Dev2PQ: Planar Quadrilateral Strip Remeshing of Developable Surfaces

publication
ACM Transactions on Graphics
authors
Floor Verhoeven, Amir Vaxman, Tim Hoffmann, Olga Sorkine-Hornung

Developable shapes can be digitally acquired by 3D scanning or freeform modeling (a). In such scenarios, the meshing is typically not aligned to principal curvature directions, which hampers practical applications, such as fabrication with flat polygonal panels. Our method remeshes an input mesh of a (piecewise) developable surface into a curvature aligned, planar polygonal mesh (e) by computing a vector field (c), from which we integrate a function whose isolines (d) align as well as possible to the locally estimated noisy rulings (b). Our vector field contains automatically-placed singularities in the planar region (d), which result in naturally placed triangular patches.

abstract

We introduce an algorithm to remesh triangle meshes representing developable surfaces to planar quad dominant meshes. The output of our algorithm consists of planar quadrilateral (PQ) strips that are aligned to principal curvature directions and closely approximate the curved parts of the input developable, and planar polygons representing the flat parts of the input that connect the PQ strips. Developable PQ-strip meshes are useful in many areas of shape modeling, thanks to the simplicity of fabrication from flat sheet material. Unfortunately, they are difficult to model due to their restrictive combinatorics. Other representations of developable surfaces, such as arbitrary triangle or quad meshes, are more suitable for interactive freeform modeling, but generally have non-planar faces or are not aligned to principal curvatures. Our method leverages the modeling flexibility of non-ruling based representations of developable surfaces, while still obtaining developable, curvature aligned PQ-strip meshes. Our algorithm optimizes for a scalar function on the input mesh, such that its isolines are extrinsically straight and align well to the locally estimated ruling directions. The condition that guarantees straight isolines is nonlinear of high order and numerically difficult to enforce in a straightforward manner. We devise an alternating optimization method that makes our problem tractable and practical to compute. Our method works automatically on any developable input, including multiple patches and curved folds, without explicit domain decomposition. We demonstrate the effectiveness of our approach on a variety of developable surfaces and show how our remeshing can be used alongside handle based interactive freeform modeling of developable shapes.

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acknowledgments

The authors are grateful to Helmut Pottmann, Michael Rabinovich and Alexander Sorkine-Hornung for illuminating discussions and guidance, and to Kaan Baki for help with video production. We also thank the authors of [Kilian et al. 2008], [Tang et al. 2016], [Stein et al. 2018] and [Jiang et al. 2020] for providing us with models from their works. This work is supported in part by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101003104) and by the Deutsche Forschungsgemeinschaft-Collaborative Research Center, TRR 109, "Discretization in Geometry and Dynamics".