Small deformations of polygons and polyhedra

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 quadratic invariant on the space of first-order deformations of a polygon. For convex polygons, this quadratic invariant has a positivity property, leading to a new proof of the infinitesimal rigidity of convex polyhedra in the Euclidean space, and also to new rigidity results for polyhedral surfaces. It can also be used to define natural metrics on the space of convex spherical (or hyperbolic) polygons with fixed edge lengths. Those metrics appear to be related to known (and interesting metrics) on the space of convex polygons with given angles in the plane.