Polygons are compound geometric objects, but when trying to understand the expected behavior of a large collection of random polygons – or even to formalize what a random polygon is – it is convenient to interpret each polygon as a point in some parameter space, essentially trading the complexity of the object for the complexity of the space. In this paper I describe such an interpretation where the parameter space is an abstract but very nice space called a Stiefel manifold and show how to exploit the geometry of the Stiefel manifold both to generate random polygons and to morph one polygon into another.