Embedding in space forms

The goal of this paper is to give explicit procedures and equations for performing metric multidimensional scaling to surfaces. More specifically, we describe a method for determining a configuration of points in a closed and orientable surface (i.e., the MDS space) for which the interpoint distances closely approximate a given set of dissimilarities. More generally, these constant sectional curvature surfaces are examples of space forms (spaces which are quotients of Euclidean, spherical, or hyperbolic space by a subgroup of the isometry group of the space). We will cast our work in this language, thereby allowing the theory to easily be generalized to higher dimensions.