Handbook of discrete and computational geometry
Handbook of discrete and computational geometry
Concrete Mathematics: A Foundation for Computer Science
Concrete Mathematics: A Foundation for Computer Science
Intersecting convex sets by rays
Proceedings of the twenty-fourth annual symposium on Computational geometry
The forest hiding problem: an illumination problem for maximal disk packings
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
ZeroTouch: an optical multi-touch and free-air interaction architecture
Proceedings of the SIGCHI Conference on Human Factors in Computing Systems
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We consider the problem of determining the minimum number Nd of unit disks that is required to block all rays emanating from a point P in the two-dimensional space, where each disk has at least a distance d to point P and to any other disk. We study the asymptotic behavior of Nd, as d tends to infinity. By deriving upper bounds and lower bounds, we prove that pi2/16 infinity} N_d/d2 2, where the upper bound is based on establishing an interesting link between unit disks positioned on a regular triangular grid and Farey sequences from number theory. By positioning point P as well as the centers of the disks on the grid points of such a triangular grid, we create hexagonal rings of disks around P. We prove that we need exactly d-1 of these hexagons to block all rays emanating from P.