Curve-pleated structures

In this paper we study pleated structures generated by folding paper along curved creases. We discuss their properties and the special case of principal pleated structures. A discrete version of pleated structures is particularly interesting because of the rich geometric properties of the principal case, where we are able to establish a series of analogies between the smooth and discrete situations, as well as several equivalent characterizations of the principal property. These include being a conical mesh, and being flat-foldable. This structure-preserving discretization is the basis of computation and design. We propose a new method for designing pleated structures and reconstructing reference shapes as pleated structures: we first gain an overview of possible crease patterns by establishing a connection to pseudogeodesics, and then initialize and optimize a quad mesh so as to become a discrete pleated structure. We conclude by showing applications in design and reconstruction, including cases with combinatorial singularities. Our work is relevant to fabrication in so far as the offset properties of principal pleated structures allow us to construct curved sculptures of finite thickness.

[1]  Martin L. Demaine,et al.  Reconstructing David Huffman's Origami Tessellations , 2013 .

[2]  Arnold Tubis,et al.  Betsy Ross Revisited: General Fold and One-Cut Regular and Star Polygons , 2016 .

[3]  Klara Mundilova,et al.  On mathematical folding of curved crease origami: Sliding developables and parametrizations of folds into cylinders and cones , 2019, Comput. Aided Des..

[4]  Otto Röschel.  Curved Folding with Pairs of Cones , 2018, Advances in Intelligent Systems and Computing.

[5]  Christian D. Santangelo,et al.  The shape and mechanics of curved-fold origami structures , 2012, 1210.0778.

[6]  Levi H. Dudte,et al.  Geometric mechanics of periodic pleated origami. , 2012, Physical review letters.

[7]  B. Audoly,et al.  “Wunderlich, Meet Kirchhoff”: A General and Unified Description of Elastic Ribbons and Thin Rods , 2014, 1403.2094.

[8]  Takeo Igarashi,et al.  Interactive Design of Planar Curved Folding by Reflection , 2011, PG.

[9]  Amir A. Zadpoor,et al.  From flat sheets to curved geometries: Origami and kirigami approaches , 2017 .

[10]  Martin Kilian,et al.  Curved folding , 2008, ACM Trans. Graph..

[11]  David A. Huffman,et al.  Curvature and Creases: A Primer on Paper , 1976, IEEE Transactions on Computers.

[12]  Mark Pauly,et al.  Computational design of deployable auxetic shells , 2011, AAG.

[13]  Olga Sorkine-Hornung,et al.  Discrete Geodesic Nets for Modeling Developable Surfaces , 2017, ACM Trans. Graph..

[14]  Johannes Wallner,et al.  Interactive Design of Developable Surfaces , 2016, ACM Trans. Graph..

[15]  Tosiyasu L. Kunii,et al.  Bending and creasing virtual paper , 1994, IEEE Computer Graphics and Applications.

[16]  L. Mahadevan,et al.  Conical dislocations in crumpling , 1999, Nature.

[17]  Erik D. Demaine,et al.  Conic Crease Patterns with Reflecting Rule Lines , 2018, ArXiv.

[18]  Erik D. Demaine,et al.  Characterization of Curved Creases and Rulings: Design and Analysis of Lens Tessellations , 2015, ArXiv.

[19]  Erik D. Demaine,et al.  (Non)Existence of Pleated Folds: How Paper Folds Between Creases , 2009, Graphs Comb..

[20]  L. Mahadevan,et al.  Conical Surfaces and Crescent Singularities in Crumpled Sheets , 1998 .

[21]  Tomohiro Tachi,et al.  Programming curvature using origami tessellations. , 2016, Nature materials.

[22]  Olga Sorkine-Hornung,et al.  The shape space of discrete orthogonal geodesic nets , 2018, ACM Trans. Graph..

[23]  Ivan Izmestiev,et al.  Classification of flexible Kokotsakis polyhedra with quadrangular base , 2014, 1411.0289.

[24]  Joseph M. Gattas,et al.  The behaviour of curved-crease foldcores under low-velocity impact loads , 2015 .

[25]  A. Kokotsakis,et al.  Über bewegliche Polyeder , 1933 .

[26]  Levi H. Dudte,et al.  Geometric mechanics of curved crease origami. , 2012, Physical review letters.

[27]  William H. Frey,et al.  Modeling buckled developable surfaces by triangulation , 2004, Comput. Aided Des..

[28]  Walter Wunderlich.  Raumkurven, die pseudogeodätische Linien zweier Kegel sind , 1950 .

[29]  Joseph S. B. Mitchell,et al.  Folding flat silhouettes and wrapping polyhedral packages: New results in computational origami , 2000, Comput. Geom..

[30]  Tomohiro Tachi,et al.  Designing One-DOF Mechanisms for Architecture by Rationalizing Curved Folding , 2011 .

