Symbolic Computations in Conformal Geometric Algebra for Three Dimensional Origami Folds

Symbolic Computations in Conformal Geometric Algebra for Three Dimensional Origami Folds

Yoshihiro Mizoguchi and Hiroyuki Ochiai (Kyushu University, JAPAN)

July 20, 2016 COA2016 @ Vienna University of Technology

hJps://github.com/KyushuUniversityMathemaQcs/MathemaQcaCGA

InsQtute of MathemaQcs for Industry, Kyushu University, JAPAN.

Math-‐for-‐Industry (MI) is a new research area that serves to create advanced technologies in response to industrial needs by innovaQng flexible & versaQle methods in mathemaQcs, staQsQcs & compuQng.

Mission: To establish a leading-‐edge research hub for mathemaQcs as applied to the real world & sciences, while nurturing research leaders & innovators to serve societal needs.

InsQtute of MathemaQcs for Industry, Kyushu University, JAPAN.

MoQvaQon

O0=FirstO; O1=Ori[O0, Ori2[P3, P1], {1}, π]; O2=Ori[O1, Ori2[P4, P3], {3}, π]; O3=Ori[O2, Ori2[P1, P4], {5}, −π]; O4=Ori[O3, Ori2[P5, P1], {4, 10}, π]; O5 = Ori[O4, Ori2[P6, P5], {13}, 0]; O6 = Ori[O5, Ori2[P7, P5], {29}, 0]; O7=Ori[O6 , Ori3[p2line[P8, P5], p2line[P9, P8], 2], {27}, π]; O8=Ori[O7, Ori3[p2line[P8, P5], p2line[P10, P8], 2], {59}, π]; O9=Ori[O8, Ori2[P11, P1], {15}, π]; O10=Ori[O9, Ori1[P7, P6], {31}, π]; O11=Ori[O10, Ori2[P7, P6], {11}, −π];

Formal Origami Design Sheet

Origami Instruc5on Sheet To provide a formal specificaQon for origami to invesQgate its geometrical properQes using formal symbolic computaQons.

Each funcQon ‘Ori#’ corresponding to a fold operaQon, is represented by an element of CGA.

Table of Contents

•  Algebra for Geometry •  CGA (Conformal Geometric Algebra)

–  DefiniQons –  Examples –  VisualizaQon –  Equality Check (Algebra)

•  Origami Folds –  FormalizaQon –  VisualizaQon –  3D Folds

•  Concolusion

Algebra for Geometry

•  Complex Numbers (2-‐dim) •  Quaternion (3-‐dim) •  Conformal Geometric Algebra (n-‐dim)

Algebra of geometric operators (1)

7

Algebra of geometric operators (2)

8

CGA (Conformal Geometric Algebra)

9

R[E] (Product)

A MathemaQca module for Conformal Geometric Algebra and Origami Folding

10

€

eφ ∗ eS = eS ∗ eφ = eS

e{0} ∗ eS =e{0}∪S 0∉ S( )0 0∈ S( )

' ( )

e{a} ∗ eS =(−1) s∈S |s<a{ } e{a}∪S a ∈{1,2,3} ∧ a ∉ S( )(−1) s∈S |s<a{ } eS −{a} a ∈{1,2,3} ∧ a ∈ S( )

' ( ,

) ,

e{∞} ∗ eS =

(−1) S e{∞}∪S 0∉ S ∧∞∉ S( )0 0∉ S ∧∞∈ S( )

−e{0} ∗ e{∞} ∗ eS −{0}( ) − 2eS −{0} 0∈ S( )

'

( ,

) ,

eT∪{a} ∗ eS = eT ∗(e{a} ∗ eS ) (∀t ∈T , t < a)

R[W] (Outer Product and Inner Product)

11

Example (1)

12

Example (2)

13

VisualizaQon (CGA)

Objects in R3

15

Operators in R3

16

-100 -50 0 50 100-100

-50

0

50

100

-100 -50 0 50 100-100

-50

0

50

100

-100 -50 0 50 100-100

-50

0

50

100

translator rotor dilator

17 €

X ∧P x, y, z( )= αS wSS ⊂ 0,1, 2, 3,∞{ }∑ = 0⇔∀S ⊂ 0,1,2,3,∞{ },αS = 0.

