Similar simplices in a d-dimensional point set
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
On the number of tetrahedra with minimum, unit, and distinct volumes in three-space
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
On the number of tetrahedra with minimum, unit, and distinct volumes in three-space
Combinatorics, Probability and Computing
Incidences in three dimensions and distinct distances in the plane
Proceedings of the twenty-sixth annual symposium on Computational geometry
Improved bounds for incidences between points and circles
Proceedings of the twenty-ninth annual symposium on Computational geometry
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We show that the number of incidences between m distinct points and n distinct circles in ℝd, for any d ≥ 3, is O(m6/11n9/11κ(m3/n)+m2/3n2/3+m+n), where κ(n)=(log n)O(α(n))2 and where α(n) is the inverse Ackermann function. The bound coincides with the recent bound of Aronov and Sharir, or rather with its slight improvement by Agarwal et al., for the planar case. We also show that the number of incidences between m points and n unrestricted convex (or bounded-degree algebraic) plane curves, no two in a common plane, is O(m4/7n17/21+m2/3n2/3+m+n), in any dimension d ≥ 3. 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 the lower bound for the number of distinct distances in a set of n points in 3-space. Another application is an improved bound for the number of incidences (or, rather, containments) between lines and reguli in three dimensions. The latter result has already been applied by Feldman and Sharir to obtain a new bound on the number of joints in an arrangement of lines in three dimensions.