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, Volume 9, Issue 4, pp 351–370 | Cite as

Locating a general minisum ‘circle’ on the plane

  • Jack BrimbergEmail author
  • Henrik Juel
  • Mark-Christoph Körner
  • Anita Schöbel
Research paper

Abstract

We approximate a set of given points in the plane by the boundary of a convex and symmetric set which is the unit circle of some norm. This generalizes previous work on the subject which considers Euclidean circles only. More precisely, we examine the problem of locating and scaling the unit circle of some given norm k with respect to given points on the plane such that the sum of weighted distances (as measured by the same norm k) between the circumference of the circle and the points is minimized. We present general results and are able to identify a finite dominating set in the case that k is a polyhedral norm.

Keywords

Facility location General norm Circle 

MSC classification (2000)

90B85 90C26 90C90 

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References

  1. Brimberg J, Juel H, Schöbel A (2007) Locating a circle on a sphere. Oper Res 55: 782–791CrossRefGoogle Scholar
  2. Brimberg J, Juel H, Schöbel A (2009a) Locating a circle on the plane using the minimax criterion. Stud Locat Anal 17: 45–60Google Scholar
  3. Brimberg J, Juel H, Schöbel A (2009b) Locating a minisum circle in the plane. Discrete Appl Math 157: 901–912CrossRefGoogle Scholar
  4. Brimberg J, Walker JH, Love RF (2007) Estimation of travel distances with the weighted p norm: some empirical results. J Transp Geogr 15: 62–72CrossRefGoogle Scholar
  5. Chernov N, Sapirstein PN (2008) Fitting circles to data with correlated noise. Comput Stat Data Anal 52: 5328–5337CrossRefGoogle Scholar
  6. Day MM (1947) Some characterizations of inner-product spaces. Trans Am Math Soc 62: 320–337CrossRefGoogle Scholar
  7. Díaz-Báñez JM, Mesa JA, Schöbel A (2004) Continuous location of dimensional structures. Eur J Oper Res 152: 22–44CrossRefGoogle Scholar
  8. Drezner Z, Steiner G, Wesolowsky GO (2002) On the circle closest to a set of points. Comput Oper Res 29: 637–650CrossRefGoogle Scholar
  9. Durier R, Michelot C (1985) Geometrical properties of the Fermat-Weber problem. Eur J Oper Res 20: 332–343CrossRefGoogle Scholar
  10. Icking C, Klein R, Ma L, Nickel S, Weißler A (2001) On bisectors for different distance functions. Discrete Appl Math 109: 139–161CrossRefGoogle Scholar
  11. Karimäki V (1991) Effective circle fitting for particle trajectories. Nucl Instrum Methods Phys Res A: Accel Spectrom Detect Assoc Equip 305: 187–192CrossRefGoogle Scholar
  12. Körner M (2010) Minisum hyperspheres. Dissertation, Georg-August-Universität GöttingenGoogle Scholar
  13. Körner M, Brimberg J, Juel H, Schöbel A Geometric fit of a point set by generalized circles. J Glob Optim, to appearGoogle Scholar
  14. Labbé M, Laporte G, Rodriguez Martin I, Gonzalez JJS (2005) Locating median cycles in networks. Eur J Oper Res 160: 457–470CrossRefGoogle Scholar
  15. Ma L (2000) Bisectors and Voronoi Diagrams for convex distance functions. Dissertation, Fernuniversität Hagen. http://wwwpi6.fernuni-hagen.de/Publikationen/tr267
  16. Martini H, Swanepoel K, Weiss G (2002) The Fermat-Toricelli problem in normed planes and spaces. J Optim Theory Appl 115: 283–314CrossRefGoogle Scholar
  17. Nickel S, Puerto J (2005) Location theory: a unified approach. Springer, BerlinGoogle Scholar
  18. Nievergelt Y (2002) A finite algorithm to fit geometrically all midrange lines, circles, planes, spheres, hyperplanes, and hyperspheres. Numerische Mathematik 91: 257–303CrossRefGoogle Scholar
  19. Nievergelt Y (2010) Median spheres: theory, algorithms, applications. Numerische Mathematik 114: 573–606CrossRefGoogle Scholar
  20. Pearce CEM (1974) Locating concentric ring roads in a city. Transp Sci 8: 142–168CrossRefGoogle Scholar
  21. Phelps RR (1989) Convex functions, monotone operators and differentiability. Lecture notes in mathematics 1364, Springer, BerlinGoogle Scholar
  22. Schöbel A (1999) Locating lines and hyperplanes. Kluwer, DordrechtGoogle Scholar
  23. Suzuki T (2005) Optimal location of orbital routes in a circular city. ISOLDE X, Sevilla and Islantilla, Spain, June 2-8Google Scholar
  24. Thisse JF (1987) Location theory, regional science, and economics. J Reg Sci 27: 519–528CrossRefGoogle Scholar
  25. Ward JE, Wendell RE (1980) A new norm for measuring distance which yields linear location problems. Oper Res 28: 836–844CrossRefGoogle Scholar
  26. Ward JE, Wendell RE, Richard E (1985) Using block norms for location modeling. Oper Res 33: 1074–1090CrossRefGoogle Scholar
  27. Wesolowsky GO (1975) Location of the median line for weighted points. Environ Plann A 7: 163–170CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Jack Brimberg
    • 1
    Email author
  • Henrik Juel
    • 2
  • Mark-Christoph Körner
    • 3
  • Anita Schöbel
    • 3
  1. 1.Royal Military College of Canada and Groupe d’Études et de Recherche en Analyse des DécisionsKingstonCanada
  2. 2.Technical University of DenmarkCopenhagenDenmark
  3. 3.Georg-August-Universität GöttingenGöttingenGermany

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