Problems and Results in Discrete and Computational Geometry

Problems and Results in Discrete and Computational Geometry

Bibliographic Details
Main Author: Smith, Justin W.
Language:English
Published: University of Cincinnati / OhioLINK 2012
Subjects:
Computer Science
pseudoline arrangement
discrete geometry
dirac conjecture
orchard problem
Online Access:http://rave.ohiolink.edu/etdc/view?acc_num=ucin1352402504
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spelling ndltd-OhioLink-oai-etd.ohiolink.edu-ucin13524025042021-08-03T05:20:30Z Problems and Results in Discrete and Computational Geometry Smith, Justin W. Computer Science pseudoline arrangement discrete geometry dirac conjecture orchard problem Let S be a set of n points in R^3 , no three collinear and not all coplanar. Ifat most n - k are coplanar and n is sufficiently large, the total number ofplanes determined is at least 1 + k * binom(n-k,2) - ((n-k)/2) * binom(k, 2). For similar conditions and sufficiently large n, (inspired by the work of P. D. T. A. Elliott in [1]) wealso show that the number of spheres determined by n points is at least 1 + binom(n-1,3) - t^{orchard}_{3} (n-1), and this bound is best possible under its hypothesis. (Byt^{orchard}_{3} , we are denoting the maximum number of three-point lines attainableby a configuration of n points, no four collinear, in the plane, i.e., the classicOrchard Problem.) New lower bounds are also given for both lines and circles.We demonstrate an infinite family of pseudoline arrangements each with nomember incident to more than (4n - 10)/9 points of intersection, where n is thenumber of pseudolines in the arrangement. We also prove a generalization ofthe Weak Dirac that holds for more general incidence structures. 2012 English text University of Cincinnati / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=ucin1352402504 http://rave.ohiolink.edu/etdc/view?acc_num=ucin1352402504 unrestricted This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws.
collection NDLTD
language English
sources NDLTD
topic Computer Science
pseudoline arrangement
discrete geometry
dirac conjecture
orchard problem
spellingShingle Computer Science
pseudoline arrangement
discrete geometry
dirac conjecture
orchard problem
Smith, Justin W.
Problems and Results in Discrete and Computational Geometry
author Smith, Justin W.
author_facet Smith, Justin W.
author_sort Smith, Justin W.
title Problems and Results in Discrete and Computational Geometry
title_short Problems and Results in Discrete and Computational Geometry
title_full Problems and Results in Discrete and Computational Geometry
title_fullStr Problems and Results in Discrete and Computational Geometry
title_full_unstemmed Problems and Results in Discrete and Computational Geometry
title_sort problems and results in discrete and computational geometry
publisher University of Cincinnati / OhioLINK
publishDate 2012
url http://rave.ohiolink.edu/etdc/view?acc_num=ucin1352402504
work_keys_str_mv AT smithjustinw problemsandresultsindiscreteandcomputationalgeometry
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