Annular breadth of hinges & hinge exit paths of annuli

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2019

Author dc:creator

• Tichenor, Scott R.

Contributors dc:contributor

• Alexander, Stephanie
• Reznick, Bruce
• Wetzel, John E
• Bishop, Richard

• exit path
• escape path
• width
• annulus
• polygonal arc
• breadth
• Wetzel
• broadworm
• Voronoi
• Rivlin

Statement dc:rights

• Copyright 2019 by Scott R. Tichenor. All rights reserved.

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/105618

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F105618