## Abstract

We study the problem of connecting two points in a simple polygon with a self-approaching path. A self-approaching path is a directed curve 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|≥|bc|. We analyze properties of self-approaching paths inside simple polygons, and characterize shortest self-approaching paths. In particular, we show that a shortest self-approaching path connecting two points in a simple polygon can be forced to follow a general class of non-algebraic curves. While this makes it difficult to design an exact algorithm, we show how to find the shortest self-approaching path under a model of computation which assumes that we can compute involute curves of high order. Lastly, we provide an efficient 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.

Original language | English |
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Article number | 101595 |

Number of pages | 24 |

Journal | Computational Geometry |

Volume | 87 |

DOIs | |

Publication status | Published - Apr 2020 |

### Funding

This work was begun at the CMO-BIRS Workshop on Searching and Routing in Discrete and Continuous Domains, October 11–16, 2015. I.K. was supported in part by the NWO under project no. 612.001.106 , and by F.R.S.-FNRS .

Funders | Funder number |
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Fonds De La Recherche Scientifique - FNRS | |

Nederlandse Organisatie voor Wetenschappelijk Onderzoek | 612.001.106 |

## Keywords

- Involute curves
- Self-approaching paths
- Shortest paths
- Simple polygons