313 research outputs found
Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere
We present two exact implementations of efficient output-sensitive algorithms
that compute Minkowski sums of two convex polyhedra in 3D. We do not assume
general position. Namely, we handle degenerate input, and produce exact
results. We provide a tight bound on the exact maximum complexity of Minkowski
sums of polytopes in 3D in terms of the number of facets of the summand
polytopes. The algorithms employ variants of a data structure that represents
arrangements embedded on two-dimensional parametric surfaces in 3D, and they
make use of many operations applied to arrangements in these representations.
We have developed software components that support the arrangement
data-structure variants and the operations applied to them. These software
components are generic, as they can be instantiated with any number type.
However, our algorithms require only (exact) rational arithmetic. These
software components together with exact rational-arithmetic enable a robust,
efficient, and elegant implementation of the Minkowski-sum constructions and
the related applications. These software components are provided through a
package of the Computational Geometry Algorithm Library (CGAL) called
Arrangement_on_surface_2. We also present exact implementations of other
applications that exploit arrangements of arcs of great circles embedded on the
sphere. We use them as basic blocks in an exact implementation of an efficient
algorithm that partitions an assembly of polyhedra in 3D with two hands using
infinite translations. This application distinctly shows the importance of
exact computation, as imprecise computation might result with dismissal of
valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages
long. The advisor was Prof. Dan Halperi
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
Minkowski Functional Description of Microwave Background Gaussianity
A Gaussian distribution of cosmic microwave background temperature
fluctuations is a generic prediction of inflation. Upcoming high-resolution
maps of the microwave background will allow detailed tests of Gaussianity down
to small angular scales, providing a crucial test of inflation. We propose
Minkowski functionals as a calculational tool for testing Gaussianity and
characterizing deviations from it. We review the mathematical formalism of
Minkowski functionals of random fields; for Gaussian fields the functionals can
be calculated exactly. We then apply the results to pixelized maps, giving
explicit expressions for calculating the functionals from maps as well as the
Gaussian predictions, including corrections for map boundaries, pixel noise,
and pixel size and shape. Variances of the functionals for Gaussian
distributions are derived in terms of the map correlation function.
Applications to microwave background maps are discussed.Comment: 24 pages with 2 figures. Submitted to New Astronom
Polygon Distances with Applications to High Performance Liquid Chromatography
Analytical chemistry uses high performance liquid chromatography (HPLC) to separate desired macromolecules from a fluid. This thesis is concerned with monolithic column environments used in HPLC separations. The monolithic column environment under consideration consists of many long polymer fibers suspended in a tube. Between the fibers are voids (interstices) which regulate the separation process. The goal of this thesis is both to identify interstices of a size such that the interstice contributes to the separation process, and to identify the boundaries of the fibers along such interstices. We model the cross section of the column environment with a collection of polygons. We define the desired interstices and boundaries in the model. We implement (using C++) an algorithmic approach to identify the desired boundaries from which we generate images and summarize relevant data
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