n modern architecture, facades and glass roofs often model smooth shapes but are realized as polyhedral surfaces. Bad approximations may be observed as wiggly meshes, even though the polyhedral mesh is close to a smooth reference surface. So what does it mean for a polyhedral surface to be smooth? In this talk, we discuss an assessment of smoothness that is based on the normal images of vertex neighborhoods. By drawing analogies to the classical theory, we derive desirable conditions for vertex neighborhoods that result in certain shapes of the normal images, and we show how reasonable requirements to the normal images of face neighborhoods lead to similar shapes of the faces of the mesh. The corresponding duality is explained by the projective invariance of our notion of smoothness. This is joint work with Caigui Jiang and Helmut Pottmann.