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
Efficient binary space partitions for hidden-surface removal and solid modeling
Discrete & Computational Geometry - Selected papers from the fifth annual ACM symposium on computational geometry, Saarbrücken, Germany, June 5-11, 1989
Counting and cutting cycles of lines and rods in space
Computational Geometry: Theory and Applications
Discrete & Computational Geometry - Special issue on ACM symposium on computational geometry, North Conway
On joints in arrangements of lines in space and related problems
Journal of Combinatorial Theory Series A
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Constructing Planar Cuttings in Theory and Practice
SIAM Journal on Computing
Lenses in arrangements of pseudo-circles and their applications
Proceedings of the eighteenth annual symposium on Computational geometry
Crossing Numbers and Hard Erdös Problems in Discrete Geometry
Combinatorics, Probability and Computing
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
On a question of bourgain about geometric incidences
Combinatorics, Probability and Computing
Incidences in three dimensions and distinct distances in the plane
Proceedings of the twenty-sixth annual symposium on Computational geometry
On lines, joints, and incidences in three dimensions
Journal of Combinatorial Theory Series A
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Given a set $L$ of $n$ lines in ${\mathbb R}^3$, joints are points in ${\mathbb R}^3$ that are incident to at least three non-coplanar lines in $L$. We show that there are at most $O(n^{5/3})$ incidences between $L$ and the set of its joints.This result leads to related questions about incidences between $L$ and a set $P$ of $m$ points in ${\mathbb R}^3$. First, we associate with every point $p \in P$ the minimum number of planes it takes to cover all lines incident to $p$. Then the sum of these numbers is at most \[ O\big(m^{4/7}n^{5/7}+m+n\big).\] Second, if each line forms a fixed given non-zero angle with the $xy$-plane – we say the lines are equally inclined – then the number of (real) incidences is at most \[ O\big(\min\big\{m^{3/4}n^{1/2}\kappa(m),\ m^{4/7}n^{5/7}\big\} + m + n\big) , \] where $\kappa(m) \,{=}\, (\log m)^{O(\alpha^2(m))}$, and $\alpha(m)$ is the slowly growing inverse Ackermann function. These bounds are smaller than the tight Szemerédi–Trotter bound for point–line incidences in $\reals^2$, unless both bounds are linear. They are the first results of this type on incidences between points and $1$-dimensional objects in $\reals^3$. This research was stimulated by a question raised by G. Elekes.