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The Steiner problem on surfaces of revolution. (English) Zbl 1305.51005

The Steiner problem consists on finding the shortest path network connecting a finite set of given points on a surface. It is well understood in the Euclidean plane and on surfaces of constant curvature. The aim of the paper is to develop an algorithm for the \(n\)-point Steiner problem on a surface of revolution with a non-decreasing generating function. Extra complexities include for this problem describing the length-minimizing geodesics and the large number of combinations of points that could lead to a minimizing configuration.
The points in the surface are first projected to the weighted plane with coordinates \((u,v)\) and metric \(\lambda(v)^2du^2+dv^2\). Then geodesics are determined in Section 4 in the cases where \(\lambda\) is constant or piecewise constant (using that the geodesic is a straight segment or a polygonal) or continuous using Clairaut’s relation. The choice of a minimal geodesic is discussed in Section 5, and applications to the \(3\)-point Steiner problem are given in Section 6, where only three cases are shown to be possible in Proposition 6.4. In Section 7, an extension of the algorithm to the \(n\)-point Steiner problem is discussed.

MSC:

51E10 Steiner systems in finite geometry
49Q10 Optimization of shapes other than minimal surfaces
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