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Applied Geometry: Foldings and Unfoldings

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Applied Geometry: Foldings and Unfoldings Applied Geometry: Foldings and Unfoldings Hellmuth Stachel [email protected] — http://www.geometrie.tuwien.ac.at/stachel ISTEC 2016, International Science and Technology Conference July 13–15, Vienna University of Technology Table of contents 1. Unfolding and folding 2. Flexible quad-meshes 3. Curved folding, Example 1 4. Curved folding, Example 2 Funding source: Partly supported by the Austrian Academy of Sciences July 13: ISTEC 2016, Vienna University of Technology 1/57 1. Unfolding and folding unfold fold There are standard procedures provided for the construction of the unfolding (development, net) of polyhedra or developable surfaces. The result is unique, apart from the placement of the different components, and it shows the intrinsic metric of the spatial structure. July 13: ISTEC 2016, Vienna University of Technology 2/57 E.g., the unfolding of the Oloid 2 3 Technische Universitaet Wien Technische Universitaet Wien Technische Universitaet Wien 3 Oloid Oloid Oloid Oloid Institut fuer Geometrie fuer Institut Institut fuer Geometrie fuer Institut 1 Developable surfaces are ruled surfaces with vanishing Gaussian curvature July 13: ISTEC 2016, Vienna University of Technology 3/57 2 3 Technische Universitaet Wien Technische Universitaet Wien Technische Universitaet Wien 3 Oloid Oloid Oloid Oloid Institut fuer Geometrie fuer Institut Institut fuer Geometrie fuer Institut Oloid: arc-length parametrization of the 1 unfolding of the circles: 2√3 √2 cos s 2(1 cos s)(1 + 2 cos s) x(s) = arccos − 9 √ (1 + cos s) 1 + cos s − √3 2 11+7cos s y(s) = ln + . 9 1 + cos s 1 + cos s July 13: ISTEC 2016, Vienna University of Technology 4/57 1. Unfolding and folding unfold fold The inverse problem, i.e., the determination of a folded structure from a given unfolding is more complex. In the smooth case we obtain a continuum of bent poses. In the polyhedral case the computation leads to a system of algebraic equations. Also here the corresponding spatial object needs not be unique. July 13: ISTEC 2016, Vienna University of Technology 5/57 1. Unfolding and folding Only if the polyhedron bounds a convex solid then the result is unique, due to Aleksandr Danilovich Alexandrov (1941). In this case, for each vertex the sum of intrinsic angles for all adjacent surfaces is < 360◦ (= convex intrinsic metric). Theorem: [Uniqueness Theorem] For any convex intrinsic metric there is a unique convex polyhedron. July 13: ISTEC 2016, Vienna University of Technology 6/57 1. Unfolding and folding If convexity is not required the unfolding of a polyhedron needs not define its spatial shape uniquely ! Definition 1: A polyhedron is called globally rigid if its intrinsic metric defines its spatial form uniquely — up to movements in space. July 13: ISTEC 2016, Vienna University of Technology 7/57 1. Unfolding and folding f 4 If convexity is not required the unfolding 7 6 f f f f 4 5 6 of a polyhedron needs not define its f spatial shape uniquely ! f 2 f 2 f 1 f 1 Definition 1: A polyhedron is called globally rigid if its intrinsic metric defines its spatial form uniquely — up to movements in space. A flipping (or snapping) polyhedron admits two sufficiently close realizations e.g., a tetrahedron – by applying a slight force. July 13: ISTEC 2016, Vienna University of Technology 7/57 1. Unfolding and folding Definition 2: A polyhedron is called (continuously) flexible if there is a continuous family of mutually incongruent polyhedra sharing the intrinsic metric. Each member of this family is called a flexion. July 13: ISTEC 2016, Vienna University of Technology 8/57 1. Unfolding and folding Definition 2: A polyhedron is called (continuously) flexible if there is a continuous family of mutually incongruent polyhedra sharing the intrinsic metric. Each member of this family is called a flexion. f 7 8 f 6 f f f 5 8 f 6 2 f 1 f f 1 f 1 Even a regular octahedron is flexible — after being re-assembled. The regular pose on the left hand side is called locally rigid. July 13: ISTEC 2016, Vienna University of Technology 8/57 4 1. Unfolding and folding B A′ Two particular examples of flexible octahedra C′ C′ where two faces are omitted. Both have an B′ A axial symmetry (types 1 B and 2) A C A′ Below: Nets of the two octahedra. B′ C C B C A′ A′ A′ A′ B B C′ A B′ C′ ′ C C′ A ′ July 13: ISTEC 2016, Vienna University of Technology 9/57 8 1. Unfolding and folding B′ According to R. Bricard (1897) t2 