Discrete Geodesic Nets for Modeling Developable Surfaces

ACM Transactions on Graphics
Michael Rabinovich, Tim Hoffmann, Olga Sorkine-Hornung

Discrete Geodesic Nets for Modeling Developable Surfaces

We propose a discrete model for developable surfaces. The strength of our model is its locality, offering a simple and consistent way to realize deformations of various developable surfaces without being limited by the topology of the surface or its decomposition into torsal developable patches. Our editing system allows for operations such as freeform handle-based editing, cutting and gluing, modeling closed and un-oriented surfaces, and seamlessly transitioning between planar, cylindrical, conical and tangent developable patches, all in a unified manner.


We present a discrete theory for modeling developable surfaces as quadrilateral meshes satisfying simple angle constraints. The basis of our model is a lesser known characterization of developable surfaces as manifolds that can be parameterized through orthogonal geodesics. Our model is simple, local, and, unlike previous works, it does not directly encode the surface rulings. This allows us to model continuous deformations of discrete developable surfaces independently of their decomposition into torsal and planar patches or the surface topology. We prove and experimentally demonstrate strong ties to smooth developable surfaces, including a theorem stating that every sampling of the smooth counterpart satisfies our constraints up to second order. We further present an extension of our model that enables a local definition of discrete isometry. We demonstrate the effectiveness of our discrete model in a developable surface editing system, as well as computation of an isometric interpolation between isometric discrete developable shapes.


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The authors would like to thank Noam Aigerman, Mario Botsch, Oliver Glauser, Roi Poranne, Katja Wolff, Christian Schüller, Jan Wezel and Hantao Zhao for illuminating discussions and help with results production. The work was supported in part by the European Research Council under Grant No.: StG-2012-306877 (ERC Starting Grant iModel) and by the Deutsche Forschungsgemeinschaft-Collaborative Research Center, TRR 109, "Discretization in Geometry and Dynamics."