Chasing Puppies: Mobile Beacon Routing on Closed Curves

TL;DR

This paper proves that a human can always catch a puppy on a smooth or polygonal closed curve by strategic movement, solving an open problem in beacon-based routing inspired by prior work.

Contribution

It establishes a strategy for the human to guarantee catching the puppy on closed curves, addressing an open problem in beacon routing.

Findings

Human can always catch the puppy in finite time on smooth curves.

The strategy applies to simple polygons under certain conditions.

Addresses an open problem from CCCG 2013.

Abstract

We solve an open problem posed by Michael Biro at CCCG 2013 that was inspired by his and others' work on beacon-based routing. Consider a human and a puppy on a simple closed curve in the plane. The human can walk along the curve at bounded speed and change direction as desired. The puppy runs with unbounded speed along the curve as long as the Euclidean straight-line distance to the human is decreasing, so that it is always at a point on the curve where the distance is locally minimal. Assuming that the curve is smooth (with some mild genericity constraints) or a simple polygon, we prove that the human can always catch the puppy in finite time.