANGUAGES; MATHEMATICAL MODELS; PARAMETER ESTIMATION; THEORY;

COMPUTATIONAL GEOMETRY;

EID: 4243191450     PISSN: 00045411     EISSN: None     Source Type: Journal    
DOI: 10.1145/972639.972641     Document Type: Article

References (31)

1. AGARWAL, P. K., ARONOV, B., AND SHARIR, M. 2003. On the complexity of many faces in arrangements of pseudo-segments and of circles. In Discrete and Computational Geometry: The Goodman-Pollack Festschrift, B. Aronov, S. Basu, J. Pach, and M. Sharir, Eds. Springer Verlag, Berlin, Germany 1-23.
2. AGARWAL, P. K., EFRAT, A., AND SHARIR, M. 1999. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput. 29, 912-953.
3. AGARWAL, P. K., AND MATOUŠEK, J. 1994. On range searching with semialgebraic sets. Disc. Comput. Geom. 11, 393-418.
4. AGARWAL, P. K., AND SHARIR, M. 2000. Arrangements and their applications. In Handbook of Computational Geometry, J.-R. Sack and J. Urrutia, Eds. Elsevier Science Publishers B.V. North-Holland, Amsterdam, 49-119.
5. ALON, N., KALAI, G., MATOUŠEK, J., AND MESHULAM, R. 2002. Transversal numbers for hypergraphs arising in geometry. Adv. Appl. Math. 29, 79-101.
6. ALON, N., AND KLEITMAN, D. 1992. Piercing convex sets and the Hadwiger-Debrunner (p, q)-problem. Adv. Math. 96, 1, 103-112.
7. ALON, N., LAST, H., PINCHASI, R., AND SHARIR, M. 2001. On the complexity of arrangements of circles in the plane. Disc. Comput. Geom. 26, 103-112.
8. ARONOV, B., AND SHARIR, M. 2002. Cutting circles into pseudo-segments and improved bounds for incidences. Disc. Comput. Geom. 28, 475-490.
9. BASU, S., POLLACK, R., AND ROY, M.-F. 2003. Algorithms in Real Algebraic Geometry. Springer-Verlag, New York.
10. BOCHNAK, J., COSTE, M., AND ROY, M.-F. 1998. Real Algebraic Geometry. Springer-Verlag, Heidelberg, Germany.
11. CHAN, T. M. 2003. On levels in arrangements of curves. Disc. Comput. Geom. 29, 275-293.
12. CHAZELLE, B., EDELSBRUNNER, H., GUIBAS, L. J., AND SHARIR, M. 1989. A singly-exponential stratification scheme for real semi-algebraic varieties and its applications. In Proceedings of the 16th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 372. Springer-Verlag, New York, 179-193.
13. CLARKSON, K., EDELSBRUNNER, H., GUIBAS, L. J., SHARIR, M., AND WELZL, E. 1990. Combinatorial complexity bounds for arrangements of curves and spheres. Disc. Comput. Geom. 5, 99-160.
14. EDELSBRUNNER, H. 1987. Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science, vol. 10. Springer-Verlag, Heidelberg, West Germany.
15. ERDOS, P. 1946. On a set of distances of n points. Amer. Math. Monthly 53, 248-250.
16. GOODMAN, J. E., AND POLLACK, R. 1993. Allowable sequences and order types in discrete and computational geometry. In New Trends in Discrete and Computational Geometry, J. Pach, Ed. Algorithms and Combinatorics, vol. 10. Springer-Verlag, New York, 103-134.
17. HANANI, C. C. A. 1934. Über wesentlich unplättbare Kurven im dreidimensionalen Raume. Fund. Math. 23, 135-142. (Hungarian with German and Russian Summaries).
18. HAUSSLER, D., AND WELZL, E. 1987. Epsilon-nets and simplex range queries. Disc. Comput. Geom. 2, 127-151.
19. HELLY, E. 1930. Über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten. Monaths. Math. und Physik 37, 281-302.
20. KEDEM, K., LIVNE, R., PACH, J., AND SHARIR, M. 1986. On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Disc. Comput. Geom. 1, 59-71.
21. LOVÁSZ, L., PACH, J., AND SZEGEDY, M. 1997. On Conway's thrackle conjecture. Disc. Comput. Geom. 18, 369-376.
22. MOLNÁR, J. 1956. Über eine Verallgemeinerung auf die Kugelfläche eines topologischen Satzes von Helly. Acta Math. Acad. Sci. 7, 107-108. (Hungarian with German and Russian Summaries).
23. PINCHASI, R. 2001. Problems in combinatorial geometry in the plane. Ph.D. dissertation, Dept. Mathematics, Hebrew Univ.
24. PINCHASI, R., AND RADOIČIĆ, R. 2003. On the number of edges in a topological graph with no self-intersecting cycle of length 4. In Proceedings of the 19th Annual ACM Symposium on Computational Geometry. ACM, New York, 98-103.
25. SHARIR, M., AND AGARWAL, P. K. 1995. Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York.
26. SNOEYINK, J., AND HERSHBERGER, J. 1991. Sweeping arrangements of curves. In Discrete and Computational Geometry: Papers from the DIMACS Special Year. American Mathematical Society, Providence, R.I., 309-349.
27. SPENCER, J., SZEMERÉDI, E., AND TROTTER, W. T. 1984. Unit distances in the Euclidean plane. In Graph Theory and Combinatorics, B. Bollobás, Ed. Academic Press, New York, NY, 293-303.
28. SZÉKELY, L. 1997. Crossing numbers and hard Erdos problems in discrete geometry. Combin., Probab., Comput. 6, 353-358.
29. TAMAKI, H., AND TOKUYAMA, T. 1998. How to cut pseudo-parabolas into segments. Disc. Comput. Geom. 19, 265-290.
30. TARDOS, G. 2003. On distinct sums and distinct distances. Adv. Math., 180, 275-289.
31. TUTTE, W. T. 1970. Toward a theory of crossing numbers. J. Combin. Theory 8, 45-53.