Distance Geometry on the Sphere

The Distance Geometry Problem asks whether a given weighted graph has a realization in a target Euclidean space R which ensures that the Euclidean distance between two realized vertices incident to a same edge is equal to the given edge weight. In this paper we look at the setting where the target space is the surface of the sphere SK−1. We show that the DGP is almost the same in this setting, as long as the distances are Euclidean. We then generalize a theorem of Gödel about the case where the distances are spherical geodesics, and discuss a method for realizing cliques geodesically on a K-dimensional sphere.