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

Y1 - 2012

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

T2 - 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012

Y2 - 27 June 2012 through 29 June 2012

ER -