On the flexibility of Kokotsakis meshes Hellmuth Stachel, Vienna University of Technology (joint work with Georg Nawratil) stachel@dmg.tuwien.ac.at — http://www.geometrie.tuwien.ac.at/stachel Workshop on Rigidity, October 11–14, 2011 Fields Institute, Toronto
Outline 1. Definition of Kokotsakis meshes 2. Three examples of flexible quad meshes 3. Transmission by one spherical four-bar 4. Composition of spherical four-bars 5. Flexibility vs. reducibility of meshes Workshop on Rigidity, October 11–14, 2011, Fields Institute, Toronto 1
1. Definition of Kokotsakis meshes A Kokotsakis mesh is a polyhedral structure consisting of an n -sided f 3 central polygon f 0 surrounded by a a 3 belt of polygons. V 3 V 2 Each side a i , i = 1 , . . . , n , of f 0 is f 4 a 4 shared by a polygon f i . Each vertex f 0 a 2 f 2 V i of f 0 is the meeting point of four V 4 faces. V 1 a 1 f 1 Each face is seen as a rigid body; only the dihedral angles can vary. Under which conditions a Kokotsakis mesh is continuously flexible ? Special case n = 4 Workshop on Rigidity, October 11–14, 2011, Fields Institute, Toronto 2
1. Definition of Kokotsakis meshes He was born on the island Crete in Greece. As a precocious child, he was accepted at the Department of Civil Engineering of Technical University of Athens already in the age of 16 . After graduation he was appointed a lecturer in the Department of Descriptive and Projective Geometry. He finished his PhD-thesis entitled “About flexible polyhedra” under the supervision of K. Caratheodori in Munich/Germany. His list of publications contains not more than 5 titles. Antonios Kokotsakis 1899–1964 Workshop on Rigidity, October 11–14, 2011, Fields Institute, Toronto 3
1. Definition of Kokotsakis meshes A Kokotsakis mesh for n = 4 is also W 2 V 3 called Neunflach [German] (nine-flat) W 1 (Kokotsakis 1931, Sauer 1932) a 3 f 3 f 2 a 2 f 0 For n = 3 the Kokotsakis mesh is V 1 a 1 V 2 equivalent to an octahedron with V 1 V 2 V 3 and W 1 W 2 W 3 as opposite f 1 triangular faces. This offers an alternative approach to W 3 R. Bricard ’s flexible octahedra . Special case: n = 3 Workshop on Rigidity, October 11–14, 2011, Fields Institute, Toronto 4
1. Definition of Kokotsakis meshes Kinematic interpretation: Σ 3 The polygons represent different I 30 V 3 systems Σ 0 , . . . , Σ n . V 2 Σ 4 The sides a i of f 0 are instan- I 40 Σ 0 taneous axes I i 0 of the relative I 20 Σ 2 motions Σ i / Σ 0 . V 4 V 1 I 10 Σ 1 The relative motions Σ i +1 / Σ i between consecutive systems are spherical four-bars mechanisms. The polygons need not be planar Workshop on Rigidity, October 11–14, 2011, Fields Institute, Toronto 5
1. Definition of Kokotsakis meshes The transmission from Σ i to the following Σ i +1 , ϕ i �→ ϕ i +1 , is realized by a spherical four-bar: B A To recall: Σ i Σ i +1 A spherical four-bar transmits the ϕ i rotation about the center A 0 by the ϕ A 0 i +1 coupler AB non-uniformly to the Σ 0 B 0 rotation about B 0 . The two arms A 0 A and B 0 B represent consecutive systems Σ i , Σ i +1 . Workshop on Rigidity, October 11–14, 2011, Fields Institute, Toronto 6
1. Definition of Kokotsakis meshes The edge lengths V 1 V 2 , . . . , V 4 V 1 of the central polygon f 0 have no Σ 3 influence on the flexibility = ⇒ I 30 V 3 V 2 Σ 4 I 40 Theorem: A Kokotsakis-mesh for I 20 f 0 = Σ 0 n = 4 is flexible if and only if the Σ 2 transmission Σ 1 �→ Σ 3 realized by V 4 V 1 the two four-bars ( V 1 , V 2 ) on the I 10 right hand side equals that via Σ 1 ( V 3 , V 4 ) on the left hand side. (we do not care about intersections between the involved quadrangles) Workshop on Rigidity, October 11–14, 2011, Fields Institute, Toronto 7
1. Definition of Kokotsakis meshes Some models of flexible Kokotsakis meshes. courtesy Nadja Posselt, Uwe Hanke, TU Dresden Workshop on Rigidity, October 11–14, 2011, Fields Institute, Toronto 8
