On the number of touching pairs in a set of planar curves

P. Györgyi; B. Hujter; S. Kisfaludi-Bak

doi:10.1016/j.comgeo.2017.10.004

On the number of touching pairs in a set of planar curves

P. Györgyi , B. Hujter , S. Kisfaludi-Bak

Algorithms, Geometry and Applications

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Abstract

Given a set of planar curves (Jordan arcs), each pair of which meets — either crosses or touches — exactly once, we establish an upper bound on the number of touchings. We show that such a curve family has O(t ²n) touchings, where t is the number of faces in the curve arrangement that contains at least one endpoint of one of the curves. Our method relies on finding special subsets of curves called quasi-grids in curve families; this gives some structural insight into curve families with a high number of touchings.

Original language	English
Pages (from-to)	29-37
Number of pages	9
Journal	Computational Geometry
Volume	67
Issue number	Januari 2018
DOIs	https://doi.org/10.1016/j.comgeo.2017.10.004
Publication status	Published - Jan 2018

Keywords

Combinatorial geometry;
Touching curves;
Pseudo-segments
Touching curves
Combinatorial geometry

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10.1016/j.comgeo.2017.10.004

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abstract = "Given a set of planar curves (Jordan arcs), each pair of which meets — either crosses or touches — exactly once, we establish an upper bound on the number of touchings. We show that such a curve family has O(t 2n) touchings, where t is the number of faces in the curve arrangement that contains at least one endpoint of one of the curves. Our method relies on finding special subsets of curves called quasi-grids in curve families; this gives some structural insight into curve families with a high number of touchings. ",

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AU - Kisfaludi-Bak, S.

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AB - Given a set of planar curves (Jordan arcs), each pair of which meets — either crosses or touches — exactly once, we establish an upper bound on the number of touchings. We show that such a curve family has O(t 2n) touchings, where t is the number of faces in the curve arrangement that contains at least one endpoint of one of the curves. Our method relies on finding special subsets of curves called quasi-grids in curve families; this gives some structural insight into curve families with a high number of touchings.

KW - Combinatorial geometry;

KW - Touching curves;

KW - Pseudo-segments

KW - Touching curves

KW - Combinatorial geometry

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SP - 29

EP - 37

JO - Computational Geometry

JF - Computational Geometry

SN - 0925-7721

IS - Januari 2018

ER -

On the number of touching pairs in a set of planar curves

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Keywords

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