A smaller cover for closed unit curves

Wacharin Wichiramala

A smaller cover for closed unit curves

Computational Geometry 2018-01-26 v1

Authors: Wacharin Wichiramala

Abstract

Forty years ago Schaer and Wetzel showed that a $\frac{1}{\pi}\times\frac {1}{2\pi}\sqrt{\pi^{2}-4}$ rectangle, whose area is about $0.122\,74,$ 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 $0.11224$ 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 $0.11023$ for this same family. This irregular hexagon is the smallest cover currently known for this family of arcs.

Cite

@article{arxiv.1801.08405,
  title  = {A smaller cover for closed unit curves},
  author = {Wacharin Wichiramala},
  journal= {arXiv preprint arXiv:1801.08405},
  year   = {2018}
}

Comments

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%20arcs

A smaller cover for closed unit curves

Abstract

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