We introduce a modified Regge calculus for general relativity on a triangulated four dimensional Riemannian manifold where the fundamental variables are areas and a certain class of angles. These variables satisfy constraints which are local in the triangulation. We expect the formulation to have applications to classical discrete gravity and non-perturbative approaches to quantum gravity.

We consider the notion of improved and perfect actions within Regge calculus. These actions are constructed in such a way that they—although being defined on a triangulation—reproduce the continuum dynamics exactly, and therefore capture the gauge symmetries of general relativity. We construct the perfect action in three dimensions with a cosmological constant, and in four dimensions for one simplex. We conclude with a discussion about Regge calculus with curved simplices, which arises naturally in this context.