http://www.flickr.com/photos/31375127@N07/

Marcelo A. Dias & Christian D. Santangelo

NSF DMR-0846582

Curved Fold Origami

http://www.flickr.com/photos/31375127@N07/

折り紙 (Origami)

http://pantspantsnopants.files.wordpress.com/2010/08/classic-origami.jpg

oru “to fold” + kami, “paper” = the art of folding paper

Folding along straight creases.

折り紙 (Origami)

http://pantspantsnopants.files.wordpress.com/2010/08/classic-origami.jpg

oru “to fold” + kami, “paper” = the art of folding paper

Folding along straight creases.

How can one explore new set of shapes?

折り紙 (Origami)

http://pantspantsnopants.files.wordpress.com/2010/08/classic-origami.jpg

oru “to fold” + kami, “paper” = the art of folding paper

Folding along straight creases.

How can one explore new set of shapes?

Folding along curved crease patterns!

Bauhaus: Weimar, Dessau, Berlin, Chicago by Hans M. Winglerhttp://erikdemaine.org/curved/history/

Student's work at the Bauhaus 1927–1928

What do we know?

Exploring new shapes...

What do we know?

Erik Demaine, et al., Curved Crease Origami

...Design and Computational Origami

Kilian at al., Curved Folding

Erik Demaine, et al., Curved Crease Origami

...Design and Computational Origami

Kilian at al., Curved Folding

Erik Demaine, et al., Curved Crease Origami

...Design and Computational Origami

Kilian at al., Curved Folding

Geometry of Folding

s 1/!g(s)

c0(s)

One flat developable surface

Geometry of Folding

s 1/!g(s)

c0(s)

One flat developable surface

!Folding

t1/!g(s)

nb

u1

N1

c(s)

Two developables connected by a crease

Geometry of Folding

s 1/!g(s)

c0(s)

One flat developable surface

!Folding

t1/!g(s)

nb

u1

N1

c(s)

Two developables connected by a crease

Inextensibility is an isometry:!

!!("g(s)) = "g(s)

Theorem : Assume that for every point p ! c0 the absolute value of the curvature of c at point !(p)is greater than that of c0 at p. Then there exist exactly two extensions of ! to isometricembeddings of a plane neighborhood of c0 to space.

Fuchs & Tabachnikov, More on Paper Folding, The American Mathematical Monthly (1999).

Working with two framest

1/!g(s)

nb

u1

N1

c(s)

Working with two framest

1/!g(s)

nb

u1

N1

c(s)

!t, n, b

"! E3

!t, u(i)

"! Tc(s)(Si), N(i) ! Tc(s)(Si)!

!(s), "(s) !g(s), !N (s), "g(s)

d

ds

!

"tnb

#

$ =

!

"0 ! 0!! 0 "0 !" 0

#

$

!

"tnb

#

$ d

ds

!

"tuN

#

$ =

!

"0 !g !N

!!g 0 "g

!!N !"g 0

#

$

!

"tuN

#

$

Fold as a ... Curve in space Curve on a surface

Frame Frenet-Serret Darboux

Triad

Scalars

Equation

Invariance of under folding, :

!(s) = 2 arccos!"g(s)"(s)

"!(s)Folding angle and curvature :!(s)

Geometrical Constraints

Concave and convex:

!(1)N (s) = !!(2)

N (s)

!1(s) = !2(s) !"(s)2

!g(s) !

!1

!2

u(1)

u(2)

nt

!(s)

!(s)/!g

J. P. Duncan & J. L. Duncan, Folded Developables Proceedings of the Royal Society of London. Series A (1982).

Helmut Pottmann & Johannes Wallner, Computational Line Geometry (2010).

