Curved Fold Origami. Marcelo A. Dias & Christian D. Santangelo NSF DMR

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1 Curved Fold Origami Marcelo A. Dias & Christian D. Santangelo NSF DMR

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3 (Origami) oru to fold + kami, paper = the art of folding paper Folding along straight creases.

4 (Origami) oru to fold + kami, paper = the art of folding paper Folding along straight creases. How can one explore new set of shapes?

5 (Origami) 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. Wingler Student's work at the Bauhaus

6 What do we know?

7 What do we know? Exploring new shapes...

8 ...Design and Computational Origami Erik Demaine, et al., Curved Crease Origami Kilian at al., Curved Folding

9 ...Design and Computational Origami Erik Demaine, et al., Curved Crease Origami Kilian at al., Curved Folding

10 ...Design and Computational Origami Erik Demaine, et al., Curved Crease Origami Kilian at al., Curved Folding

11 Geometry of Folding c 0 (s) s 1/κ g (s) One flat developable surface

12 Geometry of Folding c 0 (s) s 1/κ g (s) ϕ c(s) ˆt ˆb û 1 ˆN 1 ˆn 1/κ g (s) Folding One flat developable surface Two developables connected by a crease

13 Geometry of Folding c 0 (s) s 1/κ g (s) ϕ c(s) ˆt ˆb û 1 ˆN 1 ˆn 1/κ g (s) Folding One flat developable surface Two developables connected by a crease Inextensibility ϕ is an isometry: ϕ (κ g (s)) = κ g (s) T heorem : Assume that for every point p c 0 the absolute value of the curvature of c at point ϕ(p) is greater than that of c 0 at p. Then there exist exactly two extensions of ϕ to isometric embeddings of a plane neighborhood of c 0 to space. Fuchs & Tabachnikov, More on Paper Folding, The American Mathematical Monthly (1999).

14 Working with two frames c(s) ˆt ˆb û 1 ˆN 1 ˆn 1/κ g (s)

15 Working with two frames c(s) ˆt ˆb û 1 ˆN 1 ˆn 1/κ g (s) Fold as a... Curve in space Curve on a surface Frame Frenet-Serret Darboux Triad {ˆt, ˆn, ˆb} E 3 {ˆt, û (i)} T c(s) (S i ), ˆN(i) T c(s) (S i ) Scalars κ(s), τ(s) κ g (s), κ N (s), τ g (s) Equation d ds ˆt ˆn = ˆb 0 κ 0 ˆt κ 0 τ ˆn 0 τ 0 ˆb d ds ˆt û = ˆN 0 κ g κ N ˆt κ g 0 τ g û κ N τ g 0 ˆN

16 Geometrical Constraints J. P. Duncan & J. L. Duncan, Folded Developables Proceedings of the Royal Society of London. Series A (1982). ˆt û (2) β 2 β 1 ˆn û (1) κ g (s) Invariance of under folding, : β 1 (s) =β 2 (s) θ(s) 2 Concave and convex: Helmut Pottmann & Johannes Wallner, Computational Line Geometry (2010). ϕ κ (1) N (s) = κ(2) N (s) Geodesic torsion: τ (2) g (s) τ g (1) (s) =θ (s) θ(s) Folding angle θ(s) and curvature κ(s) : ] θ(s) = 2 arccos [ κg (s) κ(s) κ(s)/κ g

17 Mechanics of Folding Two developable Surfaces connected by a curve (fold) c(s) : ĝ (2) s v S (i) (s, v) =c(s)+v (i) ĝ (i) (s), i =1, 2 Generators on the surface: cos γ (i) (s) ˆt(s), ĝ (i) (s) ˆt ĝ (1) cot γ (i) (s) = τ g (i) (s) κ (i) N (s)

18 Mechanics of Folding Two developable Surfaces connected by a curve (fold) c(s) : ĝ (2) s v S (i) (s, v) =c(s)+v (i) ĝ (i) (s), i =1, 2 Generators on the surface: cos γ (i) (s) ˆt(s), ĝ (i) (s) ˆt ĝ (1) cot γ (i) (s) = τ g (i) (s) κ (i) N (s) Bending Energy: E el = 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) ( κ g (s) ± γ (i) (s) )

19 Mechanics of Folding Two developable Surfaces connected by a curve (fold) c(s) : ĝ (2) s v S (i) (s, v) =c(s)+v (i) ĝ (i) (s), i =1, 2 Generators on the surface: cos γ (i) (s) ˆt(s), ĝ (i) (s) ˆt ĝ (1) Bending Energy: E el = B 2 ( cot γ (i) (s) = τ g (i) (s) κ (i) N (s) dv (1) ds a (1) (H (1) ) 2 + ) dv (2) ds a (2) (H (2) ) 2 Integration along the generator = B 2 2π 0 f[θ(κ), τ; s]ds H (i) (s, v (i) )= κ (i) N (s) csc γ(i) (s) sin γ (i) (s) v (i) ( κ g (s) ± γ (i) (s) )

20 Mechanics of Folding Two developable Surfaces connected by a curve (fold) c(s) : ĝ (2) s v S (i) (s, v) =c(s)+v (i) ĝ (i) (s), i =1, 2 Generators on the surface: cos γ (i) (s) ˆt(s), ĝ (i) (s) ˆt ĝ (1) Bending Energy: E el = B 2 ( cot γ (i) (s) = τ g (i) (s) κ (i) N (s) dv (1) ds a (1) (H (1) ) 2 + ) dv (2) ds a (2) (H (2) ) 2 Integration along the generator = B 2 2π 0 f[θ(κ), τ; s]ds H (i) (s, v (i) )= κ (i) N (s) csc γ(i) (s) sin γ (i) (s) v (i) ( κ g (s) ± γ (i) (s) ) f[θ(κ), τ; s] κ N(s) 2 4 ( csc 2 γ (1) ln κ g + γ (1) [ sin γ (1) sin γ (1) w (1) ( κ g + γ (1) ) ] [ ]) + csc2 γ (2) sin γ (2) ln κ g γ (2) sin γ (2) w ( (2) κ g γ (2) )

