Smooth Origami Animation by Crease Line Adjustment
Tomohiro Tachitachi.tomohiro@gmail.com
The University of Tokyo
Crease Line Adjustment

Introduction
A folding process of simple origami can be represented bya sequence of simple folding processes [Miyazaki et al.
1996], but this method cannot be applied for complex
origami models because the folds in a complex model areconstrained around vertices and not independent of eachother.
Hence the rigid origami model that represents origami asplates connected by hinges is suitable for complex origami
models. Balkcom [2002] shows a way for calculating rigid
origami kinematics. This model has two problems, lack ofdegrees of freedom and a singularity problem that some of
the crease lines cannot begin to be folded until other creaselines are completely folded flat.
We propose a method for making a smooth andcomprehensible
origamianimationby
avoiding
above-mentioned problems by adding and adjusting creaselines on an origami model. The overall folding process isconstructed from crease line pattern.

Another way of making locally closed vertices “open” is toadjust the angle between the crease lines. Positions of thevertices are moved so that the inner angle of each locallyclosed vertex gets smaller than the outer angle. For
appropriate deformation through rigid origami simulation,
keeping of the flat-foldability of originally flat-foldable
vertices is necessary. A condition for single vertex
flat-foldability is given by Kawasaki’s theorem, which is,
Overview

ni / 2ni / 2

Our method makes locally closed vertices “open” by
Triangulation of Polygons and Crease Line Adjustment.

k =1k =1

∑ θi,2k = ∑ θi,2k −1 = π

where θ i ,1 , L , θ i , ni are the angles between adjacent creaselines around the vertex i .
Global constraint matrix is given by∂Fi ⎤ ⎡ X& ⎤⎢ ⎥ = [0] where∂Y ⎥⎦ ⎣ Y& ⎦Method⎡ ∂Fi⎢ ∂X⎣

Rigid Origami Kinematics

Fi ( X , Y ) = ∑ θ i , 2 k − ∑ θ i , 2 k −1 = 0

We use rigid origami model, which is plate-hinge modelthat contains a closed loop around each vertex, for thesimulation of origami. Change in configuration is simulated
by calculating pseudo inverse of a global constraint matrix,
which is built from conditions for single vertex rigid origami
that Belcastro and Hull [2002] presented.
Locally Closed Vertex

We call a vertex "locally closed" if there is a set of threeadjacent crease lines from the vertex that the outer two aresymmetrical about the center crease and theirmountain-valley attributes are opposite. Locally closed
vertex is not rigidly foldable before the center crease iscompletely folded flat. We call the angle between the centercrease line and one of the other two crease lines with thesame mountain-valley attribute as the center crease line
"outer angle", and the other angle "inner angle." We canmake a locally closed vertex "open" by splitting or bylessening the inner angle.

Triangulation of Polygons

Most origami models are not rigidly foldable because of
the lack the degrees of freedom. The degrees of freedom
are m-3n if the constraint matrix is full rank, where m and n
are the number of edges and vertices inside the paper
respectively. By adding k-3 crease lines on polygons with kvertices (k>3), we can add k-3 degrees of freedom to themodel. By this triangulation, some of the models become
rigidly foldable.
We chose the crease lines for the triangulation that split as
many inner angles of locally closed vertices as possible to
avoid singularity of the folds. Because the added creaselines must not intersect each other, it is not always possible tosplit the inner angles of the locally closed vertices.

|ni / 2ni / 2k =1

k =1

Under this constraint, positions of the vertices (X, Y) areadjusted to make the inner angle smaller than the outer angle
for each locally closed vertex. Although any finite angle
adjustment to the right direction can avoid the local
singularity, the progress of folding of crease line is betterbalanced when the crease line adjustment angle is larger.
We have observed through simulation of actual models thatthe animation is smooth and the configuration of the origami
model is easily comprehensible if the overall folding process
is well balanced.
For making the folding process balanced, we change theadjustment angle for each vertex according to the“imbalanceness” of the creases around the vertex. The
adjustment angle is a monotonic function of a variable{max(crease_angle) - min(crease_angle)}. This results in
no adjustment for both unfolded and flat-folded state oforigami.

Result
We have implemented this No Adjustment
method as an interactive application
and applied for complex origami
models.
The proposed method
enabled the simulation of many of
the origami models, and could makea smooth folding animation fromcrease pattern to folded shape.
There are still some origami models
that cannot be folded because of thelack of degrees of freedom and theglobally closed structure.

With Adjustment No Adjustment

With Adjustment

No Adjustment

With Adjustment

No Adjustment

With Adjustment(Both Triangulated)(Both Triangulated)(Both Triangulated)(Model: “Devil” by Jun Maekawa)

The animation of the folding processof a complex model as “Teapot andTeacup” is possible by triangulation of
the polygons and crease line
adjustment.

This model cannot be foldedcompletely flat without intersectionof the polygons and too muchdeformation of the model to beperceived as the same shape as theoriginal.

(Both Triangulated)
A folding process of fold lines
Just triangulation of squaresover other folds is modeled
naturallywithcreaseline resulted in smooth animation of“double accordion folding.”
adjustment.
By adjusting the crease lines, theconfiguration gets comprehensible
bydisplacementoftheoverlapping edges.
The adjusted crease pattern is
identical to the crease pattern of“Miura Fold.”(Completely folded, but with intersection andtoo much deformation)

Reference
BALKCOM, D. 2002. Robotic Origami Folding. PhD thesis, Carnegie Mellon University.
BELCASTRO, S. M. and HULL, T. C. 2002. A Mathematical Model for Non-Flat Origami. In Proceedings of the 3rd International Meeting of Origami Mathematics, Science and Education, 39-52.
MIYAZAKI, S., YASUDA, T., YOKOI, S., and TORIWAKI, J. 1996. An Origami Playing Simulator in the Virtual Space. The Journal of Visualization and Computer Animation, 7, 1, 25-4