Robert Connelly

Generic Global Rigidity

Abstract Suppose a finite configuration of labeled points p = (p1,. . . ,pn) in Ed is given along with certain pairs of those points determined by a graph G such that the coordinates of the points of p are generic, i.e., algebraically independent over the integers. If another corresponding configuration q = (q1,. . . ,qn) in Ed is given such that the corresponding edges of G for p and q have the same length, we provide a sufficient condition to ensure that p and q are congruent in Ed. This condition, together with recent results of Jackson and Jordán, give necessary and sufficient conditions for a graph being generically globally rigid in the plane.

Jamming in hard sphere and disk packings

Hard-particle packings have provided a rich source of outstanding theoretical problems and served as useful starting points to model the structure of granular media, liquids, living cells, glasses, and random media. The nature of “jammed” hard-particle packings is a current subject of keen interest. Elsewhere, we introduced rigorous and efficient linear-programming algorithms to assess whether a hard-sphere packing is locally, collectively, or strictly jammed, as defined by Torquato and Stillinger [J. Phys. Chem. B 105, 11849 (2001)]. One algorithm applies to ideal packings in which particles form perfect contacts. Another algorithm treats the case of jamming in packings with significant interparticle gaps. We have applied these algorithms to test jamming categories of ordered lattices as well as random packings of circular disks and spheres under periodic boundary conditions. The random packings were produced computationally with a variety of packing generation algorithms, all of which should, in princip...

Underconstrained jammed packings of nonspherical hard particles: ellipses and ellipsoids.

Continuing on recent computational and experimental work on jammed packings of hard ellipsoids [Donev, Science 303, 990 (2004)] we consider jamming in packings of smooth strictly convex nonspherical hard particles. We explain why an isocounting conjecture, which states that for large disordered jammed packings the average contact number per particle is twice the number of degrees of freedom per particle (Z[over]=2d{f}) , does not apply to nonspherical particles. We develop first- and second-order conditions for jamming and demonstrate that packings of nonspherical particles can be jammed even though they are underconstrained (hypoconstrained, Z[over]<2d{f}). We apply an algorithm using these conditions to computer-generated hypoconstrained ellipsoid and ellipse packings and demonstrate that our algorithm does produce jammed packings, even close to the sphere point. We also consider packings that are nearly jammed and draw connections to packings of deformable (but stiff) particles. Finally, we consider the jamming conditions for nearly spherical particles and explain quantitatively the behavior we observe in the vicinity of the sphere point.

Sudoku, Gerechte Designs, Resolutions, Affine Space, Spreads, Reguli, and Hamming Codes

Solving a Sudoku puzzle involves putting the symbols 1, . . . , 9 into the cells of a 9 × 9 grid partitioned into 3 × 3 subsquares, in such a way that each symbol occurs just once in each row, column, or subsquare. Such a solution is a special case of a gerechte design, in which an n×n grid is partitioned into n regions with n squares in each, and each of the symbols 1, . . . , n occurs once in each row, column, or region. Gerechte designs originated in statistical design of agricultural experiments, where they ensure that treatments are fairly exposed to localised variations in the field containing the experimental plots. In this paper we consider several related topics. In the first section, we define gerechte designs and some generalizations, and explain a computational technique for finding and classifying them. The second section looks at the statistical background, explaining how such designs are used for designing agricultural experiments, and what additional properties statisticians would like them to have. In the third section, we focus on a special class of Sudoku solutions which we call “symmetric”. They turn out to be related to some important topics in finite geometry over the 3-element field, and to ∗This research partially supported by NSF Grant Number DMS-0510625.

The rigidity of certain cabled frameworks and the second-order rigidity of arbitrarily triangulated convex surfaces

rigid, then the whole surface is rigid. It had been suspected that any (connected) polyhedral surface, convex or not, (with its triangular faces, say, held rigid) was rigid, but this has turned out to be false (see Connelly [7]). W e extend Cauchy’s theorem to show that any convex polyhedral surface, no matter how it is triangulated, is rigid. Note that vertices are allowed in the relative interior of the natural faces and edges. Here we say a triangulated surface in three-space is rigid if any continuous deformation -of the surface that keeps the distance fixed between any pair of points in each triangle (a part of the triangulation of the surface), keeps the distance fixed between any pair of points on the surface (and thus extends to an isometry of three-space). A natural face of a convex polyhedral surface is the two dimensional intersection of a support plane with the surface (see Fig. 3). Similarly natural edges and vertices are one- and zero-dimensional intersections, respectively. Thus in Cauchy’s theorem it is insisted that under a deformation of the surface the distance is fixed between any pair of points of a natural face and not just some triangle of a triangulation. Alexandrov [l] showed that if a convex polyhedral surface is triangulated with no vertices in the relative interior of a natural face (but they are allowed in natural edges), then the surface is rigid. This theorem is the basic result for the results of this paper. The author is grateful for a description of Alexandrov’s proof by Asimow and Roth [5]. Whiteley [ 151 also has a somewhat

On monotone paths among obstacles with applications to planning assemblies

We study the class of problems associated with the detection and computation of monotone paths among a set of disjoint obstacles. We give an <italic>&Ogr;</italic>(<italic>nE</italic>) algorithm for finding a monotone path (if one exists) between two points in the plane in the presence of polygonal obstacles. (Here, <italic>E</italic> is the size of the visibility graph defined by the <italic>n</italic> vertices of the obstacles.) If all of the obstacles are convex, we prove that there always exists a monotone path between any two points <italic>s</italic> and <italic>t</italic>. We give an <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic>) algorithm for finding such a path for any <italic>s</italic> and <italic>t</italic>, after an initial <italic>&Ogr;</italic>(<italic>E</italic> + <italic>n</italic> log <italic>n</italic>) preprocesing. We introduce the notions of “monotone path map”, and “shortest monotone path map” and give algorithms to compute them. We apply our results to a class of separation and assembly problems, yielding polynomial-time algorithms for planning an assembly sequence (based on separations by single translations) of arbitrary polygonal parts in two dimensions.

Blowing Up Polygonal Linkages

Abstract{Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewise-differentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular, this result settles the well-studied carpenter’s rule conjecture.

Straightening polygonal arcs and convexifying polygonal cycles

Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewise-differentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular this result settles the well-studied carpenters rule conjecture.

Realizability of Graphs

AbstractA graph is d-realizable if, for every configuration of its vertices in EN, there exists a another corresponding configuration in Ed with the same edge lengths. A graph is 2-realizable if and only if it is a partial 2-tree, i.e., a subgraph of the 2-sum of triangles in the sense of graph theory. We show that a graph is 3-realizable if and only if it does not have K5 or the 1-skeleton of the octahedron as a minor.

CHAPTER 1.7 – Rigidity

Publisher Summary This chapter discusses rigidity. There are two main categories for generalizations coming from Cauchys Theorem. One is in the category of polyhedra and similar discrete objects, such as frame works and hinged plates. The other generalization is in the category of appropriately smooth surfaces. There is also the question of exactly what kind of rigidity one is discussing, and this applies to both categories. In Cauchys theorem one thinks of each of the faces of the polytope as a rigid plate, and one stays in the configuration space of convex objects. Then the rigidity result is a statement about uniqueness in this space. On the other hand, one can also form a linearized definition of rigidity, called infinitesimal rigidity or first-order rigidity, in both the discrete and smooth categories. There are many similarities between the ideas in the discrete category and the smooth category. Other kinds of rigidity are second-order rigidity and pre-stress stability.