  Polyhedral surfaces are fundamental objects in architectural geometry and
industrial design. Whereas closeness of a given mesh to a smooth reference
surface and its suitability for numerical simulations were already studied
extensively, the aim of our work is to find and to discuss suitable assessments
of smoothness of polyhedral surfaces that only take the geometry of the
polyhedral surface itself into account. Motivated by analogies to classical
differential geometry, we propose a theory of smoothness of polyhedral surfaces
including suitable notions of normal vectors, tangent planes, asymptotic
directions, and parabolic curves that are invariant under projective
transformations. It is remarkable that seemingly mild conditions significantly
limit the shapes of faces of a smooth polyhedral surface. Besides being of
theoretical interest, we believe that smoothness of polyhedral surfaces is of
interest in the architectural context, where vertices and edges of polyhedral
surfaces are highly visible.
