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Problems and Results in Discrete and Computational Geometry

Smith, Justin W.

Abstract Details

2012, PhD, University of Cincinnati, Engineering and Applied Science: Computer Science and Engineering.
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.
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)
85 p.

Recommended Citations

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)