250,248 research outputs found
BSP-fields: An Exact Representation of Polygonal Objects by Differentiable Scalar Fields Based on Binary Space Partitioning
The problem considered in this work is to find a dimension independent algorithm for the generation of signed scalar fields exactly representing polygonal objects and satisfying the following requirements: the defining real function takes zero value exactly at the polygonal object boundary; no extra zero-value isosurfaces should be generated; C1 continuity of the function in the entire domain. The proposed algorithms are based on the binary space partitioning (BSP) of the object by the planes passing through the polygonal faces and are independent of the object genus, the number of disjoint components, and holes in the initial polygonal mesh. Several extensions to the basic algorithm are proposed to satisfy the selected optimization criteria. The generated BSP-fields allow for applying techniques of the function-based modeling to already existing legacy objects from CAD and computer animation areas, which is illustrated by several examples
Dendroidal Segal spaces and infinity-operads
We introduce the dendroidal analogs of the notions of complete Segal space
and of Segal category, and construct two appropriate model categories for which
each of these notions corresponds to the property of being fibrant. We prove
that these two model categories are Quillen equivalent to each other, and to
the monoidal model category for infinity-operads which we constructed in an
earlier paper. By slicing over the monoidal unit objects in these model
categories, we derive as immediate corollaries the known comparison results
between Joyal's quasi-categories, Rezk's complete Segal spaces, and Segal
categories.Comment: We replaced a wrong technical lemma by a correct proposition at the
begining of Section 8. This does not affect the main results of this article
(in particular, the end of Section 8 is unchanged). To appear in J. Topo
Degenerations of ideal hyperbolic triangulations
Let M be a cusped 3-manifold, and let T be an ideal triangulation of M. The
deformation variety D(T), a subset of which parameterises (incomplete)
hyperbolic structures obtained on M using T, is defined and compactified by
adding certain projective classes of transversely measured singular
codimension-one foliations of M. This leads to a combinatorial and geometric
variant of well-known constructions by Culler, Morgan and Shalen concerning the
character variety of a 3-manifold.Comment: 31 pages, 11 figures; minor changes; to appear in Mathematische
Zeitschrif
Rayleigh processes, real trees, and root growth with re-grafting
The real trees form a class of metric spaces that extends the class of trees
with edge lengths by allowing behavior such as infinite total edge length and
vertices with infinite branching degree. Aldous's Brownian continuum random
tree, the random tree-like object naturally associated with a standard Brownian
excursion, may be thought of as a random compact real tree. The continuum
random tree is a scaling limit as N tends to infinity of both a critical
Galton-Watson tree conditioned to have total population size N as well as a
uniform random rooted combinatorial tree with N vertices. The Aldous--Broder
algorithm is a Markov chain on the space of rooted combinatorial trees with N
vertices that has the uniform tree as its stationary distribution. We construct
and study a Markov process on the space of all rooted compact real trees that
has the continuum random tree as its stationary distribution and arises as the
scaling limit as N tends to infinity of the Aldous--Broder chain. A key
technical ingredient in this work is the use of a pointed Gromov--Hausdorff
distance to metrize the space of rooted compact real trees.Comment: 48 Pages. Minor revision of version of Feb 2004. To appear in
Probability Theory and Related Field
Provably Safe Robot Navigation with Obstacle Uncertainty
As drones and autonomous cars become more widespread it is becoming
increasingly important that robots can operate safely under realistic
conditions. The noisy information fed into real systems means that robots must
use estimates of the environment to plan navigation. Efficiently guaranteeing
that the resulting motion plans are safe under these circumstances has proved
difficult. We examine how to guarantee that a trajectory or policy is safe with
only imperfect observations of the environment. We examine the implications of
various mathematical formalisms of safety and arrive at a mathematical notion
of safety of a long-term execution, even when conditioned on observational
information. We present efficient algorithms that can prove that trajectories
or policies are safe with much tighter bounds than in previous work. Notably,
the complexity of the environment does not affect our methods ability to
evaluate if a trajectory or policy is safe. We then use these safety checking
methods to design a safe variant of the RRT planning algorithm.Comment: RSS 201
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