Finding Approximately Convex Ropes in the Plane

TL;DR

This paper introduces a method to find the shortest convex rope in a polygon, using multiple shooting, and demonstrates its convergence and implementation in C++ for practical computation.

Contribution

It reconstructs the convex rope problem as a shortest path problem and applies a multiple shooting method with convergence analysis.

Findings

The method converges to the shortest path under certain conditions.

Implementation in C++ demonstrates practical applicability.

The approach effectively finds convex ropes in simple polygons.

Abstract

The convex rope problem is to find a counterclockwise or clockwise convex rope starting at the vertex a and ending at the vertex b of a simple polygon P, where a is a vertex of the convex hull of P and b is visible from infinity. The convex rope mentioned is the shortest path joining a and b that does not enter the interior of P. In this paper, the problem is reconstructed as the one of finding such shortest path in a simple polygon and solved by the method of multiple shooting. We then show that if the collinear condition of the method holds at all shooting points, then these shooting points form the shortest path. Otherwise, the sequence of paths obtained by the update of the method converges to the shortest path. The algorithm is implemented in C++ for numerical experiments.