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Folding & Unfolding: Folding Polygons to Convex Polyhedra

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Folding & Unfolding: Folding Polygons to Convex Polyhedra FoldingFolding && Unfolding:Unfolding: FoldingFolding PolygonsPolygons toto ConvexConvex PolyhedraPolyhedra JosephJoseph O’RourkeO’Rourke SmithSmith CollegeCollege FoldingFolding andand UnfoldingUnfolding TalksTalks Linkage folding Tuesday Erik Demaine Paper folding Wednesday Erik Demaine Folding polygons into Saturday Joe O’Rourke convex 1 polyhedra Unfolding Saturday Joe O’Rourke polyhedra 2 OutlineOutline Reconstruction of Convex Polyhedra Cauchy to Sabitov (to an Open Problem) Folding Polygons Algorithms Examples Questions OutlineOutline11 Reconstruction of Convex Polyhedra Cauchy to Sabitov (to an Open Problem) Cauchy’s Rigidity Theorem Aleksandrov’s Theorem Sabitov’s Algorithm OutlineOutline22 Folding Polygons Algorithms Edge-to-Edge Foldings Gluing Trees; exponential lower bound Gluing Algorithm Examples Foldings of the Latin Cross Foldings of the Square Questions Transforming shapes? ReconstructionReconstruction ofof ConvexConvex PolyhedraPolyhedra graph Steinitz’s Theorem face angles edge lengths face areas Minkowski’s Theorem face normals dihedral angles inscribed/circumscribed Minkowski’sMinkowski’s TheoremTheorem (a) (b) ReconstructionReconstruction ofof ConvexConvex PolyhedraPolyhedra graph face angles Cauchy’s Theorem edge lengths face areas face normals dihedral angles inscribed/circumscribed Cauchy’sCauchy’s RigidityRigidity TheoremTheorem If two closed, convex polyhedra are combinatorially equivalent, with corresponding faces congruent, then the polyhedra are congruent; in particular, the dihedral angles at each edge are the same. Global rigidity == unique realization SameSame facialfacial structure,structure, noncongruentnoncongruent polyhedrapolyhedra SphericalSpherical polygonpolygon v v (a) (b) SignSign Labels:Labels: {+,-,0}{+,-,0} Compare spherical polygons Q to Q’ Mark vertices according to dihedral angles: {+,-,0}. Lemma: The total number of alternations in sign around the boundary of Q is ≥ 4. The spherical polygon opens. (a) Zero sign alternations; (b) Two sign alts. SignSign changeschanges EulerEuler TheoremTheorem ContradictionContradiction Lemma ≥ 4 V + + - v - + - f + + - FlexingFlexing toptop ofof regularregular octahedronoctahedron Steffen’sSteffen’s flexibleflexible polyhedronpolyhedron 14 triangles, 9 vertices http://www.mathematik.com/Steffen/ TheThe Bellow’sBellow’s ConjectureConjecture Polyhedra can bend but not breathe [Mackenzie 98] Settled in 1997 by Robert Connelly, Idzhad Sabitov, and Anke Walz Heron’s formula for area of a triangle: A2 = s(s-a)(s-b)(s-c) Francesca’s formula for the volume of a tetrahedron Sabitov: volume of a polyhedron is a polynomial in the edge lengths. OutlineOutline11 Reconstruction of Convex Polyhedra Cauchy to Sabitov (to an Open Problem) Cauchy’s Rigidity Theorem Aleksandrov’s Theorem Sabitov’s Algorithm Aleksandrov’sAleksandrov’s TheoremTheorem (1941)(1941) “For every convex polyhedral metric, there exists a unique polyhedron (up to a translation or a translation with a symmetry) realizing this metric." Pogorelov’sPogorelov’s versionversion (1973)(1973) “For every convex polyhedral metric, there exists a unique polyhedron (up to a translation or a translation with a symmetry) realizing this metric.“ “Any convex polyhedral metric given ... on a manifold homeomorphic to a sphere is realizable as a closed convex polyhedron (possibly degenerating into a doubly covered plane polygon)." AlexandrovAlexandrov GluingGluing (of(of polygons)polygons) Uses up the perimeter of all the polygons with boundary matches: No gaps. No paper overlap. Several points may glue together. At most 2 angle at any glued point. Homeomorphic to a sphere. Aleksandrov’s Theorem unique “polyhedron” FoldingFolding thethe LatinLatin CrossCross CauchyCauchy vs.vs. AleksandrovAleksandrov Cauchy: uniqueness Aleksandrov: existence and uniqueness Cauchy: faces and edges specified Aleksandrov: gluing unrelated to creases UniquenessUniqueness Cauchy: combinatorial equivalence + congruent faces => congruent Aleksandrov: “Two isometric polyhedra are equivalent” The sphere is rigid [Minding] The sphere is unique given its metric [Liebmann, Minkowski] Closed regular surfaces are rigid [Liebmann, Blaschke, Weyl] Uniqueness of these w/in certain class [Cohn-Vossen] ... “Isometric closed convex surfaces are congruent” [Pogorelov 73] AlexandrovAlexandrov ExistenceExistence11 Induction on the number of vertices n of the metric: from realization of n-1 vertex metric to n vertex metric by continuous deformation of metrics tracked by polyhedral realizations ExistenceExistence22 curvature = 2 – angs There are two vertices a and b with curvature