Curved Fold Origami Marcelo A. Dias & Christian D. Santangelo - - PowerPoint PPT Presentation

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-

• rigami.jpg
• ru “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-

• rigami.jpg
• ru “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-

• rigami.jpg
• ru “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. Wingler http://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/κg(s)

c0(s)

One flat developable surface

Geometry of Folding

s

1/κg(s)

c0(s)

One flat developable surface

ϕ

Folding

ˆ t

1/κg(s)

ˆ n

ˆ b

ˆ u1

ˆ N1

c(s)

Two developables connected by a crease

Geometry of Folding

s

1/κg(s)

c0(s)

One flat developable surface

ϕ

Folding

ˆ t

1/κg(s)

ˆ n

ˆ b

ˆ u1

ˆ N1

c(s)

Two developables connected by a crease Inextensibility is an isometry:

ϕ

ϕ∗(κg(s)) = κg(s)

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

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

Working with two frames

ˆ t

1/κg(s)

ˆ n

ˆ b

ˆ u1

ˆ N1

c(s)

Working with two frames

ˆ t

1/κg(s)

ˆ n

ˆ b

ˆ u1

ˆ N1

c(s)

• ˆ

t, ˆ n, ˆ b

• ∈ E3
• ˆ

t, ˆ u(i) ∈ Tc(s)(Si), ˆ N(i) ∈ Tc(s)(Si)⊥

κ(s), τ(s) κg(s), κN(s), τg(s)

d ds   ˆ t ˆ n ˆ b   =   κ −κ τ −τ     ˆ t ˆ n ˆ b   d ds   ˆ t ˆ u ˆ N   =   κg κN −κg τg −κN −τg     ˆ t ˆ u ˆ N  

Fold as a ... Curve in space Curve on a surface Frame Frenet-Serret Darboux Triad Scalars Equation

Invariance of under folding, :

θ(s) = 2 arccos κg(s) κ(s)

• κ(s)

Folding angle and curvature :

θ(s)

Geometrical Constraints

Concave and convex:

κ(1)

N (s) = −κ(2) N (s)

β1(s) = β2(s) ≡ θ(s) 2 κg(s)

ϕ

β1

β2

ˆ u(1)

ˆ u(2) ˆ n ˆ t

θ(s) κ(s)/κg

• 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).

τ (2)

g (s) − τ (1) g (s) = θ′(s)

Geodesic torsion:

Mechanics of Folding

Two developable Surfaces connected by a curve (fold) : Generators on the surface:

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

cos γ(i)(s) ≡ ˆ t(s), ˆ g(i)(s)

∴ cot γ(i)(s) = ∓τ (i)

g (s)

κ(i)

N (s)

ˆ g(1) ˆ g(2)

ˆ t

s v c(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)) =

κ(i)

N (s) csc γ(i)(s)

sin γ(i)(s) ∓ v(i) κg(s) ± γ(i)′(s)

• Two developable Surfaces connected by a curve (fold) :

Generators on the surface:

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

cos γ(i)(s) ≡ ˆ t(s), ˆ g(i)(s)

∴ cot γ(i)(s) = ∓τ (i)

g (s)

κ(i)

N (s)

ˆ g(1) ˆ g(2)

ˆ t

s v c(s)

Mechanics of Folding

= B 2 2π f[θ(κ), τ; 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)) =

κ(i)

N (s) csc γ(i)(s)

sin γ(i)(s) ∓ v(i) κg(s) ± γ(i)′(s)

• Two developable Surfaces connected by a curve (fold) :

Generators on the surface:

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

cos γ(i)(s) ≡ ˆ t(s), ˆ g(i)(s)

∴ cot γ(i)(s) = ∓τ (i)

g (s)

κ(i)

N (s)

ˆ g(1) ˆ g(2)

ˆ t

s v c(s)

Mechanics of Folding

= B 2 2π f[θ(κ), τ; s]ds

Integration along the generator

f[θ(κ), τ; s] ≡ κN(s)2 4

• csc2 γ(1)

κg + γ(1)′ ln

• sin γ(1)

sin γ(1) − w(1) κg + γ(1)′

• + csc2 γ(2)

κg − γ(2)′ ln

• sin γ(2)

sin γ(2) − w(2) κg − γ(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)) =

κ(i)

N (s) csc γ(i)(s)

sin γ(i)(s) ∓ v(i) κg(s) ± γ(i)′(s)

• Two developable Surfaces connected by a curve (fold) :

Generators on the surface:

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

cos γ(i)(s) ≡ ˆ t(s), ˆ g(i)(s)

∴ cot γ(i)(s) = ∓τ (i)

g (s)

κ(i)

N (s)

ˆ g(1) ˆ g(2)

ˆ t

s v c(s)

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

(i) Inextensible ribbons.

f[θ(κ), τ; s]

