Worst-case optimal algorithms for constructing visibility polygons with holes
SCG '86 Proceedings of the second annual symposium on Computational geometry
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Art gallery theorems and algorithms
Art gallery theorems and algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
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We study enumeration and visibility problems in the d-dimensional integer lattice Ldn of d-tuples of integers ≤ n. In the first part of the paper we give several useful enumeration principles and use them to study the asymptotic behavior of the number of straight lines traversing a certain fixed number of lattice vertices of Ldn, the line incidence problem and the edge visibility region. In the second part of the paper we consider an art gallery problem for point obstacles. More specifically we study the camera placement problem for the infinite lattice Ld. A lattice point is visible from a camera C (positioned at a vertex of Ld) if the line segment joining A and C crosses no other lattice vertex. For any given number s ≤ 3d of cameras we determine the position they must occupy in the lattice Ld in order to maximize their visibility.