Random spherical triangles I: Geometrical background

In this paper we identify the shape space .(S2, k) for k labelled points on the sphere S2 that gives a mathematical model applicable to data sets whose elements are, or can be represented by, configurations of labelled sequences of points on S2 and for which the fundamental properties of interest are the shapes of these configurations, and we examine the geometric structures on the space, especially the riemannian structure on .(S2, 3). In a companion paper (pp. 581.594) we investigate the statistical properties of such shapes when the k points are generated by a random mechanism.