Shortest Inspection Curves for the Sphere
- V. A. Zalgaller
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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.
- R. Bellman, “Minimisation problem,” Bull. Amer. Math. Soc., 62, 270 (1956).
- J. R. Isbell, “An optimal search pattern,” Naval Research Logistics Quart., 4, 357–359 (1957).
- V. A. Zalgaller, “On a question of Bellman,” Deposited in VINITI, 39, No. 849B (1992).
- A. S. Kronrod, Nodes and Weights of Quadrature Formulas [in Russian], Moscow, Nauka (1964).
- Shortest Inspection Curves for the Sphere
Journal of Mathematical Sciences
Volume 131, Issue 1 , pp 5307-5320
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- V. A. Zalgaller (1)
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- 1. St. Petersburg Department, Steklov Mathematical Institute, St. Petersburg, Russia