8,690 research outputs found
Static 3D Triangle Mesh Compression Overview
3D triangle meshes are extremely used to model discrete surfaces, and almost always represented with two tables: one for geometry and another for connectivity. While the raw size of a triangle mesh is of around 200 bits per vertex, by coding cleverly (and separately) those two distinct kinds of information it is possible to achieve compression ratios of 15:1 or more. Different techniques must be used depending on whether single-rate vs. progressive bitstreams are sought; and, in the latter case, on whether or not hierarchically nested meshes are desirable during reconstructio
Epitaxial Frustration in Deposited Packings of Rigid Disks and Spheres
We use numerical simulation to investigate and analyze the way that rigid
disks and spheres arrange themselves when compressed next to incommensurate
substrates. For disks, a movable set is pressed into a jammed state against an
ordered fixed line of larger disks, where the diameter ratio of movable to
fixed disks is 0.8. The corresponding diameter ratio for the sphere simulations
is 0.7, where the fixed substrate has the structure of a (001) plane of a
face-centered cubic array. Results obtained for both disks and spheres exhibit
various forms of density-reducing packing frustration next to the
incommensurate substrate, including some cases displaying disorder that extends
far from the substrate. The disk system calculations strongly suggest that the
most efficient (highest density) packings involve configurations that are
periodic in the lateral direction parallel to the substrate, with substantial
geometric disruption only occurring near the substrate. Some evidence has also
emerged suggesting that for the sphere systems a corresponding structure doubly
periodic in the lateral directions would yield the highest packing density;
however all of the sphere simulations completed thus far produced some residual
"bulk" disorder not obviously resulting from substrate mismatch. In view of the
fact that the cases studied here represent only a small subset of all that
eventually deserve attention, we end with discussion of the directions in which
first extensions of the present simulations might profitably be pursued.Comment: 28 pages, 14 figures; typos fixed; a sentence added to 4th paragraph
of sect 5 in responce to a referee's comment
Non-Euclidean geometry in nature
I describe the manifestation of the non-Euclidean geometry in the behavior of
collective observables of some complex physical systems. Specifically, I
consider the formation of equilibrium shapes of plants and statistics of sparse
random graphs. For these systems I discuss the following interlinked questions:
(i) the optimal embedding of plants leaves in the three-dimensional space, (ii)
the spectral statistics of sparse random matrix ensembles.Comment: 52 pages, 21 figures, last section is rewritten, a reference to
chaotic Hamiltonian systems is adde
Improved Compression of the Okamura-Seymour Metric
Let be an undirected unweighted planar graph. Consider a vector
storing the distances from an arbitrary vertex to all vertices of a single face in their cyclic order. The pattern of
is obtained by taking the difference between every pair of consecutive
values of this vector. In STOC'19, Li and Parter used a VC-dimension argument
to show that in planar graphs, the number of distinct patterns, denoted , is
only . This resulted in a simple compression scheme requiring space to encode the distances between and
a subset of terminal vertices . This is known as the
Okamura-Seymour metric compression problem.
We give an alternative proof of the bound that exploits planarity
beyond the VC-dimension argument. Namely, our proof relies on cut-cycle
duality, as well as on the fact that distances among vertices of are
bounded by . Our method implies the following:
(1) An space compression of the Okamura-Seymour metric,
thus improving the compression of Li and Parter to .
(2) An optimal space compression of the Okamura-Seymour
metric, in the case where the vertices of induce a connected component in
.
(3) A tight bound of for the family of Halin graphs,
whereas the VC-dimension argument is limited to showing
Single-Strip Triangulation of Manifolds with Arbitrary Topology
Triangle strips have been widely used for efficient rendering. It is
NP-complete to test whether a given triangulated model can be represented as a
single triangle strip, so many heuristics have been proposed to partition
models into few long strips. In this paper, we present a new algorithm for
creating a single triangle loop or strip from a triangulated model. Our method
applies a dual graph matching algorithm to partition the mesh into cycles, and
then merges pairs of cycles by splitting adjacent triangles when necessary. New
vertices are introduced at midpoints of edges and the new triangles thus formed
are coplanar with their parent triangles, hence the visual fidelity of the
geometry is not changed. We prove that the increase in the number of triangles
due to this splitting is 50% in the worst case, however for all models we
tested the increase was less than 2%. We also prove tight bounds on the number
of triangles needed for a single-strip representation of a model with holes on
its boundary. Our strips can be used not only for efficient rendering, but also
for other applications including the generation of space filling curves on a
manifold of any arbitrary topology.Comment: 12 pages, 10 figures. To appear at Eurographics 200
Near-Optimal Distance Emulator for Planar Graphs
Given a graph G and a set of terminals T, a distance emulator of G is another graph H (not necessarily a subgraph of G) containing T, such that all the pairwise distances in G between vertices of T are preserved in H. An important open question is to find the smallest possible distance emulator.
We prove that, given any subset of k terminals in an n-vertex undirected unweighted planar graph, we can construct in O~(n) time a distance emulator of size O~(min(k^2,sqrt{k * n})). This is optimal up to logarithmic factors. The existence of such distance emulator provides a straightforward framework to solve distance-related problems on planar graphs: Replace the input graph with the distance emulator, and apply whatever algorithm available to the resulting emulator. In particular, our result implies that, on any unweighted undirected planar graph, one can compute all-pairs shortest path distances among k terminals in O~(n) time when k=O(n^{1/3})
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