2026-09-21T10:15:27Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/105618 2023-07-11 col_2142_16340 col_2142_5131 com_2142_16339 com_2142_8903 com_2142_5130

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 application/pdf http://hdl.handle.net/2142/105618 en Alexander, Stephanie Reznick, Bruce Wetzel, John E Bishop, Richard Tichenor, Scott R. 2019-11-26T20:33:44Z 2019-11-26T20:33:44Z 2019-07-03 2019-08 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. Copyright 2019 by Scott R. Tichenor. All rights reserved. exit path escape path width annulus polygonal arc breadth Wetzel broadworm Voronoi Rivlin Annular breadth of hinges & hinge exit paths of annuli text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.