1 Pseudotriangulations, Polytopes, and How to Expand Linkages G¨ unter Rote Universitat Lliure de Berlin [joint] work of/with Bob Connelly, Erik Demaine, Paco Santos, Ileana Streinu.
2 Unfolding of polygons Theorem. Every polygonal arc in the plane can be brought into straight position, without self-overlap. Every polygon in the plane can be unfolded into convex position.
3 Infinitesimal Motion n vertices p 1 , . . . , p n . 1. (global) motion p i = p i ( t ) , t ≥ 0
3 Infinitesimal Motion n vertices p 1 , . . . , p n . 1. (global) motion p i = p i ( t ) , t ≥ 0 2. infinitesimal motion (local motion) v i = d dtp i ( t ) = ˙ p i (0) Velocity vectors v 1 , . . . , v n .
4 Expansion 2 · d dt | p i ( t ) − p j ( t ) | 2 = � v i − v j , p i − p j � =: exp ij 1 v j v i p j − p i p i p j v i · ( p j − p i ) v j · ( p j − p i ) expansion (or strain ) exp ij of the segment ij
5 The Rigidity Map M : ( v 1 , . . . , v n ) �→ (exp ij ) ij ∈ E
5 The Rigidity Map M : ( v 1 , . . . , v n ) �→ (exp ij ) ij ∈ E The rigidity matrix: the rigidity M = E matrix � �� � 2 | V |
6 Expansive Motions exp ij = 0 for all bars ij (preservation of length) exp ij ≥ 0 for all other pairs ( struts ) ij (expansiveness) [ exp ij > 0 ] (strict expansiveness)
7 Expansive motions cannot overlap
8 Proof Outline 1. Prove that expansive motions exist . 2. Select an expansive motion and provide a global motion.
8 Proof Outline 1. Prove that expansive motions exist . 1. Prove that expansive motions exist . [ 2 PROOFS ] 2. Select an expansive motion and provide a global motion.
9 Proof Outline Existence of an expansive motion � (duality) Self-stresses (rigidity) Self-stresses on planar frameworks � (Maxwell-Cremona correspondence) polyhedral terrains [ Connelly, Demaine, Rote 2000 ]
10 The Expansion Cone The set of expansive motions forms a convex polyhedral cone ¯ X 0 in R 2 n , defined by homogeneous linear equations and inequalities of the form = � v i − v j , p i − p j � ≥ 0 [ > ]
11 Bars, Struts, Frameworks, Stresses Assign a stress ω ij = ω ji ∈ R to each edge. Equilibrium of forces in vertex i : p j � ω ij ( p j − p i ) ω ij ( p j − p i ) = 0 p i j ω ij ≤ 0 for struts: Struts can only push. ω ij ∈ R for bars: Bars can push or pull.
12 Motions and Stresses Linear Programming duality: There is a strictly expansive motion if and only if there is no non-zero stress. � ω ij ( p j − p i ) = 0 , for all i � = 0 j � v i − v j , p i − p j � > 0 ω ij ∈ R , for a bar ij ω ij ≤ 0 , for a strut ij
12 Motions and Stresses Linear Programming duality: There is a strictly expansive motion if and only if there is no non-zero stress. � ω ij ( p j − p i ) = 0 , for all i � = 0 j [ M T ω = 0 ] � v i − v j , p i − p j � > 0 ω ij ∈ R , for a bar ij � = 0 � � ω ij ≤ 0 , for a strut ij Mv > 0
13 Making the Framework Planar • subdivide edges at intersection points • collapse multiple edges
14 The Maxwell-Cremona Correspondence [ 1850] 3-d lifting (polyhedral terrain) � self-stresses on a planar framework
14 The Maxwell-Cremona Correspondence [ 1850] 3-d lifting (polyhedral terrain) � self-stresses on a planar framework � orthogonal dual
15 Valley and Mountain Folds ω ij > 0 ω ij < 0 valley mountain bar or strut bar
16 Look a the highest peak! mountain → bar Every polygon has > 3 convex vertices → 3 valleys → 3 bars.
17 The general case pointed vertex There is at least one vertex with angle > π .
18 The only remaining possibility a convex polygon ✷
19 Constructing a Global Motion [ Connelly, Demaine, Rote 2000 ] • Define a point v := v ( p ) in the interior of the expansion cone, by a suitable non-linear convex objective function. • v ( p ) depends smoothly on p . • Solve the differential equation ˙ p = v ( p )
20 Constructing a Global Motion Alternative approach: Select an extreme ray of the expansion cone. Streinu [2000]: Extreme rays correspond to pseudotriangulations. [show animation]
21 Part II: Pseudotriangulations
21 Part II: Pseudotriangulations Pseudotriangulations! Assumption: Points in general position.
22 Pseudotriangles A pseudotriangulation has three convex corners and an arbitrary number of reflex vertices.
23 Pseudotriangulations/ Geodesic Triangulations Other applications: • data structures for ray shooting [ Chazelle, Edelsbrunner, Grigni, Guibas, Hershberger, Sharir, and Snoeyink 1994 ] and visibility [ Pocchiola and Vegter 1996 ] • kinetic collision detection [ Agarwal, Basch, Erickson, Gui- bas, Hershberger, Zhang 1999–2001 ] [ Kirkpatrick, Snoeyink, and Speckmann 2000 ] [ Kirkpatrick & Speckmann 2002 this afternoon ] • art gallery problems [Pocchiola and Vegter 1996b], [Speckmann and T´ oth 2001]
24 Minimum (or Pointed) Pseudotriangulations (PPT) A pointed vertex is incident to an angle > 180 ◦ . A non-crossing and pointed set of edges maximal decomposes the convex hull into n − 2 pseudotriangles using 2 n − 3 edges.