€

Fig(X ) := x, y,z( ) ∈ R3 X ∧P x, y, z( ) = 0$ % &

' ( ) .

We compute a Gröbner basis of those equaQons to draw the figure.

αS ∈ R[x, y, z]

The figure Fig(X) of an object X in R3

€

X ∈ R[E] X ∈ R[W ]or

18

VisualizaQon (1)

19

VisualizaQon (2)

Algebra Symbolic ComputaQons in CGA

Fig(X1) = Fig(X2) ?

21

€

X1 = w01 + w02 + w03 + w0∞ − w12 − 2 w13 − 6 w1∞ − w23 −5 w2∞

−4 w3∞ + w012 + w013 + w01∞ + w023 + w02∞ + w03∞ + 2 w123+6 w12∞ + 5 w13∞ + 6 w23∞ + w0123 + w012∞ + w013∞ + w023∞

−5 w123∞ + w0123∞,X2 = w0 + w1 + 2 w2 + 3 w3 + 7 w∞

The appearance of X1 and X2 are different, but the figure of X1 and X2 are same and it is a point {(1,2,3)}. Our implemented funcQon CGAEquaQonCheck can check the equaliQes of figures.

We can check the equality of figures Fig(X1) and Fig(X2) using symbolic computaQons.

FormalizaQon of Origami

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

P11

P12 P13


Origami Graph O = (Π, 〜, ≻)

Superposi5on rela5on ≻

A pair of face’s ID for two faces superposed directly. ≻ ={ (f10,f14) ,(f14,f48) ,(f16,f17) ,(f18,f26) ,(f22,f10) ,(f23,f31) ,(f26,f48) ,(f30,f22) ,(f31,f30) ,(f38,f18) ,(f39,f55) ,(f48,f16) ,(f49,f51) ,(f50,f23) ,(f50,f39) ,(f51,f50) ,(f54,f38) ,(f55,f54) }

f10f14

f18f26

f22f30

f23f31

f38f54

f39f55

f48

f50

f49f51

f16f17

Face set Π

Π = { f10 , f14 , f18 , f26 , f22 , f30 , f23 , f31 , f38 , f54 , f39 , f55 , f48 , f50 , f49 , f51 , f16 , f17 } 　 f49 ={ p2 , p13 , p12 , p4 } 　 f50 ={ p6 , p11 , p7 } 　f23 ={ p5 , p10 , p1 } 　f17 ={ p15 , p2 , p7 } etc.

The vertex coordinates sets of each polygon.

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

P11

P12 P13

Adjacency rela5on 〜

The adjacent face’s ID numbers sets to every side.

〜 = {〜10 , 〜14 , 〜18 , 〜26 , 〜22 , 〜30 , 〜23 , 〜31 , 〜38 , 〜54 , 〜39 , 〜55 , 〜48 , 〜50 , 〜49 , 〜51 , 〜16 , 〜17 } 〜10 ={ f14 , f22 , f16 } 〜14 ={ f48 , f30 , f10 } 〜22 ={ f10 , f13 , Φ } 〜16 ={ f10 , f17 , f18 } etc. Where, Φ is empty set.

f10f14

f18f26

f22f30

f23f31

f38f54

f39f55

f48

f50

f49f51

f16f17

23

We follow the E-Origami-System developed by Ida et al.

Origami data structure

Huzita-Hattori Axioms (1)

24

Huzita-Hattori Axioms (2)

25

A sequence of origami func5ons for making Kabuto.

