Journal of Mathematical Sciences

, Volume 131, Issue 1, pp 5307–5320

Shortest Inspection Curves for the Sphere

  • V. A. Zalgaller


What is the form of the shortest curve C going outside the unit sphere S in ℝ3 such that passing along C we can see all points of S from outside? How will the form of C change if we require that C has one (or both) of its endpoints on S? A solution to the latter problem also answers the following question. You are in a half-space at a unit distance from the boundary plane P, but you do not know where P is. What is the shortest space curve C such that going along C you will certainly come to P? Geometric arguments suggest that the required curves should be looked for in certain classes depending on several parameters. A computer-aided analysis yields the best curves in the classes. Some other questions are solved in a similar way. Bibliography: 4 titles.


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  1. 1.
    R. Bellman, “Minimisation problem,” Bull. Amer. Math. Soc., 62, 270 (1956).Google Scholar
  2. 2.
    J. R. Isbell, “An optimal search pattern,” Naval Research Logistics Quart., 4, 357–359 (1957).Google Scholar
  3. 3.
    V. A. Zalgaller, “On a question of Bellman,” Deposited in VINITI, 39, No. 849B (1992).Google Scholar
  4. 4.
    A. S. Kronrod, Nodes and Weights of Quadrature Formulas [in Russian], Moscow, Nauka (1964).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • V. A. Zalgaller
    • 1
  1. 1.St. Petersburg DepartmentSteklov Mathematical InstituteSt. PetersburgRussia

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