arXiv subjects

Wacharin Wichiramala

Publications and source records attributed to Wacharin Wichiramala.

5 recordsLinked to original sources

arXiv2026

Wetzel's 30-60-90 Triangle Covers Unit Arcs

John E. Wetzel conjectured that the 30-60-90 triangle T obtained by placing a square of side 1/3 on the hypotenuse covers every unit arc in the plane. We give a computer-assisted proof of this conjecture with independently checkable interval certificates. The proof reduces a hypothetical noncovered arc to a finite family of 599 closed second-order cone models, covering all representative and raw tail-order branches, and certifies a polygonal-chain lower bound greater than one in every model by interval validation of stored dual certificates. Since every certified lower endpoint exceeds 1.0048, the homothetic copy T/1.0048 still covers every unit arc. Its area is 0.260956..., below the area pi/12 approx. 0.261799 of the 30-degree unit sector, a certified area improvement over the sector cover within this convex Wetzel-cover setting.

math.MG↗

arXiv2019

How support lines touch an arc

We prove that each simple polygonal arc γ attains at most two pairs of support lines of given angle difference such that each pair has s1 < s2 < s3 that γ(s1) and γ(s3) are on one such line and γ(s2) is on the other line.

math.MG↗

arXiv2019

Wetzel's sector covers unit arcs

We settle J. Wetzel's 1970's conjecture and show that a 30{^\circ} circular sector of unit radius can accommodate every planar arc of unit length. Leo Moser asked in 1966 for the smallest (convex) region in the plane that can accommodate each arc of unit length. With area π/12, this sector is the smallest such set presently known. Moser's question has prompted a multitude of papers on related problems over the past 50 years, most remaining unanswered.

math.MG↗

arXiv2019

The $Λ$-property of a simple arc

In 2006 P. Coulton and Y. Movshovich established an unfamilar but note-worthy general property of simple, polygonal, open arcs in the plane. We give a new and quite different proof of this property, and we consider a few generalizations.

math.MG↗

arXiv2018

A smaller cover for closed unit curves

Forty years ago Schaer and Wetzel showed that a $\frac{1}π\times\frac {1}{2π}\sqrt{π^{2}-4}$ 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 Füredi 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.

cs.CG↗