Journal of Mathematical Sciences

, Volume 131, Issue 1, pp 5307–5320

Shortest Inspection Curves for the Sphere

  • V. A. Zalgaller
Article

Abstract

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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REFERENCES

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    R. Bellman, “Minimisation problem,” Bull. Amer. Math. Soc., 62, 270 (1956).Google Scholar
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    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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