arXiv subjects
Publications and source records attributed to Steven J. Gortler.
This note gives a detailed proof of the following statement. Let $d\in \mathbb{N}$ and $m,n \ge d + 1$, with $m + n \ge \binom{d+2}{2} + 1$. Then the complete bipartite graph $K_{m,n}$ is generically globally rigid in dimension $d$.
This paper studies the set of equivalent realizations of isostatic frameworks in two dimensions, and algorithms for finding all such realizations. We show that an isostatic framework has an even number of equivalent realizations that preserve edge lengths and connectivity. We enumerate the complete set of equivalent realizations for a toy framework with pinned boundary in two dimensions and study the impact of boundary length on the emergence of these realizations. To ameliorate the computational complexity of finding a solution to a large multivariate quadratic system corresponding to the constraints; alternative methods - based on constraint reduction and distance-based covering map or Cayley parameterization of the search space - are presented. The application of these methods is studied on atomic clusters, a model two-dimensional glasses, and jamming.
Let $\mathbf{p}$ be a configuration of $n$ points in $\mathbb{R}^d$ for some $n$ and some $d \ge 2$. Each pair of points defines an edge, which has a Euclidean length in the configuration. A path is an ordered sequence of the points, and a loop is a path that has the same endpoints. A path or loop, as a sequence of edges, also has a Euclidean length. In this paper, we study the question of when $\mathbf{p}$ will be uniquely determined (up to an unknowable Euclidean transform) from a given set of path or loop lengths. In particular, we consider the setting where the lengths are given simply as a set of real numbers, and are not labeled with the combinatorial data describing the paths or loops that gave rise to the lengths. Our main result is a condition on the set of paths or loops that is sufficient to guarantee such a unique determination. We also provide an algorithm, under a real computational model, for performing a reconstruction of $\mathbf{p}$ from such unlabeled lengths. To obtain our results, we introduce a new family of algebraic varieties which we call the unsquared measurement varieties. The family is parameterized by the number of points $n$ and the dimension $d$, and our results follow from a complete characterization of the linear automorphisms of these varieties for all $n$ and $d$. The linear automorphisms for the special case of $n = 4$ and $d = 2$ correspond to the so-called Regge symmetries of the tetrahedron.
We extend the mathematical theory of rigidity of frameworks (graphs embedded in $d$-dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously it must remain inside a small ball, a property we call "almost-rigidity"; (II) any other framework with the same edge lengths must lie outside a much larger ball; (III) if the framework deforms by some given amount, its edge lengths change by a minimum amount; (IV) there is a nearby framework that is prestress stable, and thus rigid. The conditions can be tested efficiently using semidefinite programming. The test is a slight extension of the test for prestress stability of a framework, and gives analytic expressions for the radii of the balls and the edge length changes. Examples illustrate how the theory may be applied in practice, and we provide an algorithm to test for rigidity or almost-rigidity. We briefly discuss how the theory may be applied to tensegrities.
We introduce a new family of algebraic varieties, $L_{d,n}$, which we call the unsquared measurement varieties. This family is parameterized by a number of points $n$ and a dimension $d$. These varieties arise naturally from problems in rigidity theory and distance geometry. In those applications, it can be useful to understand the group of linear automorphisms of $L_{d,n}$. Notably, a result of Regge implies that $L_{2,4}$ has an unexpected linear automorphism. In this paper, we give a complete characterization of the linear automorphisms of $L_{d,n}$ for all $n$ and $d$. We show, that apart from $L_{2,4}$ the unsquared measurement varieties have no unexpected automorphisms. Moreover, for $L_{2,4}$ we characterize the full automorphism group.
We provide a new type of proof for the Koebe-Andreev-Thurston (KAT) planar circle packing theorem based on combinatorial edge-flips. In particular, we show that starting from a disk packing with a maximal planar contact graph $G$, one can remove any flippable edge $e^-$ of this graph and then continuously flow the disks in the plane, such that at the end of the flow, one obtains a new disk packing whose contact graph is the graph resulting from flipping the edge $e^-$ in $G$. This flow is parameterized by a single inversive distance.
Let $\mathbf{p}$ be a configuration of $n$ points in $\mathbb{R}^d$ for some $n$ and some $d \ge 2$. Each pair of points has a Euclidean length in the configuration. Given some graph $G$ on $n$ vertices, we measure the point-pair lengths corresponding to the edges of $G$. In this paper, we study the question of when a generic $\mathbf{p}$ in $d$ dimensions will be uniquely determined (up to an unknowable Euclidean transformation) from a given set of point-pair lengths together with knowledge of $d$ and $n$. In this setting the lengths are given simply as a set of real numbers; they are not labeled with the combinatorial data that describes which point-pair gave rise to which length, nor is data about $G$ given. We show, perhaps surprisingly, that in terms of generic uniqueness, labels have no effect. A generic configuration is determined by an unlabeled set of point-pair lengths (together with $d$ and $n$) iff it is determined by the labeled edge lengths.
We study the combinatorial and rigidity properties of disk packings with generic radii. We show that a packing of $n$ disks in the plane with generic radii cannot have more than $2n-3$ pairs of disks in contact. The allowed motions of a packing preserve the disjointness of the disk interiors and tangency between pairs already in contact (modeling a collection of sticky disks). We show that if a packing has generic radii, then the allowed motions are all rigid body motions if and only if the packing has exactly $2n-3$ contacts. Our approach is to study the space of packings with a fixed contact graph. The main technical step is to show that this space is a smooth manifold, which is done via a connection to the Cauchy-Alexandrov stress lemma. Our methods also apply to jamming problems, in which contacts are allowed to break during a motion. We give a simple proof of a finite variant of a recent result of Connelly, et al. on the number of contacts in a jammed packing of disks with generic radii.
