# On properties of forbidden zones of polygons and polytopes

Ross Berkowitz, Bahman Kalantari, David Menendez, Iraj Kalantari

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

### 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  and shown to be instrumental in the characterization of mollified zone diagrams, a relaxation of zone diagrams, introduced by Asano, et al. , 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) Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 56-65 10 https://doi.org/10.1109/ISVD.2012.12 Published - Oct 3 2012 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 - Piscataway, NJ, United StatesDuration: 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

### Other

Other 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 United States Piscataway, NJ 6/27/12 → 6/29/12

### Fingerprint

Polytopes
Polygon
Circumference
Circle
Ball
Union
Diagram
Convex polygon
Voronoi Diagram
Bounded Set
Polytope
Convex Sets
Range of data
Euclidean space
Triangle
Intersection
Optimization Problem
Optimization
Arbitrary

### All Science Journal Classification (ASJC) codes

• Geometry and Topology

### Keywords

• Forbidden zone
• Mollified zone diagram
• Voronoi diagram
• Zone diagram

### Cite this

Berkowitz, R., Kalantari, B., Menendez, D., & Kalantari, I. (2012). On properties of forbidden zones of polygons and polytopes. In Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 (pp. 56-65).  (Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012). https://doi.org/10.1109/ISVD.2012.12
Berkowitz, Ross ; Kalantari, Bahman ; Menendez, David ; Kalantari, Iraj. / On properties of forbidden zones of polygons and polytopes. Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012. 2012. pp. 56-65 (Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012).
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title = "On properties of forbidden zones of polygons and polytopes",
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  and shown to be instrumental in the characterization of mollified zone diagrams, a relaxation of zone diagrams, introduced by Asano, et al. , 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.",
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Berkowitz, R, Kalantari, B, Menendez, D & Kalantari, I 2012, On properties of forbidden zones of polygons and polytopes. in 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.

Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012. 2012. p. 56-65 6257657 (Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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T1 - On properties of forbidden zones of polygons and polytopes

AU - Berkowitz, Ross

AU - Kalantari, Bahman

AU - Menendez, David

AU - Kalantari, Iraj

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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  and shown to be instrumental in the characterization of mollified zone diagrams, a relaxation of zone diagrams, introduced by Asano, et al. , 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  and shown to be instrumental in the characterization of mollified zone diagrams, a relaxation of zone diagrams, introduced by Asano, et al. , 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

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U2 - 10.1109/ISVD.2012.12

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M3 - Conference contribution

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

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

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Berkowitz R, Kalantari B, Menendez D, Kalantari I. On properties of forbidden zones of polygons and polytopes. In Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012. 2012. p. 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