Rigidity Theory and Some Applications

1. This presentation will probably involve audience discussion, which will create action items. Use PowerPoint to keep track of these action items during your presentation • In Slide Show, click on the right mouse button • Select “Meeting Minder” • Select the “Action Items” tab • Type in action items as they come up • Click OK to dismiss this box • This will automatically create an Action Item slide at the end of your presentation with your points entered. Rigidity Theory andSome Applications Brigitte Servatius WPI

2. Ingredients: Universal (ball) Joint V: vertices Rigid Rod (bar) E: edges Framework: embedding graph

3. Basic Definitions: Deformation: Continuous 1-parameter family of frameworks. Length of the bars is preserved

4. Trivial Deformations • Deformations from isometries • Trivial degrees of freedom

5. Non-trivial Deformations Watt Engine Peaucellier Engine

6. Non-trivial Deformations Watt Engine Peaucellier Engine

7. Non-trivial Deformations Watt Engine Peaucellier Engine

8. Non-trivial Deformations Watt Engine Peaucellier Engine

9. Rigidity: • Rigid: Every deformation is locally trivial. • Globally Rigid:

10. Rigidity in dimension 0 is pointless Ambient Dimension • Tendency: the rigidity of a framework decreases as the dimension of the ambient space increases. • The complete graph is rigid in all dimensions.

11. Infinitesimal Analysis Quadratic: Linear: |E| equations in m|V| unknowns: Evaluate at 0: There is always a subspace of trivial solutions Rank depends only on the dimesnion: m(m+1)/2 Infinitesimal Motion (Flex): Non-Trivial Solution Infinitesimally Rigid: Only trivial Solutions

12. (It may be rigid anyway…) Infinitesimally Rigid Rigid Infinitesimal Rigidity Unknowns Infinitesimally Rigid: Only trivial Solutions Infinitesimally Rigid Rigid

13. CONTRADICTION!!! The last red bar insists on an infinitesimal rotation centered on its pinned vertex. First let’s eliminate the trivial solutions by pinning the bottom vertices. The equation at the left vertical rod forces the velocity at the top corner to lie along the horizontal direction. The equation at the right vertical rod forces the velocity at the top left corner to also lie along the horizontal direction. The top bar forces the two horizontal vectors to be equal in magnitude and direction. The remaining vertices of the top triangle force the third vertex velocity to match the infinitesimal rotation. Visual Linear Algebra Is the Framework Infinitesimally rigid?

14. The three connecting edges happen to be concurrent. Dilate the larger triangle. The blue displacement vectors satisfy the equation at the left. Displacing the points results in a PARALLEL REDRAWING of the original framework. The vector condition is familiar… The blue redrawing displacements correspond a red flex. Conclusion: The original framework did have an infinitesimal motion. Parallel Redrawings Is the Framework Infinitesimally rigid?

15. The Rigidity Matrix A framework is infinitesimally rigid in m-space if and only if its rigidity matrix has rank

16. Euler Conjecture “A closed spacial figure allows no changes as long as it is not ripped apart” 1766.

17. Cauchy’s Theorem - 1813 “If there is an isometry between the surfaces of two strictly convex polyhedra which is an isometry on each of the faces, then the polyhedra are congruent”. The 2-skeleton of a strictly Convex 3D polyhedron is rigid. Like Me!

18. Animation by Franco Saliola, York University using STRUCK. Bricard Octahedra - 1897 By Cauchy’s Theorem, an octahedron is rigid. If the 1-skeleton is knotted ...

19. More Euler Spin-offs… • Alexandrov – 1950 • If the faces of a strictly convex polyhedron are triangulated, the resulting 1-skeleton is rigid. • Gluck – 1975 • Every closed simply connected ployhedral surface in 3-space is rigid. • Connelly – 1975 • Non-convex counterexample to Euler’s Conjecture. • Asimov & Roth - 1978 • The 1-skelelton of any convex 3D polyhedron with a non-triangular face is non-rigid.

20. More Euler Spin-offs… • Alexandrov – 1950 • If the faces of a strictly convex polyhedron are triangulated, the resulting 1-skeleton is rigid. • Gluck – 1975 • Every closed simply connected ployhedral surface in 3-space is rigid. • Connelly – 1975 • Non-convex counterexample to Euler’s Conjecture. • Asimov & Roth - 1978 • The 1-skelelton of any convex 3D polyhedron with a non-triangular face is non-rigid.

21. More Euler Spin-offs… • Alexandrov – 1950 • If the faces of a strictly convex polyhedron are triangulated, the resulting 1-skeleton is rigid. • Gluck – 1975 • Every closed simply connected ployhedral surface in 3-space is rigid. • Connelly – 1975 • Non-convex counterexample to Euler’s Conjecture. • Asimov & Roth - 1978 • The 1-skelelton of any convex 3D polyhedron with a non-triangular face is non-rigid. “Jitterbug” Photo: Richard Hawkins

22. Combinatorial Rigidity • Infinitesimal rigidity of a framework depends on the embedding. • An embedding is generic if small perturbations of the vertices do not change the rigidity properties. • Generic embeddings are an open dense subset of all embeddings.

23. Generic Embeddings Generic embedding – think random embedding. Theorem: If some generic framework is rigid, then ALL generic embeddings of the graph are also rigid. A graph is generically rigid (in dimension m) if it has any infinitesimally rigid embedding.

24. The Rigid World Generically Rigid Rigid Infinitesimally Rigid

25. Generic Rigidity in Dimension 1: • All embeddings on the line are generic. • Rigidity is equivalent to connectivity

26. Generic Rigidity in Dimension 2: • Laman’s Theorem • G = (V,E) is rigid iff G has a subset F of edges satisfying • |F| = 2|V| - 3 and • |F’| < 2|V(F’) - 3 for subsets F’ of F • This condition says that: • G has enough edges to be rigid • G has no overbraced subgraph.

27. Generic Rigidity in the Plane: The following are equivalent: • Generic Rigidity • Laman’s Condition • 3T2: The edge set contains the union three trees such that • Each vertex belongs to two trees • No two subtrees span the same vertex set • G has as subgraph with a Henneberg construction.

28. Mat1

29. Mat2

30. Mat3

31. Mat4

32. Mat4

33. B Mat 1 A b a B C c A c C a b B

34. Henneberg Moves • Zero Extension: • One Extension:

35. No vertices of degree 2 • NO TRIANGLES! • No vertices of degree 2 • NO TRIANGLES! Henneberg Moves • Zero Extension: • One Extension:

36. Applications • Computer Modeling • Cad • Geodesy (mapping) • Robotics • Navigation • Molecular Structures • Glasses • DNA • Structural Engineering • Tensegrities

37. Applications: CAD • Combinatorial (discrete) results preferred • Generic results not sufficient

38. Glass Model • Edge length ratio at most 3:1 • No small rigid subgraphs • 1st order phase transition

39. Cycle Decompositions • The graph decomposes into disjoint Hamiltonian cycles • The are many “different” ones:

40. Applications Molecular Structures Ribbon Model

41. Applications Molecular Structures Ball and Joint Model PROTASE

42. Applications Molecular Structures Ball and Joint Model HIV

43. Applications Tensegrities Bob Connelly Kenneth Snelson

44. Applications Tensegrities Tensegrities Photo by Kenneth Snelson

45. Open Problems • 3D – characterize generic rigidity • 2D • Find a “good” algorithm to detect rigid subgraphs of a large graph. • Find good recursive constructions of 3-connected dependent graphs. • Rigidity of random regular graphs • CAD: How do you properly mix length, direction, and angle constraints.