Cross-Ratios and Angles Determine a Polygon

We prove that a unique simple polygon is determined, up to similarity, by the 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 conjecture of Driscoll and Vavasis, and shows the correctness of a key step of their algorithm for computing Schwarz—Christoffel transformations mapping a disk to a polygon. Received April 10, 1998, and in revised form March 24, 1999.