Cross-ratios and angles determine a polygon

We prove that a unique simple polygon is determined, up to similarity, by t he interior angles at its vertices and the cross-ratios of diagonals of any given triangulation. (The cross-ratio of a diagonal is the product of the ratio of edge lengths for the two adjacent triangles.) This establishes a c onjecture of Driscoll and Vavasis, and shows the correctness of a key step of their algorithm for computing Schwarz-Christoffel transformations mappin g a disk to a polygon.