Related papers: A combinatorial approach to discrete geometry

Cells of Voronoi diagrams in two dimensions are usually considered as having edges of zero width. However, this is not the case in several experimental situations in which the thickness of the edges of the cells is relatively large. In this…

Calculus and geometry are ubiquitous in the theoretical modelling of scientific phenomena, but have historically been very challenging to apply directly to real data as statistics. Diffusion geometry is a new theory that reformulates…

We discuss certain aspects of the combinatorial approach to the differential geometry of non-abelian gerbes, due to W. Messing and the author (arXiv:math.AG/0106083), and give a more direct derivation of the associated cocycle equations.…

We present an analytically explicit study of optimal discrete quantization on spherical geometries equipped with the geodesic metric, focusing on highly symmetric configurations on the unit sphere $\mathbb S^2$. Three discrete uniform…

This article provides a gentle, visual introduction to the basic concepts of differential geometry appropriate for students familiar with special relativity. Visual methods are used to explain basics of differential geometry and build…

Based on a local approximation of the Riemannian distance on a manifold by a computationally cheap dissimilarity measure, a time discrete geodesic calculus is developed, and applications to shape space are explored. The dissimilarity…

We discuss a link between graph theory and geometry that arises when considering graph dynamical systems with odd interactions. The equilibrium set in such systems is not a collection of isolated points, but rather a union of manifolds,…

We propose a computation of curvature of arbitrary two-dimensional surfaces of three-dimensional objects, which is a contribution to discrete gravity with potential applications in network geometry. We begin by linking each point of the…

Starting from the hypothesis that both physics, in particular space-time and the physical vacuum, and the corresponding mathematics are discrete on the Planck scale we develop a certain framework in form of a class of ' cellular networks'…

Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…

We present medial parametrization, a new approach to parameterizing any compact planar domain bounded by simple closed curves. The basic premise behind our proposed approach is to use two close Voronoi sites, which we call dipoles, to…

This is a survey article written for the Jahresberichte der DMV. Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and…

We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties…

Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck scale, one of the many problems one has to face in this enterprise is to find the…

We extend Cellular Automata to time-varying discrete geometries. In other words we formalize, and prove theorems about, the intuitive idea of a discrete manifold which evolves in time, subject to two natural constraints: the evolution does…

We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors. Recently this technique…

We describe how to approximate the Riemann curvature tensor as well as sectional curvatures on possibly infinite-dimensional shape spaces that can be thought of as Riemannian manifolds. To this end, we extend the variational time…

The geometry of the moduli space of stable spin curves is studied, with emphasis on its combinatorial properties. In this context, the standard graph theoretic framework is not just a book-keeping device: some purely combinatorial results…

A combinatorial tiling of the sphere is naturally given by an embedded graph. We study the case that each tile has exactly five edges, with the ultimate goal of classifying combinatorial tilings of the sphere by geometrically congruent…

The Morse-Smale complex is a standard tool in visual data analysis. The classic definition is based on a continuous view of the gradient of a scalar function where its zeros are the critical points. These points are connected via gradient…