# Shortest Inspection Curves for the Sphere

- Received:

DOI: 10.1007/s10958-005-0403-9

- Cite this article as:
- Zalgaller, V.A. J Math Sci (2005) 131: 5307. doi:10.1007/s10958-005-0403-9

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## Abstract

What is the form of the shortest curve C going outside the unit sphere S in ℝ^{3} 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.