• Corpus ID: 218684772

Combinatorics of intervals in the plane I: trapezoids

@article{Benedetto2020CombinatoricsOI,
  title={Combinatorics of intervals in the plane I: trapezoids},
  author={Daniel Di Benedetto and J{\'o}zsef Solymosi and E. P. White},
  journal={ArXiv},
  year={2020},
  volume={abs/2005.09003}
}
We study arrangements of intervals in $\mathbb{R}^2$ for which many pairs form trapezoids. We show that any set of intervals forming many trapezoids must have underlying algebraic structure, which we characterise. This leads to some unexpected examples of sets of intervals forming many trapezoids, where an important role is played by degree 2 curves. 

References

SHOWING 1-8 OF 8 REFERENCES

Combinatorial Geometry and Its Algorithmic Applications

TLDR
This volume provides a comprehensive up-to-date survey of several core areas of combinatorial geometry, and describes the beginnings of the subject, going back to the nineteenth century, and explains why counting incidences and estimating the combinatorially complexity of various arrangements of geometric objects became the theoretical backbone of computational geometry in the 1980s and 1990s.

An optimal algorithm for intersecting line segments in the plane

The authors present the first optimal algorithm for the following problem: given n line segments in the plane, compute all k pairwise intersections in O(n log n+k) time. Within the same asymptotic

On the Erdős distinct distances problem in the plane

In this paper, we prove that a set of N points in R 2 has at least c N log N distinct distances, thus obtaining the sharp exponent in a problem of Erd} os. We follow the setup of Elekes and Sharir

On Vertical Visibility in Arrangements of Segments and the Queue Size in the Bentley-Ottmann Line Sweeping Algorithm

TLDR
There are similar upper and lower bounds on the maximum size of the queue in the original implementation of the Bentley–Ottmann algorithm for reporting all intersections between the segments in S, i.e., the implementation where future events are not deleted from the queue.

Planar realizations of nonlinear davenport-schinzel sequences by segments

  • Ady Wiernik
  • Mathematics
    27th Annual Symposium on Foundations of Computer Science (sfcs 1986)
  • 1986
TLDR
A construction of a set G of n segments for which YG consists of Ω(nα(n)) subsegments is presented, proving that the Hart-Sharir bound is tight in the worst case.

Characterizations of Orthodiagonal Quadrilaterals

We prove ten necessary and sufficient conditions for a convex quadrilateral to have perpendicular diagonals. One of these is a quite new eight point circle theorem and three of them are metric

Characterizations of orthodiagonal quadrilaterals

Quadrilaterals in which the diagonals are perpendicular are considered. Several characterizations of this property are presented and proved.

Binary space partitions for line segments with a limited number of directions

TLDR
It is shown that there exists a BSP of size O(kn) if the line segments have at most <i>k</i> different orientations, and the smallest BSP can be as big as Ω(<i>n</i) log <i*n/i>/log log<i+i>) in the worst case.