## 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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### References:

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