The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces

Spatial Analysis in curved spaces with Non-Euclidean Geometry

The ultimate goal of spatial information, both as part of technology and as science, is to answer questions and issues related to space, place, and location. Therefore, geometry is widely used for description, storage, and analysis. Undoubtedly, one of the most essential features of spatial information is geometric features, and one of the most obvious types of analysis is the geometric type an...

Separating Collections of Points in Euclidean Spaces

Given two disjoint convex sets A and B in Ed, a hyperplane h in Ed separates them if A lies on one of the half spaces defined by h while B lies on the complementary half space. Given a collection F of convex sets in Ed we say F is separated by a set of hyperplanes H if every pair of elements of F i s separated by some hyperplane of H. We deal here with the case that the convex sets are all poin...

Metric Derivations on Euclidean and Non-euclidean Spaces

As introduced by Weaver, (metric) derivations extend the notion of differential operators on Euclidean spaces and provide a linear differentiation theory that is well-defined on all metric spaces with Borel measures. The main result of this paper is a rigidity theorem for measures on Euclidean spaces that support a maximal number of derivations. The proof relies substantially on ideas from geom...

Windows into non-Euclidean spaces

The non-Euclidean Euclidean algorithm

In this paper we demonstrate how the geometrically motivated algorithm to determine whether a twogenerator real Möbius group acting on the Poincaré plane is or is not discrete can be interpreted as a non-Euclidean Euclidean algorithm. That is, the algorithm can be viewed as an application of the Euclidean division algorithm to real numbers that represent hyperbolic distances. In the case that t...