{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/105618"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/105618","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Annular breadth of hinges & hinge exit paths of annuli","abstract":"DSpace SAF Submission Ingestion Package generated from Vireo submission #14081 on 2019-11-26 at 12:50:08","abstract_html":"DSpace SAF Submission Ingestion Package generated from Vireo submission #14081 on 2019-11-26 at 12:50:08","abstract_has_math":false,"creators":["Tichenor, Scott R."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Alexander, Stephanie","Reznick, Bruce","Wetzel, John E","Bishop, Richard"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-11-26T20:33:44Z","date_published":"2019-11-26T20:33:44Z","updated_at":"2026-07-22T22:24:44Z","subjects":["exit path","escape path","width","annulus","polygonal arc","breadth","Wetzel","broadworm","Voronoi","Rivlin"],"languages":["en"],"rights":["Copyright 2019 by Scott R. Tichenor. All rights reserved."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/105618","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Alexander, Stephanie","Reznick, Bruce","Wetzel, John E","Bishop, Richard"]},{"key":"dc:creator","label":"Author","values":["Tichenor, Scott R."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2019-11-26T20:33:44Z","2019-07-03","2019-08"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["exit path","escape path","width","annulus","polygonal arc","breadth","Wetzel","broadworm","Voronoi","Rivlin"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2019 by Scott R. Tichenor. All rights reserved."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/105618"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["DSpace SAF Submission Ingestion Package generated from Vireo submission #14081 on 2019-11-26 at 12:50:08","Made available in DSpace on 2019-11-26T20:33:44Z (GMT). No. of bitstreams: 4 TICHENOR-DISSERTATION-2019.pdf: 7142131 bytes, checksum: 8213cdc95f89b64b2dd7f06f3379e282 (MD5) Scott Tichenor Dissertation.url: 156 bytes, checksum: f2af9d5271e339b791ea670c3877593a (MD5) LICENSE.txt: 4211 bytes, checksum: 3d298a5de90d60fd96939c23c6c9ef71 (MD5) PROQUEST_LICENSE.txt: 4557 bytes, checksum: 170f8b59760dd782291ceb3ffdc5a8c1 (MD5) Previous issue date: 2019-07-03","Given a compact set $\\textsf{S}\\subset\\mathds{R}^2$, we define the annular width function for $\\textsf{S}$, denoted $w(E)$, as the width of the annulus of support of $\\textsf{S}$ centered at $E\\in\\overline{\\mathds{R}^2}$, where $\\overline{\\mathds{R}^2}$ is an extension of the real plane $\\mathds{R}^2$. The annular breadth of $\\textsf{S}$ is defined as the absolute minimum of $w(E)$. We find the $2$-segment polygonal arc with the greatest annular breadth. For a given set $\\textsf{S}\\subset\\mathds{R}^2$, an exit path of $\\textsf{S}$ is a curve that cannot be covered by the interior of $\\textsf{S}$. Given an annulus, we find its shortest $1$- or $2$-segment polygonal arc exit path(s). Bezdek and Connelly provided a lengthy and technically demanding proof that \\emph{All orbiforms of width} $1$ \\emph{are translation covers of the set of closed planar curves of length} $2$ \\emph{or less}. We provide a short and simple proof that \\emph{All orbiforms of width} $1$ \\emph{are covers of the set of all planar curves of length} $1$ \\emph{or less}. We also provide a proof that \\emph{The Reuleaux triangle of width} $1$ \\emph{is a cover of the set of all closed curves of length} $2$ using a recent of Wichiramala.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-11-26 without embargo terms","The student, Scott Tichenor, accepted the attached license on 2019-06-25 at 16:09.","The student, Scott Tichenor, submitted this Dissertation for approval on 2019-06-25 at 16:26.","This Dissertation was approved for publication on 2019-07-03 at 15:18."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Annular breadth of hinges & hinge exit paths of annuli"]}]}],"canonical_facts":{"dc:contributor":["Alexander, Stephanie","Reznick, Bruce","Wetzel, John E","Bishop, Richard"],"dc:creator":["Tichenor, Scott R."],"dc:date":["2019-11-26T20:33:44Z","2019-07-03","2019-08"],"dc:description":["DSpace SAF Submission Ingestion Package generated from Vireo submission #14081 on 2019-11-26 at 12:50:08","Made available in DSpace on 2019-11-26T20:33:44Z (GMT). No. of bitstreams: 4 TICHENOR-DISSERTATION-2019.pdf: 7142131 bytes, checksum: 8213cdc95f89b64b2dd7f06f3379e282 (MD5) Scott Tichenor Dissertation.url: 156 bytes, checksum: f2af9d5271e339b791ea670c3877593a (MD5) LICENSE.txt: 4211 bytes, checksum: 3d298a5de90d60fd96939c23c6c9ef71 (MD5) PROQUEST_LICENSE.txt: 4557 bytes, checksum: 170f8b59760dd782291ceb3ffdc5a8c1 (MD5) Previous issue date: 2019-07-03","Given a compact set $\\textsf{S}\\subset\\mathds{R}^2$, we define the annular width function for $\\textsf{S}$, denoted $w(E)$, as the width of the annulus of support of $\\textsf{S}$ centered at $E\\in\\overline{\\mathds{R}^2}$, where $\\overline{\\mathds{R}^2}$ is an extension of the real plane $\\mathds{R}^2$. The annular breadth of $\\textsf{S}$ is defined as the absolute minimum of $w(E)$. We find the $2$-segment polygonal arc with the greatest annular breadth. For a given set $\\textsf{S}\\subset\\mathds{R}^2$, an exit path of $\\textsf{S}$ is a curve that cannot be covered by the interior of $\\textsf{S}$. Given an annulus, we find its shortest $1$- or $2$-segment polygonal arc exit path(s). Bezdek and Connelly provided a lengthy and technically demanding proof that \\emph{All orbiforms of width} $1$ \\emph{are translation covers of the set of closed planar curves of length} $2$ \\emph{or less}. We provide a short and simple proof that \\emph{All orbiforms of width} $1$ \\emph{are covers of the set of all planar curves of length} $1$ \\emph{or less}. We also provide a proof that \\emph{The Reuleaux triangle of width} $1$ \\emph{is a cover of the set of all closed curves of length} $2$ using a recent of Wichiramala.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-11-26 without embargo terms","The student, Scott Tichenor, accepted the attached license on 2019-06-25 at 16:09.","The student, Scott Tichenor, submitted this Dissertation for approval on 2019-06-25 at 16:26.","This Dissertation was approved for publication on 2019-07-03 at 15:18."],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/105618"],"dc:language":["en"],"dc:rights":["Copyright 2019 by Scott R. Tichenor. All rights reserved."],"dc:subject":["exit path","escape path","width","annulus","polygonal arc","breadth","Wetzel","broadworm","Voronoi","Rivlin"],"dc:title":["Annular breadth of hinges & hinge exit paths of annuli"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:44Z"}