Annular breadth of hinges & hinge exit paths of annuli

Tichenor, Scott R.

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https://hdl.handle.net/2142/105618 Copy

Description

Title

Annular breadth of hinges & hinge exit paths of annuli

Author(s)

Tichenor, Scott R.

Issue Date

2019-07-03

Director of Research (if dissertation) or Advisor (if thesis)

Alexander, Stephanie

Doctoral Committee Chair(s)

Reznick, Bruce

Committee Member(s)

• Wetzel, John E
• Bishop, Richard

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

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.

Graduation Semester

2019-08

Type of Resource

text

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

Copyright and License Information

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

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