Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Combinatorial complexity bounds for arrangements of curves and spheres
Discrete & Computational Geometry - Special issue on the complexity of arrangements
On the number of congruent simplices in a point
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Point-line incidences in space
Proceedings of the eighteenth annual symposium on Computational geometry
Lenses in arrangements of pseudo-circles and their applications
Proceedings of the eighteenth annual symposium on Computational geometry
On levels in arrangements of curves
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
On the Number of Incidences Between Points and Curves
Combinatorics, Probability and Computing
Crossing Numbers and Hard Erdös Problems in Discrete Geometry
Combinatorics, Probability and Computing
Distinct distances in three and higher dimensions
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Distinct Distances in Three and Higher Dimensions
Combinatorics, Probability and Computing
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(MATH) We show that the number of incidences between m distinct points and n distinct circles in $\reals^3$ is O(m 4/7 n 17/21+m 2/3 n 2/3+m+n); the bound is optimal for m n 3/2. This result extends recent work on point-circle incidences in the plane, but its proof requires a different analysis. The bound improves upon a previous bound, noted by Akutsu et al. [2] and by Agarwal and Sharir [1], but it is not as sharp (when m is small) as the recent planar bound of Aronov and Sharir [3]. Our analysis extends to yield the same bound (a) on the number of incidences between m points and n circles in any dimension d&rhoe; 3, and (b) on the number of incidences between m points and n arbitrary convex plane curves in $\reals^d$, for any d&rhoe; 3, provided that no two curves are coplanar. Our results improve the upper bound on the number of congruent copies of a fixed tetrahedron in a set of n points in 4-space, and were already used to obtain a lower bound for the number of distinct distances in a set of n points in 3-space.