Abstract
Following the description of the unfolding polyhedra method for origami design in the previous chapter, here the focus switches to an origami design method applicable to a much wider spectrum of three-dimensional goal shapes. This chapter presents the tuck-folding method to solve the following origami design problem: given a goal shape represented as a polygonal mesh (termed as the goal mesh), find the shape and fold pattern of a planar sheet that can be folded to match the goal mesh, and a history of folding motion from the planar configuration of the sheet to the configuration that matches the goal mesh. The method generates a sheet comprised of the faces of the goal mesh in addition to introduced regions having two rigid faces and three creased folds. The creased folds are used to tuck-fold the added regions to form the shape of the goal mesh. We also address the implementation of the tuck-folding method in a computational environment.
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- 1.
The goal mesh \(\mathcal {M}\) is a connected, orientable, 2-manifold polygonal mesh.
- 2.
Two surfaces are said to be topologically equivalent (or homeomorphic) if there exists a continuous map between the surfaces and such a map has a continuous inverse [11]. For example, a hemispherical surface and a circle are topologically equivalent, as we can continuously map one surface into the other. On the other hand, a closed spherical surface and a circle are not topologically equivalent, as we need to introduce additional boundaries to the closed spherical surface in order to continuously map it into a circle.
- 3.
Another strategy is to cover the holes using compliant membranes such that the deformation of these membranes does not significantly affect the folding motion of the designed origami sheet. Nevertheless, we do not consider such a strategy in this chapter for the sake of simplicity.
- 4.
A possible extension of the proposed design method could consider the replacement of certain edge modules by single creased folds and thus allow for simplification of the designed origami sheets and reduction of the total number of folds. However, here the approach of [2, 8] is taken where an edge module is applied for each interior edge of \(\mathcal {M}\) due to its wide applicability to a range of origami design problems. The aforementioned extension is strongly recommended for future studies.
- 5.
The set of the integer numbers is denoted as \(\mathbb {Z}\).
- 6.
The Macaulay brackets are denoted as 〈⋅〉 and defined as: \(\langle y\rangle = \left \{ \begin {array}{r l} y; & y \geq 0 \\[0.5em] 0; & y < 0 \end {array} \right .\).
- 7.
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Peraza Hernandez, E.A., Hartl, D.J., Lagoudas, D.C. (2019). Tuck-Folding Method for the Design of Origami Structures with Creased Folds. In: Active Origami. Springer, Cham. https://doi.org/10.1007/978-3-319-91866-2_4
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DOI: https://doi.org/10.1007/978-3-319-91866-2_4
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