Self-Approaching Paths in Simple Polygons

Authors Prosenjit Bose, Irina Kostitsyna, Stefan Langerman

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Prosenjit Bose
Irina Kostitsyna
Stefan Langerman

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Prosenjit Bose, Irina Kostitsyna, and Stefan Langerman. Self-Approaching Paths in Simple Polygons. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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| >= |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 follow 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.
  • self-approaching path
  • simple polygon
  • shortest path
  • involute curve


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  1. O. Aichholzer, F. Aurenhammer, C. Icking, R. Klein, E. Langetepe, and G. Rote. Generalized self-approaching curves. Discrete Applied Mathematics, 109(1-2):3-24, 2001. URL:
  2. S. Alamdari, T. M. Chan, E. Grant, A. Lubiw, and V. Pathak. Self-approaching Graphs. In 20th International Symposium on Graph Drawing (GD), pages 260-271, 2012. URL:
  3. E. M. Arkin, R. Connelly, and J. S. B. Mitchell. On monotone paths among obstacles with applications to planning assemblies. In 5th Annual Symposium on Computational Geometry (SCG), pages 334-343. ACM Press, 1989. URL:
  4. M. A. Bender and M. Farach-Colton. The LCA Problem Revisited. In Latin American Symposium on Theoretical Informatics, pages 88-94, 2000. URL:
  5. M. Biro, J. Iwerks, I. Kostitsyna, and J. S. B. Mitchell. Beacon-Based Algorithms for Geometric Routing. In 13th Algorithms and Data Structures Symposium (WADS), pages 158-169. Springer, 2013. URL:
  6. P. Bose, I. Kostitsyna, and S. Langerman. Self-approaching paths in simple polygons. Preprint,, 2017.
  7. B. Chazelle, H. Edelsbrunner, M. Grigni, L. Guibas, J. Hershberger, M. Sharir, and J. Snoeyink. Ray shooting in polygons using geodesic triangulations. Algorithmica, 12(1):54-68, 1994. URL:
  8. H. Dehkordi, F. Frati, and J. Gudmundsson. Increasing-Chord Graphs On Point Sets. In 22nd International Symposium on Graph Drawing, pages 464-475. Springer, 2014. URL:
  9. L. E. Dubins. On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents. American Journal of Mathematics, 79(3):497-516, 1957. URL:
  10. J. Gao and L. Guibas. Geometric algorithms for sensor networks. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 370(1958):27-51, 2012. URL:
  11. L. Guibas, J. Hershberger, D. Leven, M. Sharir, and R. Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2(1-4):209-233, 1987. URL:
  12. C. Icking and R. Klein. Searching for the kernel of a polygon - a competitive strategy. In 11th Annual Symposium on Computational Geometry (SCG), pages 258-266. ACM Press, 1995. URL:
  13. C. Icking, R. Klein, and E. Langetepe. Self-approaching curves. Mathematical Proceedings of the Cambridge Philosophical Society, 125(3):441-453, 1999. URL:
  14. M. Laczkovich. The removal of π from some undecidable problems involving elementary functions. Proceedings of the American Mathematical Society, 131(07):2235-2241, 2003. URL:
  15. J. S. B. Mitchell, C. Piatko, and E. M. Arkin. Computing a shortest k-link path in a polygon. In 33rd Annual Symposium on Foundations of Computer Science, pages 573-582. IEEE, 1992. URL:
  16. M. Nöllenburg, R. Prutkin, and I. Rutter. On self-approaching and increasing-chord drawings of 3-connected planar graphs. Journal of Computational Geometry, 7(1):47-69, 2016. URL:
  17. G. Rote. Curves with increasing chords. Mathematical Proceedings of the Cambridge Philosophical Society, 115(01):1, 1994. URL:
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