Cayley-Menger coordinates

Abstract

A major obstacle in applying distance geometry techniques is the analytical complexity of the Cayley-Menger determinants that are used to characterize euclidian spaces in terms of distances between points. In this paper we show that, with the aid of a theorem of Jacobi, the complex Cayley-Menger determinants can be replaced by simpler determinants, and we derive the concept of Cayley-Menger coordinates, a coordinate system in terms of which each point of En is characterized by n + 1 distances to n + 1 points of a reference. We also show that this coordinate system provides a natural norm for the incomplete embedding problem. This paper provides the tools to treat the problem of filling out an incomplete distance matrix so that our previous procedure can then be used to embed the corresponding structure in a three-dimensional space.