Steiner's problem and fagnano's result on the sphere

The Steiner problem on the regular tetrahedron

“…The algorithm is complete (see figure for example). Another observation relevant to our discussion of the Steiner problem on the regular tetrahedron is that no geodesic passes through the vertices of a narrow cone [Lee et al 2011]. Since a small neighborhood of a vertex is a narrow cone, no shortest path network will pass through any vertices of -.…”

“…Algorithms exist to find the solution for the Steiner problem on certain surfaces of constant curvature. The problem was studied in [Weng 2001;Litwhiler and Aly 1980;Brazil et al 1998] for on curved surfaces, including spheres. March and Halverson [2005] studied Steiner trees in hyperbolic space.…”

“…March and Halverson [2005] studied Steiner trees in hyperbolic space. Lee et al [2011] studied the Steiner problem on wide and narrow cones. Penrod [2007] and May and Mitchell [2007] developed algorithms to solve Steiner problems on the flat torus.…”

Four-point Fermat location problems revisited. New proofs and extensions of old results

“…In [14], Fagnano's result was obtained for four points on a same hemisphere, by way of an analytical reasoning. Lemma 2 allows to obtain a much more complete answer on the sphere, as follows.…”

“…The stereographic projection arguments used in [14] may quite easily be extended to prove the following absorbed case result on the sphere (one may wonder why this case was not considered there, but only the floating case, which may be solved more completely using much simpler means).…”