Discrete & Computational Geometry

, Volume 62, Issue 1, pp 1–28 | Cite as

Unconstrained and Curvature-Constrained Shortest-Path Distances and Their Approximation

  • Ery Arias-CastroEmail author
  • Thibaut Le Gouic


We study shortest paths and their distances on a subset of a Euclidean space, and their approximation by their equivalents in a neighborhood graph defined on a sample from that subset. In particular, we recover and extend the results of Bernstein et al. (Graph approximations to geodesics on embedded manifolds, Tech. Rep., Department of Psychology, Stanford University, 2000). We do the same with curvature-constrained shortest paths and their distances, establishing what we believe are the first approximation bounds for them.


Intrinsic distances Minimizing curves Shortest paths Curvature Reach Neighborhood graph Approximation of distances Motion planning 

Mathematics Subject Classification

51F99 53C22 65D99 68U05 



The authors wish to thank Stephanie Alexander, I. David Berg, Richard Bishop, Dmitri Burago, Bruce Driver, and Bruno Pelletier for very helpful discussions. The paper was carefully read by two anonymous referees, to whom we are grateful. Some of the symbolic calculations were done with Wolfram|Alpha ( This work was partially supported by the US National Science Foundation (DMS 0915160, DMS 1513465).


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaSan DiegoUSA
  2. 2.École Centrale de MarseilleMarseilleFrance
  3. 3.Higher School of EconomicsNational Research UniversityMoscowRussian Federation

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