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

Curved Fold Origami Marcelo A. Dias & Christian D. Santangelo NSF DMR-0846582 http://www.flickr.com/photos/31375127@N07/

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

折り紙 (Origami) oru “to fold” + kami, “paper” = the art of folding paper Folding along straight creases. http://pantspantsnopants.files.wordpress.com/2010/08/classic- origami.jpg

折り紙 (Origami) oru “to fold” + kami, “paper” = the art of folding paper Folding along straight creases. http://pantspantsnopants.files.wordpress.com/2010/08/classic- origami.jpg How can one explore new set of shapes?

折り紙 (Origami) oru “to fold” + kami, “paper” = the art of folding paper Folding along straight creases. http://pantspantsnopants.files.wordpress.com/2010/08/classic- origami.jpg 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?

What do we know? Exploring new shapes...

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

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

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

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

Geometry of Folding ˆ N 1 ˆ b c 0 ( s ) ˆ ϕ n ˆ t ˆ u 1 c ( s ) 1 / κ g ( s ) s 1 / κ g ( s ) Folding Two developables connected by a crease One flat developable surface

Geometry of Folding ˆ N 1 ˆ b c 0 ( s ) ˆ ϕ n ˆ t ˆ u 1 c ( s ) 1 / κ g ( s ) s 1 / κ g ( s ) Folding Two developables connected by a crease One flat developable surface Inextensibility is an isometry: ϕ ϕ ∗ ( κ g ( s )) = κ g ( s ) Theorem : 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).

ˆ N 1 ˆ b ˆ n ˆ t ˆ u 1 Working with two frames c ( s ) 1 / κ g ( s )

ˆ N 1 ˆ b ˆ n ˆ t ˆ u 1 Working with two frames c ( s ) 1 / κ g ( s ) Fold as a ... Curve in space Curve on a surface Frame Frenet-Serret Darboux Triad � � � u ( i ) � ˆ ˆ N ( i ) ∈ T c ( s ) ( S i ) ⊥ n , ˆ ∈ E 3 ˆ t , ˆ ∈ T c ( s ) ( S i ) , t , ˆ b Scalars κ ( s ) , τ ( s ) κ g ( s ) , κ N ( s ) , τ g ( s ) ˆ ˆ ˆ ˆ             t t 0 0 t 0 t κ κ g κ N Equation d d  =  = n ˆ n ˆ u ˆ u ˆ 0 0 − κ τ − κ g τ g           ds ds ˆ ˆ ˆ ˆ 0 0 0 b b N N − τ − κ N − τ g

J. P. Duncan & J. L. Duncan, Folded Geometrical Constraints Developables Proceedings of the Royal Society of London. Series A (1982). Helmut Pottmann & Johannes Wallner, Computational Line Geometry (2010). u (2) ˆ Invariance of under folding, : κ g ( s ) ϕ β 2 β 1 ( s ) = β 2 ( s ) ≡ θ ( s ) ˆ ˆ β 1 t n 2 u (1) ˆ Concave and convex: κ (1) N ( s ) = − κ (2) N ( s ) Geodesic torsion: τ (2) g ( s ) − τ (1) g ( s ) = θ ′ ( s ) θ ( s ) Folding angle and curvature : θ ( s ) κ ( s ) � κ g ( s ) � θ ( s ) = 2 arccos κ ( s ) κ ( s ) / κ g

Mechanics of Folding s g (2) ˆ Two developable Surfaces connected by a curve (fold) : c ( s ) v S ( i ) ( s, v ) = c ( s ) + v ( i ) ˆ g ( i ) ( s ) , i = 1 , 2 g (1) ˆ ˆ t cos γ ( i ) ( s ) ≡ � ˆ g ( i ) ( s ) � Generators on the surface: t ( s ) , ˆ ∴ cot γ ( i ) ( s ) = ∓ τ ( i ) g ( s ) κ ( i ) N ( s )

Mechanics of Folding s g (2) ˆ Two developable Surfaces connected by a curve (fold) : c ( s ) v S ( i ) ( s, v ) = c ( s ) + v ( i ) ˆ g ( i ) ( s ) , i = 1 , 2 g (1) ˆ ˆ t cos γ ( i ) ( s ) ≡ � ˆ g ( i ) ( s ) � Generators on the surface: t ( s ) , ˆ ∴ cot γ ( i ) ( s ) = ∓ τ ( i ) g ( s ) κ ( i ) N ( s ) Bending Energy: �� � � E el = B a (1) ( H (1) ) 2 + � � dv (1) ds dv (2) ds a (2) ( H (2) ) 2 2 κ ( i ) N ( s ) csc γ ( i ) ( s ) H ( i ) ( s, v ( i ) ) = sin γ ( i ) ( s ) ∓ v ( i ) � � κ g ( s ) ± γ ( i ) ′ ( s )

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

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

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

κ g ( s ) = 0 f [ θ ( κ ) , τ ; s ] (i) Inextensible ribbons. � 2 � 1 + w ( τ / κ ) ′ 1 + τ 2 � � 1 f [ κ , κ ′ , τ , τ ′ ; s ] = κ 2 w ( τ / κ ) ′ log 1 − w ( τ / κ ) ′ κ 2 E. L. Starostin et al., Nature Materials (2007) � 2 � 2 � 1 + τ (1)2 � 1 + τ (2)2 lim ( s ) ( s ) (ii) w → 0 ≈ w κ (1)2 g + w κ (2)2 g f [ θ ( κ ) , τ ; s ] ( s ) ( s ) N N κ (1)2 κ (2)2 ( s ) ( s ) N N � 2 1 + τ 2 κ g ( s ) = 0 � Sadowsky, M f [ κ , τ ; s ] = κ 2 Sitzungsber. Preuss. Akad. Wiss. κ 2 22 , 412–415 (1930).

Phenomenological Energy � � κ g Creasing the paper Preferred Angle: θ 0 = 2 arccos κ g + ∆ κ θ ( s ) θ 0 κ ( s ) / κ g κ g + ∆ κ κ g � � θ � � θ 0 �� 2 ˜ f [ θ ( κ ) , τ ; s ] = f [ θ ( κ ) , τ ; s ] + ǫ cos − cos 2 2 � �� � Phenomenological Term

R. Capovilla et al., J. Phys. A: Math. Gen. 35 (2002) 6571-6587 Balance Equations n + ǫ 2 ˆ δ c ( s ) = ǫ � ˆ c ( s ) → c ( s ) + δ c ( s ) t + ǫ 1 ˆ b � dsf [ κ , τ , κ ′ , τ ′ , ... ; s ] E = � � ds D i ds Q ′ δ E = EL ( f ) ǫ i + 1 ǫ ′ Q = f ǫ � + Q i 0 ǫ i + Q i i + ... F ′ + � Translational and Ω × F = 0 rotational invariance M ′ + � Ω × M + ˆ t × F = 0

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

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

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

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

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