Enumeration and visibility problems in integer lattices
Evangelos Kranakis, Michel Pocchiola · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · 1990
We study enumeration and visibility problems in the d-dimensional integer lattice L~ 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 L~, 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 lat- tice 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.