Pii: S0925-7721(00)00010-9

Densest lattice packings of 3-polytopes

Based on Minkowski’s work on critical lattices of 3-dimensional convex bodies we present an efficient algorithm for computing the density of a densest lattice packing of an arbitrary 3-polytope. As an application we calculate densest lattice packings of all regular and Archimedean polytopes.

Computing fence designs for orienting parts

A common task in automated manufacturing processes is to orient parts prior to assembly. We consider sensorless orientation of a polygonal part by a sequence of fences. We show that any polygonal part can be oriented by a sequence of fences placed along a conveyor belt, thereby settling a conjecture by Wiegley et al. [17], and present the rst polynomial-time algorithm to compute the shortest su...

Pii: S0925-7721(00)00023-7

It is shown that in any polygonal art gallery of n sides it is possible to place bn/3c point guards whose range of vision is 180◦ so that every interior point of the gallery can be seen by at least one of them. The guards can be stationed at any point of the art gallery. This settles an open problem posed by J. Urrutia.  2000 Elsevier Science B.V. All rights reserved.

Intersection of Unit-balls and Diameter of a Point Set in 3

We describe an algorithm for computing the intersection of n balls of equal radius in IR 3 which runs in time O(n lg 2 n). The algorithm can be parallelized so that the comparisons that involve the radius of the balls are performed in O(lg 3 n) batches. Using parametric search, these algorithms are used to obtain an algorithm for computing the diameter of a set of n points in IR 3 (the maximum ...

Translating a Convex Polygon to Contain a Maximum Number of Points