# Incidences between points and circles in three and higher dimensions

@inproceedings{Aronov2002IncidencesBP, title={Incidences between points and circles in three and higher dimensions}, author={Boris Aronov and Vladlen Koltun and Micha Sharir}, booktitle={SCG '02}, year={2002} }

(MATH) We show that the number of incidences between <i>m</i> distinct points and <i>n</i> distinct circles in $\reals^3$ is <i>O</i>(<i>m</i> <sup>4/7</sup> <i>n</i> <sup>17/21</sup>+<i>m</i> <sup>2/3</sup> <i>n</i> <sup>2/3</sup>+<i>m</i>+<i>n</i>); the bound is optimal for <i>m n</i> <sup>3/2</sup>. 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…

## 29 Citations

Improved bounds for incidences between points and circles

- Computer ScienceSoCG '13
- 2013

An improved upper bound is established for the number of incidences between m points and n arbitrary circles in three dimensions if the set of circles is required to be "truly three-dimensional" in the sense that no sphere or plane contains more than q of the circles.

On the number of tetrahedra with minimum, unit, and distinct volumes in three-space

- Computer ScienceSODA '07
- 2007

An O(n<sup>3</sup>) time algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby extend an early algorithm of Edelsbrunner, O'Rourke, and Seidel.

Distinct distances in three and higher dimensions

- Computer ScienceSTOC '03
- 2003

Improving an old result of Clarkson et al., we show that the number of distinct distances determined by a set <i>P</i> of <i>n</i> points in three-dimensional space is…

Incidences between Points and Circles in Three and
Higher Dimensions

- MathematicsDiscret. Comput. Geom.
- 2005

The 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 theNumber of distinct distances in aSet of n points in 3-space.

Incidences with k-non-degenerate sets and their applications

- Mathematics, Computer ScienceJ. Comput. Geom.
- 2014

It is proved that, for every $\varepsilon>0$, the number of incidences between a set of $m$ points and a $k$-non-degenerate set of £n$ planes is O(m^{4/5+\varpsilon}n^{3/5}k^{2/5] + n + mk).

Incidence Problems in Plane and Higher Dimensions

- Mathematics
- 2005

This is a survey paper of recent results about the problem of counting number of incidences between points and curves in the plane. Also this discusses some results for this problem in higher…

Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances

- Mathematics, Computer ScienceSODA
- 2017

The results provide a "grand generalization" of most of the previous studies of (special instances of) previous works for both curves and surfaces and obtain new and improved bounds for two special cases.

Ramsey-type theorems for lines in 3-space

- MathematicsDiscret. Math. Theor. Comput. Sci.
- 2016

Geometric Ramsey-type statements on collections of lines in 3-space give guarantees on the size of a clique or an independent set in (hyper)graphs induced by incidence relations between lines, points, and reguli in3-space.

Incidences in Three Dimensions and Distinct Distances in the Plane†

- MathematicsCombinatorics, Probability and Computing
- 2011

This work adapts the recent new algebraic analysis technique of Guth and Katz, as further developed by Elekes, Kaplan and Sharir, to obtain sharp bounds on the number of incidences between a certain class of helices or parabolas and points in ℝ3 and obtains the upper bound O(s3).

Non-Degenerate Spheres in Three Dimensions

- MathematicsCombinatorics, Probability and Computing
- 2011

The previous bound given in [1] on the number of k-rich η-non-degenerate spheres in 3-space with respect to any set of n points in ℝ3 is improved.

## References

SHOWING 1-10 OF 32 REFERENCES

Point-line incidences in space

- MathematicsSCG '02
- 2002

These bounds are smaller than the tight Szemerédi-Trotter bound for point-line incidences in $\reals^2$, unless both bounds are linear.

Lenses in arrangements of pseudo-circles and their applications

- Computer Science, MathematicsJACM
- 2004

A linear upper bound on the number of empty lenses in an arrangement of <i>n</i> pseudo-circles is established with the property that any two curves intersect precisely twice, and improved bounds for the number and complexity of incidences, the complexity of a single level, and thecomplexity of many faces are obtained.

Combinatorial complexity bounds for arrangements of curves and spheres

- MathematicsDiscret. Comput. Geom.
- 1990

Upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike are presented and it is proved that the maximum number of edges boundingm cells in an arrangement ofn lines is Θ(m2/3n 2/3 +n), and that it isO(m3/2β(m) forn unit-circles.

Improved Bounds for Planar k -Sets and Related Problems

- MathematicsDiscret. Comput. Geom.
- 1998

This is the first considerable improvement on this bound after its early solution approximately 27 years ago and applies to improve the current bounds on the combinatorial complexities of k -levels in the arrangement of line segments, convex polygons in the union of n lines, parametric minimum spanning trees, and parametric matroids in general.

Extremal problems in discrete geometry

- MathematicsComb.
- 1983

Several theorems involving configurations of points and lines in the Euclidean plane are established, including one that shows that there is an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points.

On Levels in Arrangements of Curves

- MathematicsDiscret. Comput. Geom.
- 2003

The first subquadratic upper bound is given (roughly O( nk^ 1-1/(9· 2s-3) ) ) for curves that are graphs of polynomial functions of an arbitrary fixed degree s .

Lenses in arrangements of pseudo-circles and their applications

- MathematicsSCG '02
- 2002

It is shown that any collection of n pseudo-circles can be cut into $\bx$ arcs so that any two intersect at most once, provided that the given pseudo- circles are x-monotone and admit an algebraic representation by three real parameters.

Towards a Theory of Geometric Graphs

- Mathematics
- 2004

On the complexity of the linkage reconfiguration problem by H. Alt, C. Knauer, G. Rote, and S. Whitesides Falconer conjecture, spherical averages and discrete analogs by G. Arutyunyants and A.…

How to Cut Pseudoparabolas into Segments

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 1998

This work investigates how to cut pseudoparabolas into the minimum number of curve segments so that each pair of segments intersect at most once, and gives an Ω( n4/3 ) lower bound and O(n5/3) upper bound on the number of cuts.

Crossing Numbers and Hard Erd} os Problems in Discrete Geometry

- Mathematics
- 1997

We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the…