Small deformations of polygons

Abstract

We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued quadratic invariant $b$ on the space of those isometric deformations which, for convex polygons, has a remarkable positivity property. We give two geometric applications. The first is an isoperimetric statement for hyperbolic polygons: among the convex hyperbolic polygons with given edge lengths, there is a unique polygon with vertices on a circle, a horocycle, or on one connected component of the space of points at constant distance from a geodesic, and it has maximal area. The second application is a new proof of the infinitesimal rigidity of convex polyhedra in the Euclidean space, and a new rigidity result for polyhedral surfaces in the Minkowski space. Finally we indicate how the invariant $b$ can be used to define natural metrics on the space of convex spherical (or hyperbolic) polygons with fixed edge lengths. Those metrics are related to known (and interesting) metrics on the space of convex polygons with given angles in the plane.