### Abstract

We establish several combinatorial bounds on the complexity (number of vertices and edges) of the complement of the union (also known as the common exterior) of k convex polygons in the plane, with a total of n edges. We show: (1) The maximum complexity of the entire common exterior is Θ(na(k) + k^{2}). ^{2} (2) The maximum complexity of a single cell of the common exterior is Θ(na(k)). (3) The complexity of m distinct cells in the common exterior is O(m^{2/3}k^{2/3}log^{1/3}(k^{2}/m) + n log k) and can be Ω(m^{2/3}k^{2/3} + na(k)) in the worst case.

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

Pages (from-to) | 139-149 |

Number of pages | 11 |

Journal | Computational Geometry: Theory and Applications |

Volume | 8 |

Issue number | 3 |

State | Published - Aug 1997 |

### Fingerprint

### Keywords

- Arrangements
- Combinatorial complexity
- Common exterior
- Convex polygons

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

*Computational Geometry: Theory and Applications*,

*8*(3), 139-149.

**The common exterior of convex polygons in the plane.** / Aronov, Boris; Sharir, Micha.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 8, no. 3, pp. 139-149.

}

TY - JOUR

T1 - The common exterior of convex polygons in the plane

AU - Aronov, Boris

AU - Sharir, Micha

PY - 1997/8

Y1 - 1997/8

N2 - We establish several combinatorial bounds on the complexity (number of vertices and edges) of the complement of the union (also known as the common exterior) of k convex polygons in the plane, with a total of n edges. We show: (1) The maximum complexity of the entire common exterior is Θ(na(k) + k2). 2 (2) The maximum complexity of a single cell of the common exterior is Θ(na(k)). (3) The complexity of m distinct cells in the common exterior is O(m2/3k2/3log1/3(k2/m) + n log k) and can be Ω(m2/3k2/3 + na(k)) in the worst case.

AB - We establish several combinatorial bounds on the complexity (number of vertices and edges) of the complement of the union (also known as the common exterior) of k convex polygons in the plane, with a total of n edges. We show: (1) The maximum complexity of the entire common exterior is Θ(na(k) + k2). 2 (2) The maximum complexity of a single cell of the common exterior is Θ(na(k)). (3) The complexity of m distinct cells in the common exterior is O(m2/3k2/3log1/3(k2/m) + n log k) and can be Ω(m2/3k2/3 + na(k)) in the worst case.

KW - Arrangements

KW - Combinatorial complexity

KW - Common exterior

KW - Convex polygons

UR - http://www.scopus.com/inward/record.url?scp=0040360922&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040360922&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0040360922

VL - 8

SP - 139

EP - 149

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 3

ER -