25 Characterization of Pointed Pseudotriangulations An edge set with any two of the following properties: • 2 n − 3 edges (or n − 2 faces) • decomposition into pseudotriangles • non-crossing, and every vertex is pointed. [Streinu 2002]
26 Characterization of Trees An edge set with any two of the following properties: • n − 1 edges • connected • acyclic
27 Characterization of Pointed Pseudotriangulations An edge set with any two of the following properties: • 2 n − 3 edges (or n − 2 faces) • decomposition into pseudotriangles • non-crossing, and every vertex is pointed.
27 Characterization of Pointed Pseudotriangulations An edge set with any two of the following properties: • 2 n − 3 edges (or n − 2 faces) • decomposition into pseudotriangles • non-crossing, and every vertex is pointed. Caveat: Removing edges from a trian- gulation does not necessarily lead to a pointed pseudotriangulation.
28 Rigidity Properties of Pseudotriangulations • Pseudotriangulations are minimally rigid. • a Henneberg-type construction • Removing a hull edge gives an expansive mechanism with 1 degree of freedom. [Streinu 2002]
29 Flipping of Edges Any interior edge can be flipped against another edge. That edge is unique. before after
29 Flipping of Edges Any interior edge can be flipped against another edge. That edge is unique. before after The flip graph is connected. Its diameter is O ( n 2 ) . [ Br¨ onnimann, Kettner, Pocchiola, Snoeyink 2001 ]
30 Part III: Cones and Polytopes [Rote, Santos, Streinu 2002] • The expansion cone ¯ X 0 = { exp ij ≥ 0 } • The perturbed expansion cone = the PPT polyhedron ¯ X f = { exp ij ≥ f ij } • The PPT polytope X f = { exp ij ≥ f ij , exp ij = f ij for ij on boundary }
31 Pinning of Vertices Trivial Motions: Motions of the point set as a whole (translations, rotations). Pin a vertex and a direction. (“tie-down”) v 1 = 0 v 2 � p 2 − p 1 This eliminates 3 degrees of freedom.
32 Extreme Rays of the Expansion Cone Pseudotriangulations with one convex hull edge removed yield expansive mechanisms. [Streinu 2000] Rigid substructures can be identified.
33 A Polyhedron for Pseudotriangulations Wanted: A perturbation of the constraints “ exp ij ≥ 0 ” such that the vertices are in 1-1 correspondence with pseudotrian- gulations.
34 Heating up the Bars ∆ T = | x | 2 � | x | 2 ds Length increase ≥ x ∈ p i p j
35 Heating up the Bars ∆ T = | x | 2 � | x | 2 ds Length increase ≥ x ∈ p i p j
35 Heating up the Bars ∆ T = | x | 2 � | x | 2 ds Length increase ≥ x ∈ p i p j � | x | 2 ds exp ij ≥ | p i − p j | · x ∈ p i p j
35 Heating up the Bars ∆ T = | x | 2 � | x | 2 ds Length increase ≥ x ∈ p i p j � | x | 2 ds exp ij ≥ | p i − p j | · x ∈ p i p j exp ij ≥ | p i − p j | 2 · ( | p i | 2 + � p i , p j � + | p j | 2 ) · 1 3
36 Heating up the Bars — Points in Convex Position ⇒
37 The Perturbed Expansion Cone = PPT Polyhedron ¯ X f = { ( v 1 , . . . , v n ) | exp ij ≥ f ij } • f ij := | p i − p j | 2 · ( | p i | 2 + � p i , p j � + | p j | 2 ) • f ′ ij := [ a, p i , p j ] · [ b, p i , p j ] [ x, y, z ] = signed area of the triangle xyz a, b : two arbitrary points.
38 Tight Edges For v = ( v 1 , . . . , v n ) ∈ ¯ X f , E ( v ) := { ij | exp ij = f ij } is the set of tight edges at v . Maximal sets of tight edges ≡ vertices of ¯ X f .
39 What are good values of f ij ? Which configurations of edges can occur in a set of tight edges? We want: • no crossing edges • no 3-star with all angles ≤ 180 ◦ It is sufficient to look at 4-point subsets.
40 Good Values f ij for 4 points f ij is given on six edges. Any five values exp ij determine the last one. Check if the resulting value exp ij of the last edge is feasible ( exp ij ≥ f ij ) → checking the sign of an expression.
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