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