[31]  Johannes Wallner,et al.  Geometric Modeling with Conical Meshes and Developable Surfaces , 2006, ACM Trans. Graph..

[32]  Keenan Crane,et al.  Developability of triangle meshes , 2018, ACM Trans. Graph..

[33]  Tomohiro Tachi.  Geometric Considerations for the Design of Rigid Origami Structures , 2010 .

[34]  Tomohiro Tachi,et al.  Origamizing Polyhedral Surfaces , 2010, IEEE Transactions on Visualization and Computer Graphics.

[35]  Tim Hoffmann,et al.  On the Integrability of Infinitesimal and Finite Deformations of Polyhedral Surfaces , 2008 .

[36]  Tomohiro Tachi.  Generalization of rigid foldable quadrilateral mesh origami , 2009 .

[37]  Serge Tabachnikov,et al.  MORE ON PAPERFOLDING , 1999 .

[38]  Larry L. Howell,et al.  Rigidly Foldable Origami Twists , 2015 .

[39]  Basile Audoly,et al.  A non‐linear rod model for folded elastic strips , 2013, 1306.5035.

[40]  Tomohiro Tachi,et al.  Composite Rigid-Foldable Curved Origami Structure , 2013 .

[41]  Jiayao Ma,et al.  Elastic buckling shape control of thin-walled cylinder using pre-embedded curved-crease origami patterns , 2019, International Journal of Mechanical Sciences.

[42]  Joseph S. B. Mitchell,et al.  Folding flat silhouettes and wrapping polyhedral packages: new results in computational origami , 1999, SCG '99.

[43]  Hai Wang,et al.  Design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns , 2017, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[44]  Y. Pomeau,et al.  Crumpled paper , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[45]  A. Bobenko,et al.  Discrete Differential Geometry: Integrable Structure , 2008 .

[46]  Vlad Andrei Raducanu,et al.  Structural Architectural Elements Made of Curved Folded Sheet Metal , 2016, Proceedings of the 34th International Conference on Education and Research in Computer Aided Architectural Design in Europe (eCAADe) [Volume 2].

[47]  Rainer Barthel,et al.  Designing Grid Structures Using Asymptotic Curve Networks , 2018 .

[48]  Erik D. Demaine,et al.  Fast , Interactive Origami Simulation using GPU Computation , 2018 .

[49]  Arthur Lebée,et al.  From Folds to Structures, a Review , 2015 .

[50]  Byoungkwon An,et al.  Folding Angle Regulation by Curved Crease Design for Self-Assembling Origami Propellers , 2015 .

[51]  Jean-Pierre Raskin,et al.  Overcurvature describes the buckling and folding of rings from curved origami to foldable tents , 2012, Nature Communications.

[52]  Johannes Wallner,et al.  Freeform surfaces from single curved panels , 2008, ACM Trans. Graph..

[53]  Jun Mitani,et al.  COLUMN-SHAPED ORIGAMI DESIGN BASED ON MIRROR REFLECTIONS , 2012 .

[54]  Jun Mitani,et al.  A Design Method for 3D Origami Based on Rotational Sweep , 2009 .

[55]  Erik D. Demaine,et al.  Origamizer: A Practical Algorithm for Folding Any Polyhedron , 2017, SoCG.

[56]  Lin-zhi Wu,et al.  Fabrication and mechanical behaviors of carbon fiber reinforced composite foldcore based on curved-crease origami , 2019, Composites Science and Technology.

[57]  Rui Peng,et al.  Origami of thick panels , 2015, Science.

[58]  Yoël Forterre,et al.  How a curved elastic strip opens. , 2014, Physical review letters.

[59]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[60]  Erik D. Demaine,et al.  Reconstructing David Huffman’s Legacy in Curved-Crease Folding , 2016 .

[61]  Larry L. Howell,et al.  Rigidly foldable origami gadgets and tessellations , 2015, Royal Society Open Science.

[62]  K A Seffen,et al.  Compliant shell mechanisms , 2012, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[63]  Eitan Grinspun,et al.  Flexible Developable Surfaces , 2012, Comput. Graph. Forum.

[64]  H. Piaggio.  Differential Geometry of Curves and Surfaces , 1952, Nature.

[65]  Leif Kobbelt,et al.  OpenMesh: A Generic and Efficient Polygon Mesh Data Structure , 2002 .

[66]  D. Sussman,et al.  Algorithmic lattice kirigami: A route to pluripotent materials , 2015, Proceedings of the National Academy of Sciences.

[67]  Richard Duks Koschitz.  Computational design with curved creases : David Huffman's approach to paperfolding , 2014 .

[68]  Leonidas J. Guibas,et al.  Geodesic patterns , 2010, ACM Trans. Graph..