Annular breadth of hinges & hinge exit paths of annuli

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

Tichenor, Scott R.

关键词: exit path;    escape path;    width;    annulus;    polygonal arc;    breadth;    Wetzel;    broadworm;    Voronoi;    Rivlin;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/105618/TICHENOR-DISSERTATION-2019.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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.