### Abstract

Let C be a collection of n Jordan regions in the plane in general position, such that each pair of their boundaries intersect in at most s points, where s is a constant. If the boundaries of two sets in C cross exactly twice, then their intersection points are called regular vertices of the arrangement A(C). Let R(C) denote the set of regular vertices on the boundary of the union of C. We present several bounds on |R(C)|, depending on the type of the sets of C. (i) If each set of C is convex, then |R(C)| = O(n^{1.5+ε}) for any ε > 0.^{1} (ii) If no further assumptions are made on the sets of C, then we show that there is a positive integer r that depends only on s such that |R(C)| = O(n^{2-1/r}). (iii) If C consists of two collections C_{1} and C_{2} where C_{1} is a collection of m convex pseudo-disks in the plane (closed Jordan regions with the property that the boundaries of any two of them intersect at most twice), and C_{2} is a collection of polygons with a total of n sides, then |R(C)| = O(m^{2/3}n^{2/3} + m + n), and this bound is tight in the worst case.

Original language | English (US) |
---|---|

Pages (from-to) | 203-220 |

Number of pages | 18 |

Journal | Discrete and Computational Geometry |

Volume | 25 |

Issue number | 2 |

State | Published - 2001 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

*Discrete and Computational Geometry*,

*25*(2), 203-220.

**On the number of regular vertices of the union of Jordan regions.** / Aronov, Boris; Efrat, A.; Halperin, D.; Sharir, M.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 25, no. 2, pp. 203-220.

}

TY - JOUR

T1 - On the number of regular vertices of the union of Jordan regions

AU - Aronov, Boris

AU - Efrat, A.

AU - Halperin, D.

AU - Sharir, M.

PY - 2001

Y1 - 2001

N2 - Let C be a collection of n Jordan regions in the plane in general position, such that each pair of their boundaries intersect in at most s points, where s is a constant. If the boundaries of two sets in C cross exactly twice, then their intersection points are called regular vertices of the arrangement A(C). Let R(C) denote the set of regular vertices on the boundary of the union of C. We present several bounds on |R(C)|, depending on the type of the sets of C. (i) If each set of C is convex, then |R(C)| = O(n1.5+ε) for any ε > 0.1 (ii) If no further assumptions are made on the sets of C, then we show that there is a positive integer r that depends only on s such that |R(C)| = O(n2-1/r). (iii) If C consists of two collections C1 and C2 where C1 is a collection of m convex pseudo-disks in the plane (closed Jordan regions with the property that the boundaries of any two of them intersect at most twice), and C2 is a collection of polygons with a total of n sides, then |R(C)| = O(m2/3n2/3 + m + n), and this bound is tight in the worst case.

AB - Let C be a collection of n Jordan regions in the plane in general position, such that each pair of their boundaries intersect in at most s points, where s is a constant. If the boundaries of two sets in C cross exactly twice, then their intersection points are called regular vertices of the arrangement A(C). Let R(C) denote the set of regular vertices on the boundary of the union of C. We present several bounds on |R(C)|, depending on the type of the sets of C. (i) If each set of C is convex, then |R(C)| = O(n1.5+ε) for any ε > 0.1 (ii) If no further assumptions are made on the sets of C, then we show that there is a positive integer r that depends only on s such that |R(C)| = O(n2-1/r). (iii) If C consists of two collections C1 and C2 where C1 is a collection of m convex pseudo-disks in the plane (closed Jordan regions with the property that the boundaries of any two of them intersect at most twice), and C2 is a collection of polygons with a total of n sides, then |R(C)| = O(m2/3n2/3 + m + n), and this bound is tight in the worst case.

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UR - http://www.scopus.com/inward/citedby.url?scp=0035584446&partnerID=8YFLogxK

M3 - Article

VL - 25

SP - 203

EP - 220

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 2

ER -