58,154 research outputs found
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
Sixteen space-filling curves and traversals for d-dimensional cubes and simplices
This article describes sixteen different ways to traverse d-dimensional space
recursively in a way that is well-defined for any number of dimensions. Each of
these traversals has distinct properties that may be beneficial for certain
applications. Some of the traversals are novel, some have been known in
principle but had not been described adequately for any number of dimensions,
some of the traversals have been known. This article is the first to present
them all in a consistent notation system. Furthermore, with this article, tools
are provided to enumerate points in a regular grid in the order in which they
are visited by each traversal. In particular, we cover: five discontinuous
traversals based on subdividing cubes into 2^d subcubes: Z-traversal (Morton
indexing), U-traversal, Gray-code traversal, Double-Gray-code traversal, and
Inside-out traversal; two discontinuous traversals based on subdividing
simplices into 2^d subsimplices: the Hill-Z traversal and the Maehara-reflected
traversal; five continuous traversals based on subdividing cubes into 2^d
subcubes: the Base-camp Hilbert curve, the Harmonious Hilbert curve, the Alfa
Hilbert curve, the Beta Hilbert curve, and the Butz-Hilbert curve; four
continuous traversals based on subdividing cubes into 3^d subcubes: the Peano
curve, the Coil curve, the Half-coil curve, and the Meurthe curve. All of these
traversals are self-similar in the sense that the traversal in each of the
subcubes or subsimplices of a cube or simplex, on any level of recursive
subdivision, can be obtained by scaling, translating, rotating, reflecting
and/or reversing the traversal of the complete unit cube or simplex.Comment: 28 pages, 12 figures. v2: fixed a confusing typo on page 12, line
MICC: A tool for computing short distances in the curve complex
The complex of curves of a closed orientable surface of
genus is the simplicial complex having its vertices,
, are isotopy classes of essential curves in . Two
vertices co-bound an edge of the -skeleton, , if there
are disjoint representatives in . A metric is obtained on
by assigning unit length to each edge of
. Thus, the distance between two vertices, ,
corresponds to the length of a geodesic---a shortest edge-path between and
in . Recently, Birman, Margalit and the second author
introduced the concept of {\em initially efficient geodesics} in
and used them to give a new algorithm for computing the
distance between vertices. In this note we introduce the software package MICC
({\em Metric in the Curve Complex}), a partial implementation of the initially
efficient geodesic algorithm. We discuss the mathematics underlying MICC and
give applications. In particular, we give examples of distance four vertex
pairs, for and 3. Previously, there was only one known example, in genus
, due to John Hempel.Comment: 19 pages, 9 figures, Version 2 has updated figures and reference
From 3D Models to 3D Prints: an Overview of the Processing Pipeline
Due to the wide diffusion of 3D printing technologies, geometric algorithms
for Additive Manufacturing are being invented at an impressive speed. Each
single step, in particular along the Process Planning pipeline, can now count
on dozens of methods that prepare the 3D model for fabrication, while analysing
and optimizing geometry and machine instructions for various objectives. This
report provides a classification of this huge state of the art, and elicits the
relation between each single algorithm and a list of desirable objectives
during Process Planning. The objectives themselves are listed and discussed,
along with possible needs for tradeoffs. Additive Manufacturing technologies
are broadly categorized to explicitly relate classes of devices and supported
features. Finally, this report offers an analysis of the state of the art while
discussing open and challenging problems from both an academic and an
industrial perspective.Comment: European Union (EU); Horizon 2020; H2020-FoF-2015; RIA - Research and
Innovation action; Grant agreement N. 68044
Recommended from our members
SFF-Oriented Modeling and Process Planning of Functionally Graded Materials Using a Novel Equal Distance Offset Approach
This paper deals with the modeling and process planning of solid freeform fabrication
(SFF) of 3D functionally graded materials (FGMs). A novel approach of representation and
process planning of FGMs, termed as equal distance offset (EDO), is developed. In EDO, a
neutral arbitrary 3D CAD model is adaptively sliced into a series of 2D layers. Within each
layer, 2D material gradients are designed and represented via dividing the 2D shape into
several sub-regions enclosed by iso-composition contours. If needed, the material
composition gradient within each of sub-regions can be further determined by applying the
equal distance offset algorithm to each sub-region. Using this approach, an arbitrary-shaped
3D FGM object with linear or non-linear composition gradients can be represented and
fabricated via suitable SFF machines.Mechanical Engineerin
Critical Percolation Exploration Path and SLE(6): a Proof of Convergence
It was argued by Schramm and Smirnov that the critical site percolation
exploration path on the triangular lattice converges in distribution to the
trace of chordal SLE(6). We provide here a detailed proof, which relies on
Smirnov's theorem that crossing probabilities have a conformally invariant
scaling limit (given by Cardy's formula). The version of convergence to SLE(6)
that we prove suffices for the Smirnov-Werner derivation of certain critical
percolation crossing exponents and for our analysis of the critical percolation
full scaling limit as a process of continuum nonsimple loops.Comment: 45 pages, 14 figures; revised version following the comments of a
refere
Optimal Filling of Shapes
We present filling as a type of spatial subdivision problem similar to
covering and packing. Filling addresses the optimal placement of overlapping
objects lying entirely inside an arbitrary shape so as to cover the most
interior volume. In n-dimensional space, if the objects are polydisperse
n-balls, we show that solutions correspond to sets of maximal n-balls. For
polygons, we provide a heuristic for finding solutions of maximal discs. We
consider the properties of ideal distributions of N discs as N approaches
infinity. We note an analogy with energy landscapes.Comment: 5 page
Stable commutator length in word-hyperbolic groups
In this paper we obtain uniform positive lower bounds on stable commutator
length in word-hyperbolic groups and certain groups acting on hyperbolic spaces
(namely the mapping class group acting on the complex of curves, and an
amalgamated free product acting on the Bass-Serre tree). If G is a word
hyperbolic group which is delta hyperbolic with respect to a symmetric
generating set S, then there is a positive constant C depending only on delta
and on |S| such that every element of G either has a power which is conjugate
to its inverse, or else the stable commutator length is at least equal to C. By
Bavard's theorem, these lower bounds on stable commutator length imply the
existence of quasimorphisms with uniform control on the defects; however, we
show how to construct such quasimorphisms directly.
We also prove various separation theorems, constructing homogeneous
quasimorphisms (again with uniform estimates) which are positive on some
prescribed element while vanishing on some family of independent elements whose
translation lengths are uniformly bounded.
Finally, we prove that the first accumulation point for stable commutator
length in a torsion-free word hyperbolic group is contained between 1/12 and
1/2. This gives a universal sense of what it means for a conjugacy class in a
hyperbolic group to have a small stable commutator length, and can be thought
of as a kind of "homological Margulis lemma".Comment: 27 pages, 1 figures; version 4: incorporates referee's suggestion
- ā¦