### Abstract

Given a region R in a Euclidean space and a distinguished point p ∈ R, the forbidden zone, F(R, p), is the union of all open balls with center in R, having p Abstract: a common boundary point. For a polytope, the forbidden zone is the union of open balls centered at its vertices. The notion of forbidden zone was defined in [1] and shown to be instrumental in the characterization of mollified zone diagrams, a relaxation of zone diagrams, introduced by Asano, et al. [2], itself a variation of Voronoi diagrams. In this article we focus on properties of F(P, p) where P is a convex polygon. We derive formulas for the area and circumference of F(P, p) when p is fixed, and for minimum areas and circumferences when p is allowed to range in P. Moreover, we give formulas for the area and circumference of a flower-shaped region corresponding to intersecting circles in F(P, p), and for optimal values as p ranges in P. We also extend our formulas for p ∉ P. We then develop a formula for the area of the intersection of circles having a common boundary point. The optimization problems associate interesting centers to a polygon, even to a triangle, different from their classical versions. Aside from geometric interest, applications could exist. Finally, we extend some of the above results and optimizations to arbitrary polytopes and bounded convex sets.

Original language | English (US) |
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Title of host publication | Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 |

Pages | 56-65 |

Number of pages | 10 |

DOIs | |

State | Published - Oct 3 2012 |

Event | 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 - Piscataway, NJ, United States Duration: Jun 27 2012 → Jun 29 2012 |

### Publication series

Name | Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 |
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### Other

Other | 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 |
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Country | United States |

City | Piscataway, NJ |

Period | 6/27/12 → 6/29/12 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Keywords

- Forbidden zone
- Mollified zone diagram
- Voronoi diagram
- Zone diagram

### Cite this

*Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012*(pp. 56-65). [6257657] (Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012). https://doi.org/10.1109/ISVD.2012.12

}

*Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012.*, 6257657, Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012, pp. 56-65, 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012, Piscataway, NJ, United States, 6/27/12. https://doi.org/10.1109/ISVD.2012.12

**On properties of forbidden zones of polygons and polytopes.** / Berkowitz, Ross; Kalantari, Bahman; Menendez, David; Kalantari, Iraj.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - On properties of forbidden zones of polygons and polytopes

AU - Berkowitz, Ross

AU - Kalantari, Bahman

AU - Menendez, David

AU - Kalantari, Iraj

PY - 2012/10/3

Y1 - 2012/10/3

N2 - Given a region R in a Euclidean space and a distinguished point p ∈ R, the forbidden zone, F(R, p), is the union of all open balls with center in R, having p Abstract: a common boundary point. For a polytope, the forbidden zone is the union of open balls centered at its vertices. The notion of forbidden zone was defined in [1] and shown to be instrumental in the characterization of mollified zone diagrams, a relaxation of zone diagrams, introduced by Asano, et al. [2], itself a variation of Voronoi diagrams. In this article we focus on properties of F(P, p) where P is a convex polygon. We derive formulas for the area and circumference of F(P, p) when p is fixed, and for minimum areas and circumferences when p is allowed to range in P. Moreover, we give formulas for the area and circumference of a flower-shaped region corresponding to intersecting circles in F(P, p), and for optimal values as p ranges in P. We also extend our formulas for p ∉ P. We then develop a formula for the area of the intersection of circles having a common boundary point. The optimization problems associate interesting centers to a polygon, even to a triangle, different from their classical versions. Aside from geometric interest, applications could exist. Finally, we extend some of the above results and optimizations to arbitrary polytopes and bounded convex sets.

AB - Given a region R in a Euclidean space and a distinguished point p ∈ R, the forbidden zone, F(R, p), is the union of all open balls with center in R, having p Abstract: a common boundary point. For a polytope, the forbidden zone is the union of open balls centered at its vertices. The notion of forbidden zone was defined in [1] and shown to be instrumental in the characterization of mollified zone diagrams, a relaxation of zone diagrams, introduced by Asano, et al. [2], itself a variation of Voronoi diagrams. In this article we focus on properties of F(P, p) where P is a convex polygon. We derive formulas for the area and circumference of F(P, p) when p is fixed, and for minimum areas and circumferences when p is allowed to range in P. Moreover, we give formulas for the area and circumference of a flower-shaped region corresponding to intersecting circles in F(P, p), and for optimal values as p ranges in P. We also extend our formulas for p ∉ P. We then develop a formula for the area of the intersection of circles having a common boundary point. The optimization problems associate interesting centers to a polygon, even to a triangle, different from their classical versions. Aside from geometric interest, applications could exist. Finally, we extend some of the above results and optimizations to arbitrary polytopes and bounded convex sets.

KW - Forbidden zone

KW - Mollified zone diagram

KW - Voronoi diagram

KW - Zone diagram

UR - http://www.scopus.com/inward/record.url?scp=84866784208&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866784208&partnerID=8YFLogxK

U2 - 10.1109/ISVD.2012.12

DO - 10.1109/ISVD.2012.12

M3 - Conference contribution

AN - SCOPUS:84866784208

SN - 9780769547244

T3 - Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012

SP - 56

EP - 65

BT - Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012

ER -