1. Definition of Kokotsakis meshes 024 � Theorem: ( A. Kokotsakis (1932)) 234 f 7 A Kokotsakis mesh is infinitesimally f 6 23 flexible ⇐ ⇒ the points of intersection f 3 123 between the traces of ( f 1 , f 3 ) , ( f 5 , f 6 ) 03 4 3 and ( f 7 , f 8 ) are collinear. � 134 13 013 This is equivalent to the collinearity of f 2 f 4 f 0 02 the intersection points ( f 2 , f 4 ) , ( f 6 , f 7 ) 24 04 01 and ( f 8 , f 5 ) . f 5 f 1 f 8 14 12 The principle of “averaging” gives rise to snapping Kokotsakis meshes. 124 Workshop on Rigidity, October 11–14, 2011, Fields Institute, Toronto 9
1. Definition of Kokotsakis meshes In discrete differential geometry there is a interest on polyhedral structures composed of quadrilaterals (quadrilateral surfaces). If all quadrilaterals are planar, they form a discrete conjugate net = quad mesh . Theorem: [ Bobenko, Hoffmann, Schief 2008] A discrete conjugate net in general position H. Pottmann, Y. Liu, J. Wallner, is continuously flexible ⇐ ⇒ all its 3 × 3 A. Bobenko, W. Wang: complexes are continuously flexible. Geometry of Multi-layer Freeform Structures for Architecture. ACM Trans. Graphics 26 (3) (2007), Bobenko et al., 2008: SIGGRAPH 2007 “. . . the complete classification of flexible discrete conjugate nets (“quad meshes”) has not been achieved yet” Workshop on Rigidity, October 11–14, 2011, Fields Institute, Toronto 10
1. Definition of Kokotsakis meshes Also the folding of the roof at cabrios is based on a flexible quad mesh courtesy: Nadja Posselt Diploma thesis, TU Dresden 2010 Workshop on Rigidity, October 11–14, 2011, Fields Institute, Toronto 11
2. Three examples of flexible quad meshes Miura-Ori folding Miura-ori is a Japanese folding technique named after Prof. Koryo mountain folds full valley folds dashed Miura, The University of Tokyo. It is used for solar panels because it can be unfolded into its rectangular Miura-Ori shape by pulling on one corner only. folding On the other hand it is used as kernel Unfolded miura-ori; to stiffen sandwich structures. dashs are valley folds , full lines are mountain folds Workshop on Rigidity, October 11–14, 2011, Fields Institute, Toronto 12
2. Three examples of flexible quad meshes Miura-ori, a Japanese folding technique Miura-ori, a Japanese folding technique Miura-ori, a Japanese folding technique Miura-ori, a Japanese folding technique Miura-ori, a Japanese folding technique Miura-ori, a Japanese folding technique M i u r a - o r i , M a J a i u p a n r a - e s e o r f o l i , d i n g a t e c J a h n p a i q u e M i u n e r a - s e o r i , f o M a J l d i u a p i n a r a g n e s e - o Miura-ori, a Japanese folding technique t e f o r i c h l d i n , a n i g t e J q u c h n a p e i q u a n e M e s i e u f o r l d a - i n g o t r Miura-ori, a Japanese folding technique e c i , h n M a i q u e M J a p i i u a u r n a Miura-ori, a Japanese folding technique e - o s M r r e a i , f a o - l d i J i u Miura-ori, a Japanese folding technique o a p n Miura-ori, a Japanese folding technique a g r n t r Miura-ori, a Japanese folding technique e e a i s c , e h - f n Miura-ori, a Japanese folding technique a o i q Miura-ori, a Japanese folding technique o l d u i e r n J g i a t , e Miura-ori, a Japanese folding technique p c a h n a i q J u n e a Miura-ori, a Japanese folding technique e p s M M M M a e n i i i i f u u u u e o r r r r s l a a a a e d - - - - o o o o f i n M M M M r r r r o i i i i g , , , , l i i i i d u u u u a a a a i t r r r r e J J J J n a a a a a a a a c - - - - g p p p p h o o o o a a a a n r r r r t i i i i n n n n e , , , , i e e e e c q a a a a s s s s h u e e e e J J J J n a a a a e f f f f p p p p o o o o i q a a a a l l l l d d d d u n n n n i i i i e e e e n n n n e s s s s g g g g e e e e t t t t f f f f e e e e o o o o c c c c l l l l d d d d h h h h i i i i n n n n n n n n i i i i g g g g q q q q u u u u t t t t e e e e e e e e c c c c h h h h n n n n i i i i q q q q u u u u e e e e Workshop on Rigidity, October 11–14, 2011, Fields Institute, Toronto 13
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