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ucin1352402504.pdf (664.17 KB)
ETD Abstract Container
Abstract Header
Problems and Results in Discrete and Computational Geometry
Author Info
Smith, Justin W.
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1352402504
Abstract Details
Year and Degree
2012, PhD, University of Cincinnati, Engineering and Applied Science: Computer Science and Engineering.
Abstract
Let S be a set of n points in R^3 , no three collinear and not all coplanar. If at most n - k are coplanar and n is sufficiently large, the total number of planes 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]) we also 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. (By t^{orchard}_{3} , we are denoting the maximum number of three-point lines attainable by a configuration of n points, no four collinear, in the plane, i.e., the classic Orchard Problem.) New lower bounds are also given for both lines and circles. We demonstrate an infinite family of pseudoline arrangements each with no member incident to more than (4n - 10)/9 points of intersection, where n is the number of pseudolines in the arrangement. We also prove a generalization of the Weak Dirac that holds for more general incidence structures.
Committee
George Purdy, PhD (Committee Chair)
Kenneth Berman, PhD (Committee Member)
Raj Bhatnagar, PhD (Committee Member)
Yizong Cheng, PhD (Committee Member)
Carla Purdy, PhD (Committee Member)
Pages
85 p.
Subject Headings
Computer Science
Keywords
pseudoline arrangement
;
discrete geometry
;
dirac conjecture
;
orchard problem
;
;
;
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Citations
Smith, J. W. (2012).
Problems and Results in Discrete and Computational Geometry
[Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1352402504
APA Style (7th edition)
Smith, Justin.
Problems and Results in Discrete and Computational Geometry.
2012. University of Cincinnati, Doctoral dissertation.
OhioLINK Electronic Theses and Dissertations Center
, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1352402504.
MLA Style (8th edition)
Smith, Justin. "Problems and Results in Discrete and Computational Geometry." Doctoral dissertation, University of Cincinnati, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1352402504
Chicago Manual of Style (17th edition)
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Document number:
ucin1352402504
Download Count:
510
Copyright Info
© 2012, all rights reserved.
This open access ETD is published by University of Cincinnati and OhioLINK.