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We define Euclid polynomials E k + 1 (λ) = E k (λ) E k (λ) - 1 + 1 and E 1 (λ) = λ + 1 in analogy to Euclid numbers e k = E k (1) . We show how to construct companion matrices E k , so E k (λ) = det λ I - E k , of height 1 (and thus of minimal height over all integer companion matrices for E k (λ) ). We prove various properties of these objects, and give experimental confirmation of some unproved properties. [ABSTRACT FROM AUTHOR]

Voronoi diagrams for a set of geometric objects is a partition of the plane (or space in higher dimensions) into disjoint regions each dominated by some given object under a predetermined criterion. In this paper we are interested in various measures associated with criteria on goodness of an input line segment with respect to each point in the plane as the “point of view.” These measures basically show how well a segment or information displayed on the segment can be seen from the point. Mathematically, the measures are defined in terms of the shapes of the triangle determined by the point and the line segment. We study the combinatorial and algorithmic complexities of those Voronoi diagrams. We also study an associated optimization problem: find a point that maximizes the smallest measure value over the measures with respect to all the given line segments. We give sufficient conditions for an optimal point to lie on a Voronoi edge and present a heuristic optimization algorithm for those measures having this property. [ABSTRACT FROM AUTHOR]