{"id":{"repo_id":"ohiolink","oai_identifier":"oai:etd.ohiolink.edu:ucin1352402504"},"canonical_url":"https://search.dev.ndltd.org/etd/ohiolink/oai:etd.ohiolink.edu:ucin1352402504","repository":{"repo_id":"ohiolink","name":"OhioLINK","base_url":"https://etd.ohiolink.edu/acprod/odb_etd/ws/oai/oai"},"display":{"title":"Problems and Results in Discrete and Computational Geometry","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Smith, Justin W."],"institution":"University of Cincinnati","degree_name":"PhD","degree_level":"doctoral","degree_discipline":"Engineering and Applied Science: Computer Science and Engineering","degree_department":null,"school":null,"contributors":["Purdy, George"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012","date_published":"2012","updated_at":"2026-07-24T03:36:23Z","subjects":["Computer Science","pseudoline arrangement","discrete geometry","dirac conjecture","orchard problem"],"languages":["English"],"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."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://rave.ohiolink.edu/etdc/view?acc_num=ucin1352402504","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Purdy, George"]},{"key":"dc:creator","label":"Author","values":["Smith, Justin W."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012"]},{"key":"dc:publisher","label":"Institution","values":["University of Cincinnati / OhioLINK"]},{"key":"dc:type","label":"Dc Type","values":["Electronic Thesis or Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Engineering and Applied Science: Computer Science and Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["PhD"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Cincinnati"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Computer Science","pseudoline arrangement","discrete geometry","dirac conjecture","orchard problem"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]},{"key":"dc:rights","label":"Dc Rights","values":["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."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://rave.ohiolink.edu/etdc/view?acc_num=ucin1352402504"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf","p.85","664.2 KB"]},{"key":"dc:title","label":"Title","values":["Problems and Results in Discrete and Computational Geometry"]}]}],"canonical_facts":{"dc:contributor":["Purdy, George"],"dc:creator":["Smith, Justin W."],"dc:date":["2012"],"dc:description":["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."],"dc:format":["application/pdf","p.85","664.2 KB"],"dc:identifier":["http://rave.ohiolink.edu/etdc/view?acc_num=ucin1352402504"],"dc:language":["English"],"dc:publisher":["University of Cincinnati / OhioLINK"],"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."],"dc:subject":["Computer Science","pseudoline arrangement","discrete geometry","dirac conjecture","orchard problem"],"dc:title":["Problems and Results in Discrete and Computational Geometry"],"dc:type":["Electronic Thesis or Dissertation"],"thesis:degree_discipline":["Engineering and Applied Science: Computer Science and Engineering"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["PhD"],"thesis:institution_name":["University of Cincinnati"]},"updated_at":"2026-07-24T03:36:23Z"}