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Journal of Global Optimization

, Volume 51, Issue 1, pp 115–132 | Cite as

Geometric fit of a point set by generalized circles

  • Mark-Christoph KörnerEmail author
  • Jack Brimberg
  • Henrik Juel
  • Anita Schöbel
Open Access
Article

Abstract

In our paper we approximate a set of given points by a general circle. More precisely, given two norms k 1 and k 2 and a set of points in the plane, we consider the problem of locating and scaling the unit circle of norm k 1 such that the sum of weighted distances between the circumference of the circle and the given points is minimized, where the distance is measured by a norm k 2. We present results for the general case. In the case that k 1 and k 2 are both polyhedral norms, we are able to solve the problem by investigating a finite candidate set.

Keywords

Circle location Dimensional facility Minisum Polyhedral norms 

Mathematics Subject Classification (2000)

62J02 65D10 90C26 90B85 97N50 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. 1.
    Brimberg J., Juel H., Schöbel A.: Locating a circle on a sphere. Oper. Res. 55, 782–791 (2007)CrossRefGoogle Scholar
  2. 2.
    Brimberg, J., Juel, H., Körner, M., Schöbel, A.: Locating a general minisum circle on the plane. Preprint series of the Institute for Numerical and Applied Mathematics, University of Göttingen, vol. 10 (2009)Google Scholar
  3. 3.
    Brimberg, J., Juel, H., Schöbel, A.: Locating a circle on the plane using the minimax criterion. Stud. Locat. Anal. (17), 45–60 (2009)Google Scholar
  4. 4.
    Brimberg J., Juel H., Schöbel A.: Locating a minisum circle in the plane. Discrete Appl. Math. 157, 901–912 (2009)CrossRefGoogle Scholar
  5. 5.
    Brimberg J., Walker J.H., Love R.F.: Estimation of travel distances with the weighted ℓp norm: some empirical results. Journal of Transport Geography 15, 62–72 (2007)CrossRefGoogle Scholar
  6. 6.
    Chan N.N.: On circular functional relationships. Journal of the Royal Statistical Society. Series B (Methodological) 27, 45–56 (1965)Google Scholar
  7. 7.
    Chernov N., Sapirstein P.N.: Fitting circles to data with correlated noise. Computational Statistics and Data Analysis 52, 5328–5337 (2008)CrossRefGoogle Scholar
  8. 8.
    Drezner Z.: Facility Location, A Survey of Applications and Methods. Springer, Berlin (1995)Google Scholar
  9. 9.
    Drezner Z., Steiner G., Wesolowsky G.O.: On the circle closest to a set of points. Computers and Operations Research 29, 637–650 (2002)CrossRefGoogle Scholar
  10. 10.
    Drezner Z., Suzuki A.: The big triangle small triangle method for the solution of nonconvex facility location problems. Operations Research 52, 128–135 (2004)CrossRefGoogle Scholar
  11. 11.
    Hamacher H.W., Drezner Z.: Facility Location: Applications and Theory. Springer, Berlin (2001)Google Scholar
  12. 12.
    Karimäki V.: Effective circle fitting for particle trajectories. Nucl. Instrum. Methods Phys. Res. A Accel. Spectrom. Detec. Assoc. Equip. 305, 187–192 (1991)CrossRefGoogle Scholar
  13. 13.
    Körner, M., Brimberg, J., Juel, H., Schöbel, A.: General minisum circle location. In: Proceedings of the 21th Canadian Conference on Computational Geometry, pp. 111–114 (2009)Google Scholar
  14. 14.
    Labbé M., Laporte G., Rodriguez Martin I., Gonzalez J.J.S.: Locating median cycles in networks. Eur. J. Oper. Res. 160, 457–470 (2005)CrossRefGoogle Scholar
  15. 15.
    Love R.F., Morris J.G., Wesolowsky G.O.: Facilities Location—Models & Methods. North-Holland, New York (1988)Google Scholar
  16. 16.
    Nievergelt Y.: A finite algorithm to fit geometrically all midrange lines, circles, planes, spheres, hyperplanes, and hyperspheres. Numerische Mathematik 91, 257–303 (2002)CrossRefGoogle Scholar
  17. 17.
    Nievergelt Y.: Median spheres: theory, algorithms, applications. Numerische Mathematik 114, 573–606 (2010)CrossRefGoogle Scholar
  18. 18.
    Pearce C.E.M.: Locating concentric ring roads in a city. Transp. Sci. 8, 142–168 (1974)CrossRefGoogle Scholar
  19. 19.
    Plastria F., Carrizosa E.: Gauge distances and median hyperplanes. J. Optim. Theor. Appl. 110, 173–182 (2001)CrossRefGoogle Scholar
  20. 20.
    Plastria F.: GBSSS: the generalized big square small square method for planar single-facility location. Eur. J. Oper. Res. 62, 163–174 (1992)CrossRefGoogle Scholar
  21. 21.
    Scholz, D.: Geometric branch & bound methods in global optimization: theory and applications to facility location problems. Dissertation, University of Göttingen (2010)Google Scholar
  22. 22.
    Scholz, D., Schöbel, A.: The theoretical and empirical rate of convergence for geometric branch-and-bound methods. J. Glob. Optim. (2010) doi: 10.1007/s10898-009-9502-3
  23. 23.
    Schöbel A.: Locating Lines and Hyperplanes. Kluwer, Dordrecht (1999)Google Scholar
  24. 24.
    Schöbel A., Scholz D.: The big cube small cube solution method for multidimensional facility location problems. Comput. Oper. Res. 37, 115–122 (2010)CrossRefGoogle Scholar
  25. 25.
    Suzuki, T.: Optimal location of orbital routes in a circular city. ISOLDE X, Sevilla and Islantilla, Spain, June 2–8 (2005)Google Scholar
  26. 26.
    Ward J.E., Wendell R.E.: A new norm for measuring distance which yields linear location problems. Oper. Res. 28, 836–844 (1980)CrossRefGoogle Scholar
  27. 27.
    Ward J.E., Wendell R.E., Richard E.: Using Block norms for location modeling. Oper. Res. 33, 1074–1090 (1985)CrossRefGoogle Scholar
  28. 28.
    Wesolowsky G.O.: Location of the median line for weighted points. Environ. Plan. A 7, 163–170 (1975)CrossRefGoogle Scholar
  29. 29.
    Witzgall, C.: Optimal location of a central facility: mathematical models and concepts. National Bureau of Standards Report 8388. Gaithersburg, Maryland (1964)Google Scholar

Copyright information

© The Author(s) 2010

Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • Mark-Christoph Körner
    • 1
    Email author
  • Jack Brimberg
    • 2
    • 3
  • Henrik Juel
    • 4
  • Anita Schöbel
    • 5
  1. 1.Institute for Numerical and Applied MathematicsGeorg-August-Universität GöttingenGöttingenGermany
  2. 2.Royal Military College of CanadaKingstonCanada
  3. 3.Groupe d’Études et de Recherche en Analyse des DécisionsMontréalCanada
  4. 4.Technical University of DenmarkLyngbyDenmark
  5. 5.Georg-August-Universität GöttingenGöttingenGermany

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