Represent a Folded Object

The following covers the math foundation needed to understand how a geometric model of an arbitrary polyhedron, whose plane faces approximates a generic object. Three levels of complexity, each building on the previous, form the framework needed to represent a crease pattern. Beginning with the flat foldable model and then generalizing into a structural and then finite element approach, the following apply mathematic rigor to modeling by defining the terminology used in a crease pattern.

Crease patterns are developed on the unit square, but can be generalized into rectangular shapes. Additionally, though origami is traditionally free of cutting (such is the realm of kirigami) the math used in modeling and simulator allows for such abstraction.

Geometric Model for Origami

Typically, an origami crease pattern is applied to a square sheet of paper. In the math representation, this allows for normalization to the unit square. Tachi in Freeform Origami defines as:

... art of folding, or more formally, isometrically transforming, a sheet of paper into various forms without stretching, cutting, or gluing another piece of paper to it.

The developability of a around each interior vertex has gauss area of 0 around each vertex$k$ by summing the over each incident sector angle. Fold angles need to be expressed in a reversible pattern: as there is a distinct sense of top or bottom side, the duality between finished and unfinished side is preserved. As shape is not preserved in the isometric transformation, conditions for sector angles and crease lengths need to be established:

$$\overbrace{ G_{k}=2 \pi-\sum_{i} \theta_{k, i}=0 }^\text{Maekawa's theorem} \qquad \overbrace{ \sum_{i} \theta_{k, i}-\sum_{i} \theta_{k, i}^{0}=0, \quad \sum_{i} \ell_{j, i}-\sum_{i} \ell_{j, i}^{0}=0 }^\text{Boundary Conditions}.$$

Discretizing to a finite number sector angles around each vertex ironically allows for a more flexible interpretation than using a smooth surface model which is more complex than the polyhedral approximation. Each incident sector angle between the unfolded and folded states must be the same. Additionally the length of each fold is also the same. So, for each interior vertex

$$F_{k}=\sum_{i} \sigma_{k, i} \theta_{k, i}=0 \rightarrow \left[\frac{\partial \mathbf{F}}{\partial \mathbf{x}}\right] d \mathbf{x}=\mathbf{0}$$

description for valid deformation. A valid deformation is a crease pattern represented as the vector equation $\mathbf{F}(\mathbf{x})=\mathbf{0}$ which is solved through the motion of deformation as the matrix $[\partial \mathbf{F} / \partial \mathbf{x}]$ is a rectangular matrix with dimensions $n_{\mathrm{c}} \times 3 n_{\mathrm{ver}}$ or the number of constraints and 3 times the number of interior vertices.

Maekawa's Theorem

Consider the crease pattern where one creases a square sheet in half once along the vertical and then along the horizontal. Unfolding would show a single node surrounded by 3 valley creases and one mountain. In this flat foldable state, folding once in half along the vertical and horizontal produces a self similar shape of another square. This is not a tessellation as the boundary was not preserved.

Tw consecutive mountain creases in origami simulator at 99% folded from originally flat state

An important consequence of Maekawa's Theorem on a flat foldable crease pattern, the number of mountain and valley folds will always differ by two. This is because any additional fold which runs through the node will always add both a mountain and valley, effectively "canceling" the additional folds. The description of the infinitesimal deformation $d{\bf x}$ as the kernel of the the$n_c \times 3n_{\text{vert}}$ Jacobian matrix and is found using a pseudo-inverse (akin to a stress measure).

$$d \mathbf{x}=\left(\mathbf{I}-\left[\frac{\partial \mathbf{F}}{\partial \mathbf{x}}\right]^{+}\left[\frac{\partial \mathbf{F}}{\partial \mathbf{x}}\right]\right) d \mathbf{x}_{0}$$

Flat Foldability

A crease pattern is flat foldable if each fold line can be folded completely such that each mountain and valley reach a fold angle of $\pm \pi$radians, respectively. From the flat sheet to a flat model, all nodes of a flat foldable crease pattern must have a valid set of coordinates as well as valid overlapping ordering of facets around each node in the deformed configuration

Structural Approach

A structural approach is presented by Schenk and Guest (2010), whose framework

$$\mathbf{A} \mathbf{t}=\mathbf{f} \quad \mathbf{C} \mathbf{d}=\mathbf{e} \quad \mathbf{G} \mathbf{e}=\mathbf{t}$$

• $\mathbf{A}$ ... equilibrium matrix
• $\mathbf{C}$ ... compatibility matrix
• $\mathbf{G}$ ... axial bar stiffness
• $\mathbf{f}$ ... nodal forces
• $\mathbf{t}$ ... bar tension
• $\mathbf{e}$... bar extensions

The first equation comes from truss analysis, by method of joints: equilibrium around a node requires that the tensions acting on node must sum to zero.The second equation comes from generalized Hooke's law, such that the displacements are related to the strains. In origami, the application of a crease pattern to a sheet of paper, is an exploration in the kernel or nullspace of compatiblity matrix, ie. the set of nodal displacements such that bar extensions are zero.

$$\begin{align} &F=\sin (\theta)=\sin (\theta(\mathbf{p}))=\ldots\\ &J=\frac{1}{\cos (\theta)} \sum \frac{\partial F}{\partial p_{i}} d p_{i}=d \theta\\ & \left[\begin{array}{l} \mathbf{C} \\ \mathbf{J} \end{array}\right] \mathbf{d}=\left[\begin{array}{c} \mathbf{e} \\ d \theta \end{array}\right]\\ \end{align}$$

Stiffness analysis

$$\begin{aligned} \mathbf{K} \mathbf{d} &=\mathbf{f} \\ \mathbf{K} &=\mathbf{A} \mathbf{G} \mathbf{C}=\mathbf{C}^{T} \mathbf{G} \mathbf{C} \\ &\mathbf{K}=\left[\begin{array}{l} \mathbf{C} \\ \mathbf{J} \end{array}\right]^{T}\left[\begin{array}{cc} \mathbf{G} & 0 \\ 0 & \mathbf{G}_{J} \end{array}\right]\left[\begin{array}{l} \mathbf{C} \\ \mathbf{J} \end{array}\right] \end{aligned}$$

Finite Element Formulation

In realtime simulation, a highly parallelized method of the finite element method (FEM) is used. Using a crease pattern, the Origami Simulator adds additional edges such that the crease pattern is triangularized, such that each polygonal face is separated into individual triangular faces. These additional creases are not forced to a positive or negative fold angle, but elastically constrained to zero. This allows for flexibility which would otherwise not be possible for rigid facets.