less than . Connect by shortest path . Cut manifold along and insert double that leaves curvature unchanged. Adjust shape of until a and b both disappear: n-1 vertices. Realize by induction, introduce nearby pseudovertex, track sufficiently close metrics. (a)(a) aa flattened;flattened; (b)(b) bb flattened.flattened. c c A=60 θ θ=B=30 α=240 β=300 (a) (b) D-FormsD-Forms SmoothSmooth closedclosed convexconvex curvescurves ofof samesame perimeter.perimeter. GlueGlue perimetersperimeters together.together. D-formD-form c1 c2 Helmut Pottmann and Johannes Wallner. Computational Line Geometry. Springer-Verlag, 2001. Fig 6.49, p.401 PottmannPottmann && WallnerWallner When is a D-form is the convex hull of a space curve? Always When is it free of creases? Always OutlineOutline11 Reconstruction of Convex Polyhedra Cauchy to Sabitov (to an Open Problem) Cauchy’s Rigidity Theorem Aleksandrov’s Theorem Sabitov’s Algorithm Sabitov’sSabitov’s AlgorithmAlgorithm Given edge lengths of triangulated convex polyhedron, computes vertex coordinates in time exponential in the number of vertices. SabitovSabitov VolumeVolume PolynomialPolynomial 2N 2(N-1) 2(N-2) V + a1(l)V + a2(l)V + ... 0 + aN(l)V = 0 2 Tetrahedron: V + a1(l) = 0 l = vector of six edge lengths 2 2 2 a1(l) = ijk (li) (lj) (lk) /144 Francesca’s formula Volume of polyhedron is root of polynomial 22NN possiblepossible rootsroots (a) (b) GeneralizedGeneralized PolyhedraPolyhedra Polynomial represents volume of generalized polyhedra any simplicial 2-complex homeomorphic to an orientable manifold of genus ≥ 0 mapped to R3 by continuous function linear on each simplex. Need not be embeddable: surface can self-intersect. SabitovSabitov ProofProof11 e v1 d v2 SabitovSabitov ProofProof22 vol(P) = vol(P’) - vol(T), = M1 ... [many steps] ... polynomial = 0 unknowns: edge lengths vector l V = vol(P) unknown diagonal d SabitovSabitov ProofProof33 unknowns: edge lengths l [given] V = vol(P) [“known” from volume polynomial] unknown diagonal d Try all roots for V, all roots for d: candidates for length of d & dihedral at e. Repeat for all e. Check the implied dihedral angles for the convex polyhedron. Open:Open: PracticalPractical AlgorithmAlgorithm forfor CauchyCauchy RigidtyRigidty Find either C a polynomial-time algorithm, C or even a numerical approximation procedure, that takes as C input the combinatorial structure and edge lengths of a triangulated convex polyhedron, and C outputs coordinates for its vertices. OutlineOutline22 Folding Polygons Algorithms Edge-to-Edge Foldings Gluing Trees; exponential lower bound Gluing Algorithm Examples Foldings of the Latin Cross Foldings of the Square Questions Transforming shapes? FoldingFolding PolygonsPolygons toto ConvexConvex PolyhedraPolyhedra When can a polygon fold to a v8 v7 polyhedron? v9 v6 “Fold” = close up v10 v5 perimeter, no overlap, no gap : v11 v4 v12 v3 When does a v13 v2 polygon have an Aleksandrov v0 v1 (a) (b) gluing? UnfoldableUnfoldable PolygonPolygon b a c FoldabilityFoldability isis “rare”“rare” Lemma: The probability that a random polygon of n vertices can fold to a polytope approaches 0 as n J 1. PerimeterPerimeter HalvingHalving v4 x v3 v5 v2 y v1 Edge-to-EdgeEdge-to-Edge GluingsGluings Restricts gluing of whole edges to whole edges. [Lubiw & O’Rourke, 1996] VideoVideo [Demaine, Demaine, Lubiw, JOR, Pashchenko (Symp. Computational Geometry, 1999)] GluingGluing TreesTrees v4 1/3 v5 v3 e4 e3 e e e5 4 3 e2 v4 v5 v3 2/3 v1 v2 v1 e1 v2 1 1 e5 e2 (a) (b) (c) 1 e1 FoldingFolding ofof nonconvexnonconvex pentagonpentagon v4 v5 v3 e e e5 4 3 e2 e e 4 3 v e1 4 v1 e1 v2 e5 (a) e2 v 2 v5 v1 v3 ExponentialExponential NumberNumber ofof GluingGluing TreesTrees α β x (a) y β αβαβ αβ αβαβα (b) x ... y αβαβ αβ αβαβαβ β β β α α βαα α β βα x ... (c) α α y αβαβαβ βα α β β ExponentialExponential NumberNumber ofof GluingGluing TreesTrees GeneralGeneral GluingGluing AlgorithmAlgorithm No edge-to-edge assumption. Implementations: Anna Lubiw, Koichi Hirata (independently) Exponential-time, dynamic programming flavor. Open:Open: Polynomial-timePolynomial-time FoldingFolding DecisionDecision AlgorithmAlgorithm Given a polygon P of n vertices, determine in time polynomial in n if P has an Aleksandrov folding, and so can fold to some convex polyhedron. TwoTwo CaseCase StudiesStudies The Latin Cross The Square FoldingFolding thethe LatinLatin CrossCross 85 distinct gluings 6 5 6 7 4 6 4 7 4 Reconstruct shapes 8 3 7 3 8 3 2 F43/P7 F54/P8 8 2 7 2 8 F19/P9 2 by ad hoc 7 1 1 6 1 1 7 1 techniques 6 2 5 2 6 1 5 8 5 3 4 3 5 8 5 23 incongruent 4 3 4 convex polyhedra 5 44 4 5 4 5 3 43 4 3 6 3 3 5 7 3 F53/P10 2 F3/P11 2 5 F12/P12 2 7 6 1 1 8 1 6 1 7 2 8 1 7 1 8 6 6 2 7 2 8 7 5 6 TheThe 2323 convexconvex polyhedrapolyhedra foldablefoldable fromfrom thethe LatinLatin
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