Relaxed triangle inequality for the orbital similarity criterion by Southworth and Hawkins and its variants

  • D. V. MilanovEmail author
  • Yu. V. Milanova
  • K. V. Kholshevnikov
Original Article


In this article, we prove the relaxed triangle inequality for Southworth and Hawkins, Drummond and Jopek orbital similarity criteria on the set of non-rectilinear Keplerian orbits with the eccentricity bounded above. We give estimates of the minimal coefficients in the inequality for each criterion and show that one of the calculated coefficients is exactly minimal. The obtained inequalities can be used for the acceleration of algorithms involving pairwise distances calculations between orbits. We present an algorithm for calculation of all distances not exceeding a fixed number in a quasi-metric space and demonstrate that the algorithm is faster than the complete calculation on the set of meteors orbits. Finally, we estimate the correlation dimensions of the set of main belt asteroids orbits and meteors orbits with respect to various orbital metrics and quasi-metrics.


Orbital similarity criterion Space of Keplerian orbits Quasi-metric Relaxed triangle inequality Clustering algorithm Distance matrix Correlation integral Correlation dimension 



This work is supported by the Russian Science Foundation, Grant 18-12-00050. We express our gratitude to the anonymous referee for useful and detailed comments, which improved the article.

Compliance with ethical standards

Conflict of interest

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the paper submitted.


  1. Baggaley, W., Galligan, D.: Cluster analysis of the meteoroid orbit population. Planet. Space Sci. 45(7), 865–868 (1997)ADSCrossRefGoogle Scholar
  2. Defays, D.: An efficient algorithm for a complete link method. Comput. J. 20(4), 364–366 (1977)MathSciNetCrossRefGoogle Scholar
  3. Drummond, J.: A test of comet and meteor shower associations. Icarus 45(3), 545–553 (1981)ADSCrossRefGoogle Scholar
  4. Dumitru, B., Birlan, M., Popescu, M., Nedelcu, D.: Association between meteor showers and asteroids using multivariate criteria. Astron. Astrophys. 607, A5 (2017)ADSCrossRefGoogle Scholar
  5. Grassberger, P., Procaccia, I.: Measuring the strangeness of strange attractors. Phys. D Nonlinear Phenom. 9(1–2), 189–208 (1983)ADSMathSciNetCrossRefGoogle Scholar
  6. Hoeffding W.: (1961) The strong law of large numbers for U-statistics. Tech. rep., North Carolina State University. Department of StatisticsGoogle Scholar
  7. Jenniskens, P., Nénon, Q., Albers, J., Gural, P., Haberman, B., Holman, D., Morales, R., Grigsby, B., Samuels, D., Johannink, C.: The established meteor showers as observed by CAMS. Icarus 266, 331–354 (2016)ADSCrossRefGoogle Scholar
  8. Jopek, T.: Remarks on the meteor orbital similarity D-criterion. Icarus 106(2), 603–607 (1993)ADSCrossRefGoogle Scholar
  9. Jopek, T., Froeschlé, C.: A stream search among 502 TV meteor orbits. An objective approach. Astron. Astrophys. 320, 631–641 (1997)ADSGoogle Scholar
  10. Kholshevnikov K.: On metrics in spaces of Keplerian orbits (in Russian). In: Proceedings of 45-th Int. Stud. Confs. “Physics of Cosmos”. Ekaterinburg, 1–5 Feb 2016, Ekaterinburg, Ural Federal University Press, pp 168–184 (2016)Google Scholar
  11. Kholshevnikov K., Shchepalova A.: On distances between orbits of planets and asteroids. Vestnik St. Petersburg University, Mathematics 51(3) (2018) (in press)Google Scholar
  12. Kholshevnikov, K., Kokhirova, G., Babadzhanov, P., Khamroev, U.: Metrics in the space of orbits and their application to searching for celestial objects of common origin. Mon. Not. R. Astron. Soc. 462(2), 2275–2283 (2016)ADSCrossRefGoogle Scholar
  13. Kirk, W., Shahzad, N.: Fixed points and Cauchy sequences in semimetric spaces. J. Fix. Point Theory Appl. 17(3), 541–555 (2015)MathSciNetCrossRefGoogle Scholar
  14. Lindblad, B., Southworth, R.: A Study of Asteroid Families and Streams by Computer Techniques, p. 267. NASA Special Publication, Washington DC (1971)Google Scholar
  15. Nanni, M.: Speeding-up hierarchical agglomerative clustering in presence of expensive metrics. In: Pacific-Asia Conference on Knowledge Discovery and Data Mining, p. 378. Springer, Berlin (2005)Google Scholar
  16. Sibson, R.: Slink: an optimally efficient algorithm for the single-link cluster method. Comput. J. 16(1), 30–34 (1973)MathSciNetCrossRefGoogle Scholar
  17. Southworth, R., Hawkins, G.: Statistics of meteor streams. Smithson. Contrib. Astrophys. 7, 261 (1963)ADSGoogle Scholar
  18. Welch, P.: A new search method for streams in meteor data bases and its application. Mon. Not. R. Astron. Soc. 328(1), 101–111 (2001)ADSCrossRefGoogle Scholar
  19. Xia, Q.: The geodesic problem in quasimetric spaces. J. Geom. Anal. 19(2), 452–479 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Institute of Applied Astronomy of Russian Academy of SciencesSt. PetersburgRussia

Personalised recommendations