Modelling the folding of paper into three dimensions using affine transformations

belcastro, sarah-marie; Hull, Thomas

doi:10.1016/s0024-3795(01)00608-5

Cited by 138 publications

(113 citation statements)

References 3 publications

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“…Therefore, upon changing the crease pattern, one can assume that the dominant deformation modes of an origami-folded structure do not strain facets. These deformation modes are referred to as rigid foldings [13,14]. Given their low potential energy, rigid foldings can also be called compliant, floppy or even zero-energy deformation modes.…”

Section: Introductionmentioning

confidence: 99%

Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern

2017

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Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape, curvature and elongation on a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-trivial surfaces thus inspiring applications across a wide range of domains including structural engineering, architectural design and aerospace engineering. The present paper suggests a homogenization-type two-scale asymptotic method which, combined with standard tools from differential geometry of surfaces, yields a macroscopic continuous characterization of the global deformation modes of origami tessellations and other similar periodic pin-jointed trusses. The outcome of the method is a set of nonlinear differential equations governing the parametrization, metric and curvature of surfaces that the initially discrete structure can fit. The theory is presented through a case study of a fairly generic example: the eggbox pattern. The proposed continuous model predicts correctly the existence of various fittings that are subsequently constructed and illustrated.

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Section: Introductionmentioning

confidence: 99%

Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern

2017

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“…Among the current research on origami mathematics, a great majority is focused on the flat-and/or rigidfoldability problems [10][11][12][13][14][15][16][17], whereas origami geometric design that studies the folded configurations and/or folding mechanisms of origami structures and is fundamental to many engineering applications is still intractable. Deriving parametric equations that describe the folding kinematics of a certain crease pattern is the most commonly used approach in origami design [18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Design of three-dimensional origami structures based on a vertex approach

2015

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Origami geometric design is fundamental to many engineering applications of origami structures. This paper presents a new method for the design of three-dimensional (3D) origami structures suitable for engineering use. Using input point sets specified, respectively, in the x−z and y−z planes of a Cartesian coordinate system, the proposed method generates the coordinates of the vertices of a folded origami structure, whose fold lines are then defined by straight line segments each connecting two adjacent vertices. It is mathematically guaranteed that the origami structures obtained by this method are developable. Moreover, an algorithm to simulate the unfolding process from designed 3D configurations to planar crease patterns is provided. The validity and versatility of the proposed method are demonstrated through several numerical examples ranging from Miura-Ori to cylinder and curved-crease designs. Furthermore, it is shown that the proposed method can be used to design origami structures to support two given surfaces.

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“…For rigid foldability, folding along fold lines should permit the structure to transition between states while c planar = 0 is continuously satisfied. The analytical derivations of the kinematics and geometric characteristics of foldability (including rigid foldability) have been previously discussed [34][35][36][37], however these tend to be cumbersome for verifying the rigid foldability of complex origami systems. A more straightforward method to check rigid foldability is to perform the eigenvalue analyses described in §6 with the fold stiffness (K ρ ) substantially reduced (e.g.…”

Section: Discussionmentioning

confidence: 99%

“…The folding of the tube can be performed through an analytical [34][35][36][37] through all of the vertices in the pattern until all fold angles, and the new geometric shape, are calculated. We use the numerical method in [38] to perform the folding.…”

Section: Kinematics In Reconfiguring Polygonal Tubesmentioning

confidence: 99%

Origami tubes with reconfigurable polygonal cross-sections

2016

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Thin sheets can be assembled into origami tubes to create a variety of deployable, reconfigurable and mechanistically unique three-dimensional structures. We introduce and explore origami tubes with polygonal, translational symmetric cross-sections that can reconfigure into numerous geometries. The tubular structures satisfy the mathematical definitions for flat and rigid foldability, meaning that they can fully unfold from a flattened state with deformations occurring only at the fold lines. The tubes do not need to be straight and can be constructed to follow a nonlinear curved line when deployed. The cross-section and kinematics of the tubular structures can be reprogrammed by changing the direction of folding at some folds. We discuss the variety of tubular structures that can be conceived and we show limitations that govern the geometric design. We quantify the global stiffness of the origami tubes through eigenvalue and structural analyses and highlight the mechanical characteristics of these systems. The two-scale nature of this work indicates that, from a local viewpoint, the crosssections of the polygonal tubes are reconfigurable while, from a global viewpoint, deployable tubes of desired shapes are achieved. This class of tubes has potential applications ranging from pipes and microrobotics to deployable architecture in buildings.

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Modelling the folding of paper into three dimensions using affine transformations

Cited by 138 publications

References 3 publications

Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern

Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern

Design of three-dimensional origami structures based on a vertex approach

Origami tubes with reconfigurable polygonal cross-sections

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