21 (i) f[θ(κ), τ; s] κ g (s) =0 Inextensible ribbons. f[κ, κ, τ, τ ; s] =κ 2 ( 1+ τ 2 κ 2 ) 2 ( ) 1 1+w (τ/κ) w (τ/κ) log 1 w (τ/κ) E. L. Starostin et al., Nature Materials (2007)

22 (i) f[θ(κ), τ; s] κ g (s) =0 Inextensible ribbons. f[κ, κ, τ, τ ; s] =κ 2 ( 1+ τ 2 κ 2 ) 2 ( ) 1 1+w (τ/κ) w (τ/κ) log 1 w (τ/κ) E. L. Starostin et al., Nature Materials (2007) (ii) f[θ(κ), τ; s] lim w 0 wκ (1)2 N (s) ( 1+ τ g (1)2 (s) κ (1)2 N (s) ) 2 + wκ (2)2 N (s) ( 1+ τ g (2)2 (s) κ (2)2 N (s) ) 2 κ g (s) =0 f[κ, τ; s] =κ 2 ( 1+ τ 2 κ 2 ) 2 Sadowsky, M Sitzungsber. Preuss. Akad. Wiss. 22, (1930).

23 Phenomenological Energy Creasing the paper [ ] κ Preferred Angle: g θ 0 = 2 arccos κ g + κ θ(s) θ 0 κ g + κ κ g κ(s)/κ g ( ( ) ( )) 2 θ θ0 f[θ(κ), τ; s] =f[θ(κ), τ; s]+ɛ cos cos 2 2 }{{} Phenomenological Term

24 R. Capovilla et al., J. Phys. A: Math. Gen. 35 (2002) Balance Equations c(s) c(s)+δc(s) δc(s) =ɛ ˆt + ɛ 1ˆn + ɛ 2ˆb E = dsf[κ, τ, κ, τ,...; s] δe = dsd i EL(f)ɛ i + dsq Q = fɛ + Q i 0ɛ i + Q i 1ɛ i +... Translational and rotational invariance F + Ω F =0 M + Ω M + ˆt F =0

25 Closed c 0 (s) of constant κ g c(s)

26 Closed c 0 (s) of constant κ g τ(s) sin [γ 1 (s)+γ 2 (s)] & θ (s) sin [γ 1 (s) γ 2 (s)] c(s)

27 Closed c 0 (s) of constant κ g τ(s) sin [γ 1 (s)+γ 2 (s)] & θ (s) sin [γ 1 (s) γ 2 (s)] (i) (ii) Torsion inflection: τ(s) =0 γ 1 + γ 2 = π Extreme angle: θ (s) =0 γ 1 = γ 2 = π/2 c(s)

28 Closed c 0 (s) of constant κ g τ(s) sin [γ 1 (s)+γ 2 (s)] & θ (s) sin [γ 1 (s) γ 2 (s)] (i) (ii) Torsion inflection: τ(s) =0 γ 1 + γ 2 = π Extreme angle: θ (s) =0 γ 1 = γ 2 = π/2 c(s)

29 Closed c 0 (s) of constant κ g τ(s) sin [γ 1 (s)+γ 2 (s)] & θ (s) sin [γ 1 (s) γ 2 (s)] (i) (ii) Torsion inflection: τ(s) =0 γ 1 + γ 2 = π Extreme angle: θ (s) =0 γ 1 = γ 2 = π/2 c(s)

30 Closed c 0 (s) of constant κ g τ(s) sin [γ 1 (s)+γ 2 (s)] & θ (s) sin [γ 1 (s) γ 2 (s)] (i) (ii) Torsion inflection: τ(s) =0 γ 1 + γ 2 = π Extreme angle: θ (s) =0 γ 1 = γ 2 = π/2 c(s) V. D. Sedykh, Four Vertices of a Convex Space Curve Bull. London Math. Soc. (1994) 26 (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 R 3 with nowhere vanishing curvature has at least four vertices.

31 Perturbation Theory R. Capovilla et al., J. Phys. A: Math. Gen. 35 (2002) κ(s) =κ 0 + εκ 1 (s) 4 th order ODE in κ 1 (s) τ(s) =ετ 1 (s) κ 1 (0) = α, κ 1 ( π 2 ) = β 3 th order ODE in τ 1 (s) τ 1(0) = τ p Integration of the Frenet frame for a closed curve d ds ˆt ˆn = ˆb 0 κ 0 κ 0 τ 0 τ 0 ˆt ˆn ˆb

32 β α Manifold τ p (α, β) gives the range of parameters compatible with closed curves τ p β Total energy as a β function α and. E el /B α

33 β α Manifold τ p (α, β) gives the range of parameters compatible with closed curves τ p β Total energy as a β function α and. E el /B α Minimum

34 Perturbative Solution Torsion Curvature Angle s s s w=0.2 w=0.1

35 Perturbative Solution Torsion Curvature Angle s s s w=0.2 w=0.1

36 Perturbative Solution Torsion Curvature Angle s s s w=0.2 w=0.1

37 Concluding Remarks 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. Multiple-folds: E T otal = #creases lim E el (w i ) w i max w i 0 i=1 Erik Demaine, et al., Curved Crease Origami Potential practical application and a new window to understand shape formation in nature. Exploring material properties of folded structures.