Density of Range Capturing Hypergraphs

For a finite set $X$ of points in the plane, a set $S$ in the plane, and a positive integer $k$, we say that a $k$-element subset $Y$ of $X$ is captured by $S$ if there is a homothetic copy $S'$ of $S$ such that $X\cap S' = Y$, i.e., $S'$ contains exactly $k$ elements from $X$. A $k$-uniform $S$-capturing hypergraph $H = H(X,S,k)$ has a vertex set $X$ and a hyperedge set consisting of all $k$-element subsets of $X$ captured by $S$. In case when $k=2$ and $S$ is convex these graphs are planar graphs, known as convex distance function Delaunay graphs. In this paper we prove that for any $k\geq 2$, any $X$, and any convex compact set $S$, the number of hyperedges in $H(X,S,k)$ is at most $(2k-1)|X| - k^2 + 1 - \sum_{i=1}^{k-1}a_i$, where $a_i$ is the number of $i$-element subsets of $X$ that can be separated from the rest of $X$ with a straight line. In particular, this bound is independent of $S$ and indeed the bound is tight for all "round" sets $S$ and point sets $X$ in general position with respect to $S$. This refines a general result of Buzaglo, Pinchasi and Rote stating that every pseudodisc topological hypergraph with vertex set $X$ has $O(k^2|X|)$ hyperedges of size $k$ or less.Comment: new version with a tight result and shorter proo

Combined 3D thinning and greedy algorithm to approximate realistic particles with corrected mechanical properties

The shape of irregular particles has significant influence on micro- and macro-scopic behavior of granular systems. This paper presents a combined 3D thinning and greedy set-covering algorithm to approximate realistic particles with a clump of overlapping spheres for discrete element method (DEM) simulations. First, the particle medial surface (or surface skeleton), from which all candidate (maximal inscribed) spheres can be generated, is computed by the topological 3D thinning. Then, the clump generation procedure is converted into a greedy set-covering (SCP) problem. To correct the mass distribution due to highly overlapped spheres inside the clump, linear programming (LP) is used to adjust the density of each component sphere, such that the aggregate properties mass, center of mass and inertia tensor are identical or close enough to the prototypical particle. In order to find the optimal approximation accuracy (volume coverage: ratio of clump's volume to the original particle's volume), particle flow of 3 different shapes in a rotating drum are conducted. It was observed that the dynamic angle of repose starts to converge for all particle shapes at 85% volume coverage (spheres per clump < 30), which implies the possible optimal resolution to capture the mechanical behavior of the system.Comment: 34 pages, 13 figure

Separation-Sensitive Collision Detection for Convex Objects

We develop a class of new kinetic data structures for collision detection between moving convex polytopes; the performance of these structures is sensitive to the separation of the polytopes during their motion. For two convex polygons in the plane, let $D$ be the maximum diameter of the polygons, and let $s$ be the minimum distance between them during their motion. Our separation certificate changes $O(\log(D/s))$ times when the relative motion of the two polygons is a translation along a straight line or convex curve, $O(\sqrt{D/s})$ for translation along an algebraic trajectory, and $O(D/s)$ for algebraic rigid motion (translation and rotation). Each certificate update is performed in $O(\log(D/s))$ time. Variants of these data structures are also shown that exhibit \emph{hysteresis}---after a separation certificate fails, the new certificate cannot fail again until the objects have moved by some constant fraction of their current separation. We can then bound the number of events by the combinatorial size of a certain cover of the motion path by balls.Comment: 10 pages, 8 figures; to appear in Proc. 10th Annual ACM-SIAM Symposium on Discrete Algorithms, 1999; see also http://www.uiuc.edu/ph/www/jeffe/pubs/kollide.html ; v2 replaces submission with camera-ready versio

The robustness of equilibria on convex solids

We examine the minimal magnitude of perturbations necessary to change the number $N$ of static equilibrium points of a convex solid $K$. We call the normalized volume of the minimally necessary truncation robustness and we seek shapes with maximal robustness for fixed values of $N$. While the upward robustness (referring to the increase of $N$) of smooth, homogeneous convex solids is known to be zero, little is known about their downward robustness. The difficulty of the latter problem is related to the coupling (via integrals) between the geometry of the hull \bd K and the location of the center of gravity $G$. Here we first investigate two simpler, decoupled problems by examining truncations of \bd K with $G$ fixed, and displacements of $G$ with \bd K fixed, leading to the concept of external \rm and internal \rm robustness, respectively. In dimension 2, we find that for any fixed number $N=2S$, the convex solids with both maximal external and maximal internal robustness are regular $S$-gons. Based on this result we conjecture that regular polygons have maximal downward robustness also in the original, coupled problem. We also show that in the decoupled problems, 3-dimensional regular polyhedra have maximal internal robustness, however, only under additional constraints. Finally, we prove results for the full problem in case of 3 dimensional solids. These results appear to explain why monostatic pebbles (with either one stable, or one unstable point of equilibrium) are found so rarely in Nature.Comment: 20 pages, 6 figure