Advertisement

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
Article
  • 36 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

51F99 53C22 65D99 68U05 

Notes

Acknowledgements

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 (http://www.wolframalpha.com). This work was partially supported by the US National Science Foundation (DMS 0915160, DMS 1513465).

References

  1. 1.
    Albrecht, F., Berg, I.D.: Geodesics in Euclidean space with analytic obstacle. Proc. Am. Math. Soc. 113(1), 201–207 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alexander, R., Alexander, S.: Geodesics in Riemannian manifolds-with-boundary. Indiana Univ. Math. J. 30(4), 481–488 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alexander, S.B., Berg, I.D., Bishop, R.L.: The Riemannian obstacle problem. Illinois J. Math. 31(1), 167–184 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Babaeian, A., Babaee, M., Bayestehtashk, A., Bandarabadi, M.: Nonlinear subspace clustering using curvature constrained distances. Pattern Recognit. Lett. 68(2), 118–125 (2015)CrossRefGoogle Scholar
  5. 5.
    Babaeian, A., Bayestehtashk, A., Bandarabadi, M.: Multiple manifold clustering using curvature constrained path. PloS ONE 10(9), Art. No. e0137986 (2015)Google Scholar
  6. 6.
    Bauer, U., Polthier, K., Wardetzky, M.: Uniform convergence of discrete curvatures from nets of curvature lines. Discrete Comput. Geom. 43(4), 798–823 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bernstein, M., De Silva, V., Langford, J.C., Tenenbaum, J.B.: Graph approximations to geodesics on embedded manifolds. Tech. Rep., Department of Psychology, Stanford University (2000)Google Scholar
  8. 8.
    Boissonnat, J.-D., Cérézo, A., Leblond, J.: Shortest paths of bounded curvature in the plane. In: Proceedings the IEEE International Conference on Robotics and Automation, pp. 2315–2320 (1992)Google Scholar
  9. 9.
    Burago, D., Burago, Yu., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  10. 10.
    De Silva, V., Tenenbaum, J.B.: Global versus local methods in nonlinear dimensionality reduction. Adv. Neural Inf. Process. Syst. 15, 705–712 (2002)Google Scholar
  11. 11.
    Delling, D., Sanders, P., Schultes, D., Wagner, D.: Engineering route planning algorithms. In: Lerner, J., Wagner, D., Zweig, K.A. (eds.) Algorithmics of Large and Complex Networks. Lecture Notes in Computer Science, vol. 5515, pp. 117–139. Springer, Berlin (2009)Google Scholar
  12. 12.
    Dubins, L.E.: On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. Am. J. Math. 79(3), 497–516 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93(3), 418–491 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hoffmann, T.: Discrete Differential Geometry of Curves and Surfaces. COE Lecture Note, vol. 18. Kyushu University, Fukuoka (2009)Google Scholar
  15. 15.
    Janson, L., Schmerling, E., Clark, A., Pavone, M.: Fast marching tree: a fast marching sampling-based method for optimal motion planning in many dimensions. Int. J. Robot. Res. 34(7), 883–921 (2015)CrossRefGoogle Scholar
  16. 16.
    Karaman, S., Frazzoli, E.: Optimal kinodynamic motion planning using incremental sampling-based methods. In: Proceedings of the 49th IEEE Conference on Decision and Control (CDC’10), pp. 7681–7687 (2010)Google Scholar
  17. 17.
    Karaman, S., Frazzoli, E.: Sampling-based algorithms for optimal motion planning. Int. J. Robot. Res. 30(7), 846–894 (2011)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kavraki, L.E., Kolountzakis, M.N., Latombe, J.C.: Analysis of probabilistic roadmaps for path planning. IEEE Trans. Robot. Autom. 14(1), 166–171 (1998)CrossRefGoogle Scholar
  19. 19.
    Kruskal, J.B., Seery, J.B.: Designing network diagrams. In: General Conference on Social Graphics, pp. 22–50 (1980)Google Scholar
  20. 20.
    Latombe, J.-C.: Robot Motion Planning. The Springer International Series in Engineering and Computer Science, vol. 124. Springer, New York (2012)Google Scholar
  21. 21.
    LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  22. 22.
    LaValle, S.M., Kuffner, J.J.: Randomized kinodynamic planning. Int. J. Robot. Res. 20(5), 378–400 (2001)CrossRefGoogle Scholar
  23. 23.
    Lee, J.M.: Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics, vol. 176. Springer, New York (2006)Google Scholar
  24. 24.
    Li, Y., Littlefield, Z., Bekris, K.E.: Sparse methods for efficient asymptotically optimal kinodynamic planning. In: Akin, H.L., et al. (eds.) Algorithmic Foundations of Robotics XI. Springer Tracts in Advanced Robotics, vol. 107, pp. 263–282. Springer, Cham (2015)Google Scholar
  25. 25.
    Maier, M., Hein, M., von Luxburg, U.: Optimal construction of \(k\)-nearest-neighbor graphs for identifying noisy clusters. Theoret. Comput. Sci. 410(19), 1749–1764 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Niyogi, P., Smale, S., Weinberger, S.: Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39(1), 419–441 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Reeds, J., Shepp, L.: Optimal paths for a car that goes both forwards and backwards. Pacific J. Math. 145(2), 367–393 (1990)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Schmerling, E., Janson, L., Pavone, M.: Optimal sampling-based motion planning under differential constraints: the drift case with linear affine dynamics. In: Proceedings of the 54th IEEE Conference on Decision and Control (CDC’15), pp. 2574–2581 (2015)Google Scholar
  29. 29.
    Schmerling, E., Janson, L., Pavone, M.: Optimal sampling-based motion planning under differential constraints: the driftless case. In: Proceedings of the 2015 IEEE International Conference on Robotics and Automation (ICRA’15), pp. 2368–2375 (2015)Google Scholar
  30. 30.
    Shang, Y., Ruml, W.: Improved MDS-based localization. In: Conference of the IEEE Computer and Communications Societies (INFOCOM’04), vol. 4, pp. 2640–2651 (2004)Google Scholar
  31. 31.
    Shang, Y., Ruml, W., Zhang, Y., Fromherz, M.P.: Localization from mere connectivity. In: Proceedings of the 4th ACM International Symposium on Mobile Ad Hoc Networking & Computing (MobiHoc’03), pp. 201–212. ACM, New York (2003)Google Scholar
  32. 32.
    Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)CrossRefGoogle Scholar
  33. 33.
    Thrun, S., Burgard, W., Fox, D.: Probabilistic Robotics. MIT Press, Cambridge (2005)zbMATHGoogle Scholar
  34. 34.
    Waldmann, S.: Topology: An Introduction. Springer, Cham (2014)zbMATHGoogle Scholar

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

Personalised recommendations