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) |
|---|---|
| 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 |
|---|
Other
| Other | 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 |
|---|---|
| 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
}
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 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 -