Self-approaching curves

Abstract

We present a new class of curves which are self-approaching in the following sense. For any three consecutive points a, b, c on the curve the point b is closer to c than a to c. This is a generalisation of curves with increasing chords which are self-approaching in both directions. We show a tight upper bound of 5.3331. .. for the length of a self-approaching curve over the distance between its endpoints. Keywords. Curves with increasing chords, self-approaching curves, convex hull, detour, arc length.

Generalized Self-Approaching Curves

“…If a mobile robot wants to get to the kernel of an unknown star-shaped polygon and continuously follows the angular bisector of the innermost left and right reflex vertices that are visible, the resulting path is self-approaching for ϕ = π/2; see [5]. Since the value of c(π/2) is known to be ≈ 5.3331, as already shown in Icking et al [6], one immediately obtains an upper bound for the competitive factor of the robot's strategy. Improving on this, Lee and Chwa [7] give a tight upper bound of π + 1 for this factor, and Lee et al [8] present a different strategy that achieves a factor of 3.829, while a lower bound of 1.48 is shown by López-Ortiz and Schuierer [9].…”

“…We treat only the case ϕ = π/2, where we have proper spirals. The case ϕ = π/2, where we have circular arcs, has already been treated in [6].…”

Gathering by repulsion

“…directed path Π is self-approaching if for any three consecutive points x, y, z on the path, we have the property that |xz| ≥ |yz| [7].…”

Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D

“…Sometimes the geometric properties of curves allow us to infer upper bounds on their detour [4,18,24], but these results do not lead to efficient computation of the detour of the curve.…”