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Discrete Affine Surfaces based on Quadrangular Meshes

In this paper we are interested in defining affine structures on discrete quadrangular surfaces of the affine three-space. We introduce, in a constructive way, two classes of such surfaces, called respectively indefinite and definite surfaces. The underlying meshes for indefinite surfaces are asymptotic nets satisfying a non-degeneracy condition, while the underlying meshes for definite surfaces are non-degenerate conjugate nets satisfying a certain natural condition. In both cases we associate to any of these nets several discrete affine invariant quantities: a metric, a normal and a co-normal vector fields, and a mean curvature. Moreover, we derive structural and compatibility equations which are shown to be necessary and sufficient conditions for the existence of a discrete quadrangular surface with a given affine structure.