26

P1

P2 P3

P4P1

P2 P3

P4

P1

P2 P3

P4

P5P6

P1

P2 P3

P4

P5P6

P7

P1

P5P6

P7

P8

P1

P5P6

P7

P8

P9

P1

P5P6

P7

P8

P9

P10

P1

P5P6

P7

P8

P9

P10

P1

P5P6

P7

P8

P9

P10

P1

P5P6

P7

P8

P9

P10

P11

P1

P5P6

P7

P8

P9

P10

P11

P1

P5P6

P7

P8

P9

P10

P11

A design sheet of origami is wriJen by successive applicaQons of origami funcQons.


Design Sheet

Formula

Drawing

A sequence of origami func5ons for making Kabuto.

27

P1

P2 P3

P4P1

P2 P3

P4

P1

P2 P3

P4

P5P6

P1

P2 P3

P4

P5P6

P7

P1

P5P6

P7

P8

P1

P5P6

P7

P8

P9

P1

P5P6

P7

P8

P9

P10

P1

P5P6

P7

P8

P9

P10

P1

P5P6

P7

P8

P9

P10

P1

P5P6

P7

P8

P9

P10

P11

P1

P5P6

P7

P8

P9

P10

P11

P1

P5P6

P7

P8

P9

P10

P11

A design sheet of origami is wriJen by successive applicaQons of origami funcQons.


Design Sheet

Formula

Drawing

VisualizaQon (Origami)

Visualization functions

29

3D Folds

Three-dimensional origami folds

31

3D folding using CGA

32

Simple proof using CGA

33

€

R1 =12

+w132−

a4 2

w3∞

€

R2 =12

+w132−

a2 2

w3∞

€

R3 =12

+w132−3a4 2

w3∞

R = R3 ∗R2 ∗R1

A Mathematica module for Conformal Geometric Algebra and Origami Folding

34

Conclusion

•  We implemented funcQons that realize CGA operaQons using symbolic computaQons in MathemaQca. hJps://github.com/KyushuUniversityMathemaQcs/MathemaQcaCGA

•  Our module includes funcQons for the geometric product, the inner and outer product of a CGA which s an extension of R3 with e0 and e∞.

•  We draw the figure of an element in R3 by solving an equaQon system using Gröbner basis.

•  We implemented funcQons of 2D origami folding following the 2D origami system introduced by Ida et al.

•  We implemented funcQons which compute a CGA element corresponding to a 3D origami folding.

•  We presented a simple proof of the 3D origami property using a symbolic computaQon of CGA equaQons.

Future Work

•  To implement 3D origami funcQons completely. We need to extend the data structure for 3D origami.

•  To create a database of design sheets of origami folds using a formula in CGA.

•  To find an aJracQve folding moQon defined by a formula in CGA.

•  To implement a funcQon for judging the collision of faces in 3D origami folds.

•  To invesQgate mathemaQcal properQes in CGA, especially checking equaliQes of CGA objects.

A Mathematica module for Conformal Geometric Algebra and Origami Folding

36

Reference 1)  T. Ida, Huzita's basic origami fold in geometric algebra, 16th

InternaQonal Symposium on Symbolic and Numeric Algorithms for ScienQfic CompuQng (SYNASC 2014), 11-‐13. IEEE computer society, 2014.

2)  T.Ida, H.Takahashi, M.Marin, A.Kasem and F.Ghourabi, ComputaQonal origami system EOS. Proc. 4th InternaQonal Conference of Origami Science, MathemaQcs and EducaQon (4OSME), pages 285-‐-‐293, 2009.

3)  M.Kondo, T.Matsuo, Y.Mizoguchi, and H.Ochiai. A mathemaQca module for conformal geometric algebra and origami folding. 7th InternaQonal Symposium on Symbolic ComputaQon in Souware Science (SCSS2016), volume 39 of EPiC Series in CompuQng, 68-‐80. EasyChair, 2016.

4)  C.Perwass. Geometric Algebra with ApplicaQons in Engineering, volume 4 of Geometry and CompuQng. Springer, 2009.

Education

Symbolic Computations in Conformal Geometric Algebra for Three Dimensional Origami Folds