# Self-approaching paths in simple polygons

## TL;DR

This paper investigates the properties and characterization of shortest self-approaching paths within simple polygons, introduces methods to find such paths using advanced curves, and provides an algorithm to determine if a polygon is self-approaching.

## Contribution

It offers a characterization of shortest self-approaching paths, demonstrates the complexity involving non-algebraic curves, and presents algorithms for path finding and polygon classification.

## Key findings

- Shortest self-approaching paths can involve non-algebraic curves.
- An algorithm exists to test if a polygon is self-approaching.
- Methods are developed to find self-approaching paths using high-order involute curves.

## Abstract

We study self-approaching paths that are contained in a simple polygon. A self-approaching path is a directed curve connecting two points such that the Euclidean distance between a point moving along the path and any future position does not increase, that is, for all points $a$, $b$, and $c$ that appear in that order along the curve, $|ac| \ge |bc|$. We analyze the properties, and present a characterization of shortest self-approaching paths. In particular, we show that a shortest self-approaching path connecting two points inside a polygon can be forced to use a general class of non-algebraic curves. While this makes it difficult to design an exact algorithm, we show how to find a self-approaching path inside a polygon connecting two points under a model of computation which assumes that we can calculate involute curves of high order. Lastly, we provide an algorithm to test if a given simple polygon is self-approaching, that is, if there exists a self-approaching path for any two points inside the polygon.