We show that a jammed packing of disks with generic radii, in a generic container, is such that the minimal number of contacts occurs and there is only one dimension of equilibrium stresses. We also point out some connections to packings with different radii and results in the theory of circle packings whose graph forms a triangulation of a given topological surface. We also point out a counterexample, due to F. Nazarov, to a previous conjecture that that triangulated packings with fixed numbers of disks with fixed numbers of disks for each radius claiming that such packings were the most dense.
We show that any graph that is generically globally rigid in $\mathbb{R}^d$ has a realization in $\mathbb{R}^d$ that is both generic and universally rigid. This also implies that the graph also must have a realization in $\mathbb{R}^d$ that is both infinitesimally rigid and universally rigid; such a realization serves as a certificate of generic global rigidity. Our approach involves an algorithm by Lovász, Saks and Schrijver that, for a sufficiently connected graph, constructs a general position orthogonal representation of the vertices, and a result of Alfakih that shows how this representation leads to a stress matrix and a universally rigid framework of the graph.
We prove that universal second-order rigidity implies universal prestress stability and that triangulated convex polytopes in three-space (with holes appropriately positioned) are prestress stable.
In this paper we study the property of generic global rigidity for frameworks of graphs embedded in d-dimensional complex space and in a d-dimensional pseudo-Euclidean space ($R^d$ with a metric of indefinite signature). We show that a graph is generically globally rigid in Euclidean space iff it is generically globally rigid in a complex or pseudo-Euclidean space. We also establish that global rigidity is always a generic property of a graph in complex space, and give a sufficient condition for it to be a generic property in a pseudo-Euclidean space. Extensions to hyperbolic space are also discussed.
We prove that if a framework of a graph is neighborhood affine rigid in $d$-dimensions (or has the stronger property of having an equilibrium stress matrix of rank $n-d-1$) then it has an affine flex (an affine, but non Euclidean, transform of space that preserves all of the edge lengths) if and only if the framework is ruled on a single quadric. This strengthens and also simplifies a related result by Alfakih. It also allows us to prove that the property of super stability is invariant with respect to projective transforms and also to the coning and slicing operations. Finally this allows us to unify some previous results on the Strong Arnold Property of matrices.
We describe a very simple condition that is necessary for the universal rigidity of a complete bipartite framework $(K(n,m),p,q)$. This condition is also sufficient for universal rigidity under a variety of weak assumptions, such as general position. Even without any of these assumptions, in complete generality, we extend these ideas to obtain an efficient algorithm, based on a sequence of linear programs, that determines whether an input framework of a complete bipartite graph is universally rigid or not.
In this report we describe a mesh editing system that we implemented that uses a natural stretching and bending energy defined over smooth surfaces. As such, this energy behaves uniformly under various mesh resolutions. All of the elements of our approach already exist in the literature. We hope that our discussions of these energies helps to shed light on the behaviors of these methods and provides a unified discussion of these methods.
Given a two-dimensional space endowed with a divergence function that is convex in the first argument, continuously differentiable in the second, and satisfies suitable regularity conditions at Voronoi vertices, we show that orphan-freedom (the absence of disconnected Voronoi regions) is sufficient to ensure that Voronoi edges and vertices are also connected, and that the dual is a simple planar graph. We then prove that the straight-edge dual of an orphan-free Voronoi diagram (with sites as the first argument of the divergence) is always an embedded triangulation. Among the divergences covered by our proofs are Bregman divergences, anisotropic divergences, as well as all distances derived from strictly convex $\mathcal{C}^1$ norms (including the $L_p$ norms with $1< p < \infty$). While Bregman diagrams of the {first kind} are simply affine diagrams, and their duals ({weighted} Delaunay triangulations) are always embedded, we show that duals of orphan-free Bregman diagrams of the \emph{second kind} are always embedded.
We introduce a multi-scale framework for low-level vision, where the goal is estimating physical scene values from image data---such as depth from stereo image pairs. The framework uses a dense, overlapping set of image regions at multiple scales and a "local model," such as a slanted-plane model for stereo disparity, that is expected to be valid piecewise across the visual field. Estimation is cast as optimization over a dichotomous mixture of variables, simultaneously determining which regions are inliers with respect to the local model (binary variables) and the correct co-ordinates in the local model space for each inlying region (continuous variables). When the regions are organized into a multi-scale hierarchy, optimization can occur in an efficient and parallel architecture, where distributed computational units iteratively perform calculations and share information through sparse connections between parents and children. The framework performs well on a standard benchmark for binocular stereo, and it produces a distributional scene representation that is appropriate for combining with higher-level reasoning and other low-level cues.
A framework is a graph and a map from its vertices to E^d (for some d). A framework is universally rigid if any framework in any dimension with the same graph and edge lengths is a Euclidean image of it. We show that a generic universally rigid framework has a positive semi-definite stress matrix of maximal rank. Connelly showed that the existence of such a positive semi-definite stress matrix is sufficient for universal rigidity, so this provides a characterization of universal rigidity for generic frameworks. We also extend our argument to give a new result on the genericity of strict complementarity in semidefinite programming.