On the size of lattice simplices with a single interior lattice point

Let $\mathcal{T}^d(1)$ be the set of all $d$-dimensional simplices $T$ in $\real^d$ with integer vertices and a single integer point in the interior of $T$. It follows from a result of Hensley that $\mathcal{T}^d(1)$ is finite up to affine transformations that preserve $\mathbb{Z}^d$. It is known that, when $d$ grows, the maximum volume of the simplices T \in \cT^d(1) becomes extremely large. We improve and refine bounds on the size of $T \in \mathcal{T}^d(1)$ (where by the size we mean the volume or the number of lattice points). It is shown that each $T \in \mathcal{T}^d(1)$ can be decomposed into an ascending chain of faces whose sizes are `not too large'. More precisely, if $T \in \mathcal{T}^d(1)$, then there exist faces $G_1 \subseteq ... \subseteq G_d=T$ of $T$ such that, for every $i \in \{1,...,d\}$, $G_i$ is $i$-dimensional and the size of $G_i$ is bounded from above in terms of $i$ and $d$. The bound on the size of $G_i$ is double exponential in $i$. The presented upper bounds are asymptotically tight on the log-log scale.Comment: accepted in SIAM J. Discrete Mat

Fitting a Simplicial Complex using a Variation of k-means

We give a simple and effective two stage algorithm for approximating a point cloud $\mathcal{S}\subset\mathbb{R}^m$ by a simplicial complex $K$. The first stage is an iterative fitting procedure that generalizes k-means clustering, while the second stage involves deleting redundant simplices. A form of dimension reduction of $\mathcal{S}$ is obtained as a consequence

Bijective Deformations in Rn\mathbb{R}^n via Integral Curve Coordinates

We introduce Integral Curve Coordinates, which identify each point in a bounded domain with a parameter along an integral curve of the gradient of a function $f$ on that domain; suitable functions have exactly one critical point, a maximum, in the domain, and the gradient of the function on the boundary points inward. Because every integral curve intersects the boundary exactly once, Integral Curve Coordinates provide a natural bijective mapping from one domain to another given a bijection of the boundary. Our approach can be applied to shapes in any dimension, provided that the boundary of the shape (or cage) is topologically equivalent to an $n$-sphere. We present a simple algorithm for generating a suitable function space for $f$ in any dimension. We demonstrate our approach in 2D and describe a practical (simple and robust) algorithm for tracing integral curves on a (piecewise-linear) triangulated regular grid

Barycentric coordinate neighbourhoods in Riemannian manifolds

We quantify conditions that ensure that a signed measure on a Riemannian manifold has a well defined centre of mass. We then use this result to quantify the extent of a neighbourhood on which the Riemannian barycentric coordinates of a set of $n+1$ points on an $n$-manifold provide a true coordinate chart, i.e., the barycentric coordinates provide a diffeomorphism between a neighbourhood of a Euclidean simplex, and a neighbourhood containing the points on the manifold

Augmented Semantic Signatures of Airborne LiDAR Point Clouds for Comparison

LiDAR point clouds provide rich geometric information, which is particularly useful for the analysis of complex scenes of urban regions. Finding structural and semantic differences between two different three-dimensional point clouds, say, of the same region but acquired at different time instances is an important problem. A comparison of point clouds involves computationally expensive registration and segmentation. We are interested in capturing the relative differences in the geometric uncertainty and semantic content of the point cloud without the registration process. Hence, we propose an orientation-invariant geometric signature of the point cloud, which integrates its probabilistic geometric and semantic classifications. We study different properties of the geometric signature, which are an image-based encoding of geometric uncertainty and semantic content. We explore different metrics to determine differences between these signatures, which in turn compare point clouds without performing point-to-point registration. Our results show that the differences in the signatures corroborate with the geometric and semantic differences of the point clouds.Comment: 18 pages, 6 figures, 1 tabl

Interpolation Error Estimates for Mean Value Coordinates over Convex Polygons

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in [Gillette et al., AiCM, doi:10.1007/s10444-011-9218-z], we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles of the polygon approach pi.Comment: 20 pages, revised based on referees' comment

Gradient bounds for Wachspress coordinates on polytopes

We derive upper and lower bounds on the gradients of Wachspress coordinates defined over any simple convex d-dimensional polytope P. The bounds are in terms of a single geometric quantity h_*, which denotes the minimum distance between a vertex of P and any hyperplane containing a non-incident face. We prove that the upper bound is sharp for d=2 and analyze the bounds in the special cases of hypercubes and simplices. Additionally, we provide an implementation of the Wachspress coordinates on convex polyhedra using Matlab and employ them in a 3D finite element solution of the Poisson equation on a non-trivial polyhedral mesh. As expected from the upper bound derivation, the H^1-norm of the error in the method converges at a linear rate with respect to the size of the mesh elements.Comment: 18 pages, to appear in SINU

Orthocentric simplices and their centers

A simplex is said to be orthocentric if its altitudes intersect in a common point, called its orthocenter. In this paper it is proved that if any two of the traditional centers of an orthocentric simplex (in any dimension) coincide, then the simplex is regular. Along the way orthocentric simplices in which all facets have the same circumradius are characterized, and the possible barycentric coordinates of the orthocenter are described precisely. In particular these barycentric coordinates are used to parametrize the shapes of orthocentric simplices. The substantial, but widespread, literature on orthocentric simplices is briefly surveyed in order to place the new results in their proper context, and some of the previously known results are given new proofs from the present perspective.Comment: 25 page

Error Estimates for Generalized Barycentric Interpolation

We prove the optimal convergence estimate for first order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the well-known maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions.Comment: 21 pages, 10 figures. Accepted to Advances in Computational Mathematics, April, 201

Inequalities for the lattice width of lattice-free convex sets in the plane

A closed, convex set $K$ in $\mathbb{R}^2$ with non-empty interior is called lattice-free if the interior of $K$ is disjoint with $\mathbb{Z}^2$. In this paper we study the relation between the area and the lattice width of a planar lattice-free convex set in the general and centrally symmetric case. A correspondence between lattice width on the one hand and covering minima on the other, allows us to reformulate our results in terms of covering minima introduced by Kannan and Lov\'asz. We obtain a sharp upper bound for the area for any given value of the lattice width. The lattice-free convex sets satisfying the upper bound are characterized. Lower bounds are studied as well. Parts of our results are applied in a paper by the authors and Weismantel for cutting plane generation in mixed integer linear optimization, which was the original inducement for this paper. We further rectify a result of Kannan and Lov\'asz with a new proof.Comment: to appear in Beitr\"age Algebra Geo