there are three types of flexible t1 C′ octahedra (four-sided double- A′ N pyramids). The first two have axis or plane of C symmetry. Those of type 3 admit two flat poses. In each such pose, A B the pairs (A, A′), (B, B′), and (C, C′) of opposite vertices are associated points of a strophoid . S S P July 13: ISTEC 2016, Vienna University of Technology 10/57 1. Unfolding and folding B′ According to Bricard’s construc- tion, all bisectors must pass C through the midpoint N of the A N concentric circles. C′ B′′ The two flat poses of a type-3 B A′ flexible octahedron, when ABC remains fixed. C′′ A′′ July 13: ISTEC 2016, Vienna University of Technology 11/57 1. Unfolding and folding This polyhedron called 06 “Vierhorn” is locally 05 01 03 05 01 05 rigid, but can flip 02 08 08 02 02 between its spatial shape and two flat 13 realizations in the 09 09 16 09 16 12 12 planes of symmetry 10 15 16 (W. Wunderlich, C. Schwabe). At the science exposition “Phänomena” 1984 in Zürich this polyhedron was exposed and falsely stated that this polyhedron is flexible. July 13: ISTEC 2016, Vienna University of Technology 12/57 04 1. Unfolding and folding 01 02 03 05 08 13 06 07 16 09 10 01 02 12 05 11 08 14 15 16 09 13/57 12 e totally flat configuration to another, passing through the “Vierhorn” and its unfolding Wolfram MathWorld: A flexible polyhedron which flexes from on intermediate configurations of positive volume. July 13: ISTEC 2016, Vienna University of Technology 2. Flexible quad-meshes A polygonal mesh is a simply connected subset of a polyhedral surface (sphere-like or with boundary) consisting of (not necessarily planar) polygons, edges and vertices in the Euclidean 3- space. The edges are either internal when they are shared by two faces, or they belong to the boundary of the mesh. When all polygons are quadrangles, then it is called a quadrilateral surface. July 13: ISTEC 2016, Vienna University of Technology 14/57 2. Flexible quad-meshes In modern architecture, most freeform surfaces are designed as polyhedral surfaces – like the Capital Gate in Abu Dhabi, built by the Austrian company Waagner Biro (160 m high, 18◦ inclination) July 13: ISTEC 2016, Vienna University of Technology 15/57 2. Flexible quad-meshes f 3 a3 V3 V2 f 4 a4 Under which conditions is this 3 3 f 0 a2 f 2 mesh continuously flexible ? × V4 V1 a1 f 1 July 13: ISTEC 2016, Vienna University of Technology 16/57 2. Flexible quad-meshes f 3 V3 β2 γ2 I30 1 B1 2 α2 γ 2 δ A1 B γ2 α A2 1 β1 f 0 f 2 β2 I20 α2 ϕ1 ϕ3 ϕ2 I10 V4 2 I30 I10 δ1 I20 δ δ1 β1 α1 γ1 f 1 Transmission from f 1 to f 3 via the quadrangle f 2 Composition of two spherical four-bar linkages July 13: ISTEC 2016, Vienna University of Technology 17/57 2. Flexible quad-meshes A Kokotsakis mesh is continuously flexible f ⇐⇒ 3 f f the transmission from 1 to 3 can in two I30 ways be decomposed into two spherical V3 V2 f four-bar mechanisms, one via V1 and V2, 4 I40 f I20 f the other via V4 and V3. 0 2 V4 V1 I10 The internal edges can be arranged in two f 1 ‘horizontal’ (blue) and two ‘vertical’ edge folds (red). July 13: ISTEC 2016, Vienna University of Technology 18/57 2. Flexible quad-meshes A Kokotsakis mesh is continuously flexible f ⇐⇒ 3 f f the transmission from 1 to 3 can in two I30 ways be decomposed into two spherical V3 V2 f four-bar mechanisms, one via V1 and V2, 4 I40 f I20 f the other via V4 and V3. 0 2 V4 V1 I10 The internal edges can be arranged in two f 1 ‘horizontal’ (blue) and two ‘vertical’ edge folds (red). July 13: ISTEC 2016, Vienna University of Technology 18/57 2. Flexible quad-meshes I. Planar-symmetric type (Kokotsakis 1932): The reflection in the plane of symmetry of V1 and V4 V3 V2 maps each horizontal fold onto itself while the two vertical folds are exchanged. V1 V4 II. Translational type: V3 There is a translation V1 V4 and V2 V3 mapping a4 V2 the three faces on the right→ hand side→ onto the triple on the left hand side. V V4 1 July 13: ISTEC 2016, Vienna University of Technology 19/57 6 2. Flexible quad-meshes III: Isogonal type (Kokotsakis 1932): A Kokotsakis mesh is flexible when at each vertex Vi opposite angles are either equal or complementary, i.e., αi = βi , γi = δi or Vi α = π β , γ = π δ and (n = 4) i − i i − i δi sin α1 sin γ1 sin α2 sin γ2 βi ± ± α γ sin(α1 γ1) · sin(α2 γ2) i i − − sin β3 sin γ3 sin β4 sin γ4 = ± ± sin(β3 γ3) · sin(β4 γ4) − − A quad mesh where all 3 3 complexes are of this type is continuously flexible and called Voss surface (Kokotsakis,× Graf, Sauer) July 13: ISTEC 2016, Vienna University of Technology 20/57 2.
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