κg(s) = 0

f[κ, κ′, τ, τ ′; s] = κ2

• 1 + τ 2

κ2 2 1 w (τ/κ)′ log 1 + w (τ/κ)′ 1 − w (τ/κ)′

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

(i) Inextensible ribbons.

f[θ(κ), τ; s]

κg(s) = 0

f[κ, κ′, τ, τ ′; s] = κ2

• 1 + τ 2

κ2 2 1 w (τ/κ)′ log 1 + w (τ/κ)′ 1 − w (τ/κ)′

• (ii)

f[θ(κ), τ; s]

lim

w→0

≈ wκ(1)2

N

(s)

• 1 + τ (1)2

g

(s) κ(1)2

N

(s) 2 + wκ(2)2

N

(s)

• 1 + τ (2)2

g

(s) κ(2)2

N

(s) 2

Sadowsky, M

• Sitzungsber. Preuss. Akad. Wiss.

22, 412–415 (1930).

f[κ, τ; s] = κ2

• 1 + τ 2

κ2 2

κg(s) = 0

Phenomenological Energy

˜ f[θ(κ), τ; s] = f[θ(κ), τ; s] + ǫ

• cos

θ 2

• − cos

θ0 2 2

• Phenomenological Term

Creasing the paper Preferred Angle: θ0 = 2 arccos

• κg

κg + ∆κ

• θ(s)

κ(s)/κg

κg + ∆κ κg

θ0

Balance Equations

c(s) → c(s) + δc(s) E =

• dsf[κ, τ, κ′, τ ′, ...; s]

δc(s) = ǫˆ t + ǫ1ˆ n + ǫ2ˆ b δE =

• dsDi

EL(f)ǫi +

• dsQ′

Q = fǫ + Qi

0ǫi + Qi 1ǫ′ i + ...

F′ + Ω × F = 0 M′ + Ω × M + ˆ 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)

κg

c(s)

Closed of constant

c0(s)

κg

c(s)

τ(s) ∼ sin [γ1(s) + γ2(s)] & θ′(s) ∼ sin [γ1(s) − γ2(s)]

Closed of constant

c0(s)

κg

c(s)

τ(s) ∼ sin [γ1(s) + γ2(s)] & θ′(s) ∼ sin [γ1(s) − γ2(s)] (i) Torsion inflection: τ(s) = 0 ⇒ γ1 + γ2 = π (ii) Extreme angle: θ′(s) = 0 ⇒ γ1 = γ2 = π/2

Closed of constant

c0(s)

κg

c(s)

τ(s) ∼ sin [γ1(s) + γ2(s)] & θ′(s) ∼ sin [γ1(s) − γ2(s)] (i) Torsion inflection: τ(s) = 0 ⇒ γ1 + γ2 = π (ii) Extreme angle: θ′(s) = 0 ⇒ γ1 = γ2 = π/2

Closed of constant

c0(s)

κg

c(s)

τ(s) ∼ sin [γ1(s) + γ2(s)] & θ′(s) ∼ sin [γ1(s) − γ2(s)] (i) Torsion inflection: τ(s) = 0 ⇒ γ1 + γ2 = π (ii) Extreme angle: θ′(s) = 0 ⇒ γ1 = γ2 = π/2

Closed of constant

c0(s)

κg

c(s)

τ(s) ∼ sin [γ1(s) + γ2(s)] & θ′(s) ∼ sin [γ1(s) − γ2(s)] (i) Torsion inflection: τ(s) = 0 ⇒ γ1 + γ2 = π (ii) Extreme angle: θ′(s) = 0 ⇒ γ1 = γ2 = π/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 at least four vertices.

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

κ1(0) = α, κ1 π 2

• = β

τ ′

1(0) = τp

4th

• rder ODE in

κ1(s) 3th

• rder ODE in

τ1(s)

κ(s) = κ0 + εκ1(s) τ(s) = ετ1(s)

d ds   ˆ t ˆ n ˆ b   =   κ −κ τ −τ     ˆ t ˆ n ˆ b  

Integration of the Frenet frame for a closed curve

• R. Capovilla et al., J. Phys. A: Math. Gen. 35

(2002) 6571-6587

Perturbation Theory

α β Eel/B

α

β τp

τp(α, β)

Manifold gives the range

• f parameters compatible with

closed curves Total energy as a function and .

α

β

α β Eel/B

α

β τp

τp(α, β)

Manifold gives the range

• f parameters compatible with

closed curves Total energy as a function and .

α

β

Minimum

Torsion Curvature Angle

s

Perturbative Solution

s

w=0.1 w=0.2

s

Torsion Curvature Angle

s

Perturbative Solution

s

w=0.1 w=0.2

s

Torsion Curvature Angle

s

Perturbative Solution

s

w=0.1 w=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: ET otal = lim

max ∆wi→0 #creases

• i=1

Eel(wi)∆wi

Erik Demaine, et al., Curved Crease Origami