Drapeability

Abstract

An arc in the plane is drapeable if it lies in the convex hull of a convex arc of the same length. We establish some fundamental properties of drapeability, and we investigate some sufficient conditions for an arc to be drapeable.

Besicovitch triangles extended

“…An arc inside of a triangle T is circumscribed by T if it touches all sides of T. The arcs of interest resemble a staple or letters s, w. Staple arcs, or w-arcs satisfying (4), are those in [2], s-arcs will satisfy (2) instead of (3) below.…”

“…Definition 4 We say that an arc in a strip 0 ≤ y ≤ h is a − arc, if it has consecutive points B, C, D such that B, D lie on one boundary while C lies on the other.…”

“…As in [2] we consider only simple polygonal arcs referring to the result of [4]: If a convex set covers any simple polygonal unit arc it covers any unit arc. The idea of the proof is the same as in [2].…”

Besicovitch triangles cover unit arcs

“…If the convex set has an axis of reflection symmetry, then it is enough to show that the set contains a congruent copy of each worm. If a convex set covers any simple polygonal unit arc it is a worm-cover [3].…”

The Geometry of Wide Curves in the Plane

“…This argument proves the uniqueness of the based convex solution curve if δ ≥ 1. For a deeper study of drapes and related planar curves, see the article of Maki, Wetzel, and Wichiramala[8].…”