Unfolding of surfaces

Abstract

This paper deals with the approximation of the unfolding of a smooth globally developable surface (i.e. "isometric" to a domain of script E sign2) with a triangulation. We prove the following result: let Tn be a sequence of globally developable triangulations which tends to a globally developable smooth surface S in the Hausdorff sense. If the normals of Tn tend to the normals of S, then the shape of the unfolding of Tn tends to the shape of the unfolding of S. We also provide several examples: first, we show globally developable triangulations whose vertices are close to globally developable smooth surfaces; we also build sequences of globally developable triangulations inscribed on a sphere, with a number of vertices and edges tending to infinity. Finally, we also give an example of a triangulation with strictly negative Gauss curvature at any interior point, inscribed in a smooth surface with a strictly positive Gauss curvature. The Gauss curvature of these triangulations becomes positive (at each interior vertex) only by switching some of their edges. © Springer 2006.