Authors
N. Leduc, C.Douthe, B. Vaudeville, S. Aubry, K. Leempoels, L. Hauswirth, O. Baverel
Abstract
This paper proposes to extend the notions of mean and Gaussian curvatures of smooth and discrete surfaces to semi-discrete surfaces. This class of surfaces, also called piecewise developable surfaces is characterized by a zero Gaussian curvature everywhere except for seams between two patches and vertices. The study focuses on families of surfaces with constant Gaussian curvature along the seams. As a case study of the problem, dForms, closed piecewise developable surfaces formed by two discs of different shapes but equal perimeters, are chosen.
After a topological description of the semi-discrete surfaces, based on the Gauss- Bonnet theorem, two types of dForms with constant Gaussian curvature are studied. The first type describes a two-parameter family based on symmetrical cutting patterns. The symmetry property allows the three-dimensional geometry of this subclass of dForms to be fully described. The second type is asymmetrical and makes use of the properties of the evolute of the curves of the cutting patterns to fulfil the closure condition. In this case, we also propose a method to explore the space of possible configurations using an augmented isometric flow method.
