1,771 research outputs found
Deconstructing Approximate Offsets
We consider the offset-deconstruction problem: Given a polygonal shape Q with
n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance,
as the Minkowski sum of another polygonal shape P with a disk of fixed radius?
If it does, we also seek a preferably simple-looking solution P; then, P's
offset constitutes an accurate, vertex-reduced, and smoothened approximation of
Q. We give an O(n log n)-time exact decision algorithm that handles any
polygonal shape, assuming the real-RAM model of computation. A variant of the
algorithm, which we have implemented using CGAL, is based on rational
arithmetic and answers the same deconstruction problem up to an uncertainty
parameter \delta; its running time additionally depends on \delta. If the input
shape is found to be approximable, this algorithm also computes an approximate
solution for the problem. It also allows us to solve parameter-optimization
problems induced by the offset-deconstruction problem. For convex shapes, the
complexity of the exact decision algorithm drops to O(n), which is also the
time required to compute a solution P with at most one more vertex than a
vertex-minimal one.Comment: 18 pages, 11 figures, previous version accepted at SoCG 2011,
submitted to DC
Approximate Fitting of a Circular Arc When Two Points Are Known
The task of approximating points with circular arcs is performed in many
applications, such as polyline compression, noise filtering, and feature
recognition. However, the development of algorithms that perform a significant
amount of circular arcs fitting requires an efficient way of fitting circular
arcs with complexity O(1). The elegant solution to this task based on an
eigenvector problem for a square nonsymmetrical matrix is described in [1]. For
the compression algorithm described in [2], it is necessary to solve this task
when two points on the arc are known. This paper describes a different approach
to efficiently fitting the arcs and solves the task when one or two points are
known.Comment: 15 pages, 4 figures, extended abstract published at the conferenc
Polynomial Meshes: Computation and Approximation
We present the software package WAM, written in Matlab, that generates Weakly
Admissible Meshes and Discrete Extremal Sets of Fekete and Leja type, for 2d and 3d
polynomial least squares and interpolation on compact sets with various geometries.
Possible applications range from data fitting to high-order methods for PDEs
Subperiodic Dubiner distance, norming meshes and trigonometric polynomial optimization
We extend the notion of Dubiner distance from algebraic to trigonometric polynomials on subintervals of the period, and we obtain its explicit form by the Szego variant of Videnskii inequality. This allows to improve previous estimates for Chebyshev-like trigonometric norming meshes, and suggests a possible use of such meshes in the framework of multivariate polynomial optimization on regions defined by circular arcs
Criticality for the Gehring link problem
In 1974, Gehring posed the problem of minimizing the length of two linked
curves separated by unit distance. This constraint can be viewed as a measure
of thickness for links, and the ratio of length over thickness as the
ropelength. In this paper we refine Gehring's problem to deal with links in a
fixed link-homotopy class: we prove ropelength minimizers exist and introduce a
theory of ropelength criticality.
Our balance criterion is a set of necessary and sufficient conditions for
criticality, based on a strengthened, infinite-dimensional version of the
Kuhn--Tucker theorem. We use this to prove that every critical link is C^1 with
finite total curvature. The balance criterion also allows us to explicitly
describe critical configurations (and presumed minimizers) for many links
including the Borromean rings. We also exhibit a surprising critical
configuration for two clasped ropes: near their tips the curvature is unbounded
and a small gap appears between the two components. These examples reveal the
depth and richness hidden in Gehring's problem and our natural extension.Comment: This is the version published by Geometry & Topology on 14 November
200
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