Rigidity Theory and Some Applications Brigitte Servatius WPI 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.

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

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

Trivial Deformations Deformations from isometries Trivial degrees of freedom

Non-trivial Deformations Watt Engine Peaucellier Engine

Non-trivial Deformations Watt Engine Peaucellier Engine

Non-trivial Deformations Watt Engine Peaucellier Engine

Non-trivial Deformations Watt Engine Peaucellier Engine

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

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. Rigidity in dimension 0 is pointless

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

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

Visual Linear Algebra Is the Framework Infinitesimally rigid? 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. CONTRADICTION!!! The last red bar insists on an infinitesimal rotation centered on its pinned vertex.

Parallel Redrawings Is the Framework Infinitesimally rigid? 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.

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

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

Cauchy’s Theorem “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!

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

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 – The 1-skelelton of any convex 3D polyhedron with a non-triangular face is non-rigid.

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 – The 1-skelelton of any convex 3D polyhedron with a non-triangular face is non-rigid.

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 – The 1-skelelton of any convex 3D polyhedron with a non-triangular face is non-rigid. “Jitterbug” Photo: Richard Hawkins

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.

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

The Rigid World Rigid Generically Rigid Infinitesimally Rigid

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

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.

Generic Rigidity in the Plane : 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. The following are equivalent:

Mat1

Mat2

Mat3

Mat4

B Mat 1 B a A cC b B a A c C b

Henneberg Moves Zero Extension: One Extension:

Henneberg Moves Zero Extension: One Extension: 1.No vertices of degree 2 2.NO TRIANGLES! 1.No vertices of degree 2 2.NO TRIANGLES!

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

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

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

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

Applications Molecular Structures Ribbon Model

Applications Molecular Structures PROTASE Ball and Joint Model

Applications Molecular Structures HIV Ball and Joint Model

Applications Tensegrities Bob Connelly Kenneth Snelson

Applications Tensegrities Photo by Kenneth Snelson Tensegrities

Open Problems 3D – characterize generic rigidity 2D 1.Find a “good” algorithm to detect rigid subgraphs of a large graph. 2.Find good recursive constructions of 3-connected dependent graphs. 3.Rigidity of random regular graphs 4.CAD: How do you properly mix length, direction, and angle constraints.