6 research outputs found
A smaller cover for closed unit curves]{A smaller cover for closed unit curves
Forty years ago Schaer and Wetzel showed that a rectangle, whose area is about is the
smallest rectangle that is a cover for the family of all closed unit arcs. More
recently F\"{u}redi and Wetzel showed that one corner of this rectangle can be
clipped to form a pentagonal cover having area for this family of
curves. Here we show that then the opposite corner can be clipped to form a
hexagonal cover of area less than for this same family. This
irregular hexagon is the smallest cover currently known for this family of
arcs.Comment: In the appendix, the computer code for numerical optimization is
provided together with explanation. The link to the actual file, a
Mathematica notebook, is at
www.math.sc.chula.ac.th/~wacharin/optimization/closed%20arc
Universal convex covering problems under translation and discrete rotations
We consider the smallest-area universal covering of planar objects of
perimeter 2 (or equivalently closed curves of length 2) allowing translation
and discrete rotations. In particular, we show that the solution is an
equilateral triangle of height 1 when translation and discrete rotation of
are allowed. Our proof is purely geometric and elementary. We also give
convex coverings of closed curves of length 2 under translation and discrete
rotations of multiples of and . We show a minimality of the
covering for discrete rotation of multiples of , which is an equilateral
triangle of height smaller than 1, and conjecture that the covering is the
smallest-area convex covering. Finally, we give the smallest-area convex
coverings of all unit segments under translation and discrete rotations
for all integers
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 , denoted , as the width of the annulus of support of 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 is defined as the absolute minimum of . We find the -segment polygonal arc with the greatest annular breadth.
For a given set \textsf{S}\subset\mathds{R}^2, an exit path of is a curve that cannot be covered by the interior of . Given an annulus, we find its shortest - or -segment polygonal arc exit path(s).
Bezdek and Connelly provided a lengthy and technically demanding proof that \emph{All orbiforms of width} \emph{are translation covers of the set of closed planar curves of length} \emph{or less}. We provide a short and simple proof that \emph{All orbiforms of width} \emph{are covers of the set of all planar curves of length} \emph{or less}. We also provide a proof that \emph{The Reuleaux triangle of width} \emph{is a cover of the set of all closed curves of length} using a recent of Wichiramala
Errata and Addenda to Mathematical Constants
We humbly and briefly offer corrections and supplements to Mathematical
Constants (2003) and Mathematical Constants II (2019), both published by
Cambridge University Press. Comments are always welcome.Comment: 162 page