! (2)g (s)! ! (1)

g (s) = "!(s)

Geodesic torsion:

Mechanics of FoldingTwo developable Surfaces connected by a curve (fold) :

Generators on the surface:

S(i)(s, v) = c(s) + v(i)g(i)(s), i = 1, 2

cos !(i)(s) ! "t(s), g(i)(s)#

! cot !(i)(s) = ! " (i)g (s)

#(i)N (s)

g(1)

g(2)

t

s

vc(s)

Mechanics of Folding

Bending Energy:

Eel =B

2

!"dv(1)ds

#a(1)(H(1))2 +

"dv(2)ds

#a(2)(H(2))2

$

H(i)(s, v(i)) =!(i)

N (s) csc "(i)(s)sin "(i)(s)! v(i)

!!g(s)± "(i)!(s)

"

Two developable Surfaces connected by a curve (fold) :

Generators on the surface:

S(i)(s, v) = c(s) + v(i)g(i)(s), i = 1, 2

cos !(i)(s) ! "t(s), g(i)(s)#

! cot !(i)(s) = ! " (i)g (s)

#(i)N (s)

g(1)

g(2)

t

s

vc(s)

Mechanics of Folding

=B

2

! 2!

0f [!("), # ; s]ds

Integration along the generator

Bending Energy:

Eel =B

2

!"dv(1)ds

#a(1)(H(1))2 +

"dv(2)ds

#a(2)(H(2))2

$

H(i)(s, v(i)) =!(i)

N (s) csc "(i)(s)sin "(i)(s)! v(i)

!!g(s)± "(i)!(s)

"

Two developable Surfaces connected by a curve (fold) :

Generators on the surface:

S(i)(s, v) = c(s) + v(i)g(i)(s), i = 1, 2

cos !(i)(s) ! "t(s), g(i)(s)#

! cot !(i)(s) = ! " (i)g (s)

#(i)N (s)

g(1)

g(2)

t

s

vc(s)

Mechanics of Folding

=B

2

! 2!

0f [!("), # ; s]ds

Integration along the generator

f [!("), # ; s] ! "N (s)2

4

!csc2 $(1)

"g + $(1)! ln

"sin $(1)

sin $(1) " w(1)#"g + $(1)!

$%

+csc2 !(2)

"g ! !(2)! ln

!sin !(2)

sin !(2) ! w(2)""g ! !(2)!

#$%

Bending Energy:

Eel =B

2

!"dv(1)ds

#a(1)(H(1))2 +

"dv(2)ds

#a(2)(H(2))2

$

H(i)(s, v(i)) =!(i)

N (s) csc "(i)(s)sin "(i)(s)! v(i)

!!g(s)± "(i)!(s)

"

Two developable Surfaces connected by a curve (fold) :

Generators on the surface:

S(i)(s, v) = c(s) + v(i)g(i)(s), i = 1, 2

cos !(i)(s) ! "t(s), g(i)(s)#

! cot !(i)(s) = ! " (i)g (s)

#(i)N (s)

g(1)

g(2)

t

s

vc(s)

E. L. Starostin et al., Nature Materials (2007)

(i) Inextensible ribbons.f [!("), # ; s]!g(s) = 0

f [!, !!, ", " !; s] = !2

!1 +

"2

!2

"2 1w ("/!)! log

!1 + w ("/!)!

1! w ("/!)!

"

E. L. Starostin et al., Nature Materials (2007)

(i) Inextensible ribbons.f [!("), # ; s]!g(s) = 0

f [!, !!, ", " !; s] = !2

!1 +

"2

!2

"2 1w ("/!)! log

!1 + w ("/!)!

1! w ("/!)!

"

(ii) f [!("), # ; s]lim

w!0! w"(1)2

N (s)

!1 +

# (1)2g (s)

"(1)2N (s)

"2

+ w"(2)2N (s)

!1 +

# (2)2g (s)

"(2)2N (s)

"2

Sadowsky, M Sitzungsber. Preuss. Akad. Wiss. 22, 412–415 (1930).

f [!, " ; s] = !2

!1 +

"2

!2

"2!g(s) = 0

Phenomenological Energy

f [!("), # ; s] = f [!("), # ; s] + $

!cos

!!

2

"! cos

!!0

2

""2

# $% &Phenomenological Term

Creasing the paper Preferred Angle: !0 = 2arccos!

