Smooth Origami Animation by Crease Line Adjustment
Tomohiro Tachi
tachi.tomohiro@gmail.com
The University of Tokyo
Crease Line Adjustment
Introduction
A folding process of simple origami can be represented by
a 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 are
constrained around vertices and not independent of each
other.
Hence the rigid origami model that represents origami as
plates 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 of
degrees of freedom and a singularity problem that some of
the crease lines cannot begin to be folded until other crease
lines are completely folded flat.
We propose a method for making a smooth and
comprehensible
origami
animation
by
avoiding
above-mentioned problems by adding and adjusting crease
lines on an origami model. The overall folding process is
constructed from crease line pattern.
Another way of making locally closed vertices open is to
adjust the angle between the crease lines. Positions of the
vertices are moved so that the inner angle of each locally
closed 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 Kawasakis theorem, which is,
Overview
ni / 2
ni / 2
Our method makes locally closed vertices open by
Triangulation of Polygons and Crease Line Adjustment.
k =1
k =1
Fi
X
We call a vertex "locally closed" if there is a set of three
adjacent crease lines from the vertex that the outer two are
symmetrical about the center crease and their
mountain-valley attributes are opposite. Locally closed
vertex is not rigidly foldable before the center crease is
completely folded flat. We call the angle between the center
crease line and one of the other two crease lines with the
same mountain-valley attribute as the center crease line
"outer angle", and the other angle "inner angle." We can
make a locally closed vertex "open" by splitting or by
lessening the inner angle.
Fi ( X , Y ) =
Rigid Origami Kinematics
Locally Closed Vertex
i,2k 1 =
where i ,1 , L , i , ni are the angles between adjacent crease
lines around the vertex i .
Global constraint matrix is given by
Method
We use rigid origami model, which is plate-hinge model
that contains a closed loop around each vertex, for the
simulation 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.
i,2k =
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 k
vertices (k>3), we can add k-3 degrees of freedom to the
model. 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 crease
lines must not intersect each other, it is not always possible to
split the inner angles of the locally closed vertices.
Fi X&
= [0] where
Y Y&
ni / 2
ni / 2
k =1
k =1
i,2k i,2k 1 = 0
Under this constraint, positions of the vertices (X, Y) are
adjusted 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 better
balanced when the crease line adjustment angle is larger.
We have observed through simulation of actual models that
the 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 the
adjustment 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 of
origami.
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 make
a smooth folding animation from
crease pattern to folded shape.
There are still some origami models
that cannot be folded because of the
lack of degrees of freedom and the
globally 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 process
of a complex model as Teapot and
Teacup is possible by triangulation of
the polygons and crease line
adjustment.
This model cannot be folded
completely flat without intersection
of the polygons and too much
deformation of the model to be
perceived as the same shape as the
original.
(Both Triangulated)
A folding process of fold lines
Just triangulation of squares
over other folds is modeled
naturally
with
crease
line resulted in smooth animation of
double accordion folding.
adjustment.
By adjusting the crease lines, the
configuration gets comprehensible
by
displacement
of
the
overlapping edges.
The adjusted crease pattern is
identical to the crease pattern of
Miura Fold.
(Completely folded, but with intersection and
too 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
