Problems and Results in Discrete and Computational Geometry

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.

Name thesis:degree_name

PhD

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Engineering and Applied Science: Computer Science and Engineering

Grantor dc:publisher

University of Cincinnati

Year dc:date

2012

Author dc:creator

• Smith, Justin W.

Contributors dc:contributor

• Purdy, George

• Computer Science
• pseudoline arrangement
• discrete geometry
• dirac conjecture
• orchard problem

Statement dc:rights

• 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.

Language dc:language

English

Repository record dc:identifier

http://rave.ohiolink.edu/etdc/view?acc_num=ucin1352402504

OAI identifier oai:identifier

oai:etd.ohiolink.edu:ucin1352402504