Universal Rigidity of Bar Frameworks in General Position: A Euclidean Distance Matrix Approach

A configuration p in r-dimensional Euclidean space is a finite collection of labeled points $${p}^{1},\ldots ,{p}^{n}$$ in $${\mathbb{R}}^{r}$$ that affinely span $${\mathbb{R}}^{r}$$ . Each configuration p defines a Euclidean distance matrix $${D}_{p} = ({d}_{ij})$$ = $$(\vert \vert {p}^{i} - {p}^{j}\vert {\vert }^{2})$$ , where $$\vert \vert \cdot \vert \vert $$ denotes the Euclidean norm. A fundamental problem in distance geometry is to find out whether or not a given proper subset of the entries of D p suffices to uniquely determine the entire matrix D p . This problem is known as the universal rigidity problem of bar frameworks. In this chapter, we present a unified approach for the universal rigidity of bar frameworks, based on Euclidean distance matrices (EDMs), or equivalently, on projected Gram matrices. This approach makes the universal rigidity problem amenable to semidefinite programming methodology. Using this approach, we survey some recently obtained results and their proofs, emphasizing the case where the points $${p}^{1},\ldots ,{p}^{n}$$ are in general position.