Steiner trees on curved surfaces (Q5943054)

First the author points out the differences between the Steiner trees in the plane and those on curved surfaces. Then he discusses in detail Steiner trees on spheres, in particular the properties of Steiner points (defined by the condition that all angles at this point are equal to \(2\pi /3\)). The last section is devoted to the study of Steiner points for spherical triangles. For instance, here the following theorem is proved: ``If all sides of \(\Delta abc\) are no more than \(\arccos (-1/\sqrt{3})\), then there is at most one Steiner point lying in \(\Delta abc\).'' One of the main tools are ``equiangular curves''. These are defined by the locus of the point \(p\) with constant angle \(\theta =\angle apb\) for given points \(a\) and \(b\). There are two sign errors in the discussion of equiangular curves, one in the expression for \(|bp|\) and one in the second formula for the angle \(\theta \).