"g

"g + !"

"

!(s)

!(s)/!g!g + !!

!g

!0

Balance Equations

c(s)! c(s) + !c(s)

E =!

dsf [!, ", !!, " !, ...; s]

!c(s) = "!t + "1n + "2b

!E =!

dsDiEL(f)"i +

!dsQ!

Q = f!! +Qi0!i +Qi

1!"i + ...

F! + !!! F = 0

M! + !!!M + t! F = 0

Translational and rotational invariance

R. Capovilla et al., J. Phys. A: Math. Gen. 35 (2002) 6571-6587

Closed of constant c0(s) !g

c(s)

Closed of constant c0(s) !g

c(s)

!(s) ! sin ["1(s) + "2(s)] & #!(s) ! sin ["1(s)" "2(s)]

Closed of constant c0(s) !g

c(s)

!(s) ! sin ["1(s) + "2(s)] & #!(s) ! sin ["1(s)" "2(s)]

(i) Torsion inflection:!(s) = 0! "1 + "2 = #

(ii) Extreme angle:$!(s) = 0! "1 = "2 = #/2

Closed of constant c0(s) !g

c(s)

!(s) ! sin ["1(s) + "2(s)] & #!(s) ! sin ["1(s)" "2(s)]

(i) Torsion inflection:!(s) = 0! "1 + "2 = #

(ii) Extreme angle:$!(s) = 0! "1 = "2 = #/2

Closed of constant c0(s) !g

c(s)

!(s) ! sin ["1(s) + "2(s)] & #!(s) ! sin ["1(s)" "2(s)]

(i) Torsion inflection:!(s) = 0! "1 + "2 = #

(ii) Extreme angle:$!(s) = 0! "1 = "2 = #/2

Closed of constant c0(s) !g

c(s)

!(s) ! sin ["1(s) + "2(s)] & #!(s) ! sin ["1(s)" "2(s)]

(i) Torsion inflection:!(s) = 0! "1 + "2 = #

(ii) Extreme angle:$!(s) = 0! "1 = "2 = #/2

Let the curve have nowhere vanishing curvature.

Definition: Zero-torsion points of the curve are called its verteces

Theorem: Every smooth closed connected convex curve in R3 with nowhere vanishing curvature has atleast four vertices.

V. D. Sedykh, Four Vertices of a Convex Space Curve Bull. London Math. Soc. (1994) 26 (2): 177-180.

!1(0) = ", !1

!#

2

"= $

% !1(0) = %p

4th order ODE in !1(s)

3th order ODE in "1(s)

!(s) = !0 + "!1(s)

#(s) = "#1(s)

d

ds

!

"tnb

#

$ =

!

"0 ! 0!! 0 "0 !" 0

#

$

!

"tnb

#

$

Integration of the Frenet frame for a closed curve

R. Capovilla et al., J. Phys. A: Math. Gen. 35 (2002) 6571-6587Perturbation Theory

!

!

Eel/B

!

!

!p

!p(",#)Manifold gives the range of parameters compatible with

closed curves

Total energy as a function and .! !

!

!

Eel/B

!

!

!p

!p(",#)Manifold gives the range of parameters compatible with

closed curves

Total energy as a function and .! !

Minimum

Torsion

Curvature

Angles

Perturbative Solution

s

w=0.1w=0.2

s

Torsion

Curvature

Angles

Perturbative Solution

s

w=0.1w=0.2

s

Torsion

Curvature

Angles

Perturbative Solution

s

w=0.1w=0.2

s

New and more complex set of shapes can be explored.

Geometry of developable surfaces is not enough to explain the problem. Equilibrium configuration is found as a result of the competition between uncreased and creased regions.

Potential practical application and a new window to understand shape formation in nature. Exploring material properties of folded structures.

Concluding Remarks

Multiple-folds:

ETotal = limmax !wi!0

#creases!

i=1

Eel(wi)!wi

Erik Demaine, et al., Curved Crease Origami