The Geometry of Wide Curves in the Plane

Abstract

This article treats the problem: Given the positive number Δ and two points of the plane, determine the shortest planar curves of minimal width Δ joining these points.

The length, width, and inradius of space curves

“…Further, it follows from Corollary 2.4 that L 2 ≥ πw 2 ≥ πw. Thus, again by Lemma 2.2, 2 , which completes the proof.…”

“…It has been known since Barbier in 1860 [5], and follows from the Cauchy-Crofton formula (Lemma 2.3), that for closed planar curves L/w ≥ π, where equality holds only for curves of constant width. The corresponding question for general planar curves, however, was answered only in 1961 when Zalgaller [29] produced a caliper shaped curve (Figure 2(a)) with L/w ≈ 2.2782, which has been subsequently rediscovered several times [1,19]; see [2,16]. In 1994, Zalgaller [30] studied the width problem for curves in R 3 , and produced a closed curve, "L 5 ", which he claimed to be minimal; however, our cylindrical example in Section 5 improves upon Zalgaller's curve.…”

Errata and Addenda to Mathematical Constants

“…Relevant progress is described in [1000,1001,1002,1003]. We mention, in Figure 8.3, that the quantity x = sec(ϕ) = 1.0435901095... is algebraic of degree six [1004,1005]:…”