Resources in Rigidity Theory
From MathWiki
This page connects to some preprints, etc. from Walter Whiteley and collaborators. Some of them have ended up being cited as sources of continuing work.
Table of contents 

Chapters from a 1987 Draft Rigidity Book
 Draft Chapter 1 Introduction by Ben Roth from the 1987 draft book.
 Draft Chapter 2 Basic Concepts by Bob Connelly from the 1987 draft book.
 Draft Chapter 3 Basic Concepts of Static Rigidity by Bob Connelly from the 1987 draft book.
 Draft Chapter 4 Rigidity of Frameworks on Convex Surfaces by Ben Roth from the 1987 draft book.
 Draft Chapter 6 Bipartite Frameworks by Henry Crapo from the 1987 draft book.
 Draft Chapter 7 Generic Rigidity by TiongSeng Tay and Walter Whiteley from the 1987 draft book.
 Draft Chapter 9 More Generic Rigidity by Walter Whiteley from the 1987 draft book.
 Draft Chapter 10 Tensegrity by Walter Whiteley from the 1987 draft book.
 Draft chapter on Global and SecondOrder Rigidity of Tensegrity Frameworks from the 1987 draft book.
Parallel Drawing in 3D
A set of preprint papers with connections to parallel drawings of configurations in dimension 2 and in higher dimensions:
 A preprint on Parallel Drawings in 3Space from 1986 with some basic results.
 A draft chapter on Parallel Drawings in 3Space from 1987
 Preprint on 3Space Reciprocals and Parallel Drawings from 1994, with Henry Crapo. Some of this was superseded by the work of Rybnykov, but the explicit connections to the reciprocals of Rankin is still of interest
 Preprint on Parallel Drawings in the plane from 1993, with Henry Crapo.
 Preprint on Geometry of Direction Length Frameworks from 1996. This includes reciprocal diagrams, averaging, and other geometric results for the mixed frameworks, which continue to be of interest to people in the wider community.
Inductive Techniques
There are a number of papers developing rigidity results with inductive techniques. For Plane Rigidity, this associated with Henneberg Methods. Here is the link to a preprint which surveys a number of methods for the plane, including a duality between methods, for the special case of planar graphs: Bases and Circuits for 2Rigidity
 Bases and Circuits for 2rigidity: constructions via tree partitions using inductive constructions, and with a duality for planar graphs.
 Expanded Inductive Constructions of Bases and Circuits which is an expanded version of the paper above.
Global Rigidity
There are a few older preprints on global rigidity which may be of continuing interest.
 Transfer of Global Rigidity Results Among Dimensions: Graph Powers and Coning looking at transfer to higher dimensions and other metric.
 A 2007 preprint on Localization in Sensor Networks based on an undergraduate project of Matthew Cheung using inductive constructions.
 A 2009 preprint on Global Rigidity and the Effect of Coning.pdf which later appeared in Discrete and Computational Geometry.
 Draft chapter on Global and SecondOrder Rigidity of Tensegrity Frameworks from the 1987 draft book.
Schlegel Diagrams
 Preprint on Schlegal Diagrams of Simple 4Polyhedra are all exact projections of convex 4polytopes, using reciprocal diagrams.
Multivariate Splines
 Preprint on Geometry of Bivariate Splines which are analogous to rigidity results in dimensions dâ‰¥2.
Control of Formations
 2010 Masters Thesis of Matthew Cheung on Control of Autonomous Formations with the Pebble Game Algorithm
Extension of Rigidity Results for Grids of Parallelograms
 2003 Poster and Handout from two undergrads extending the results for rigidity of parallelograms and triangles dissecting a plane polygon (a leaf).
1978 Introduction to Structural Topology
This is a scan of a the first introductory survey, broken into part. It was written with lots of Cayley Algebra and examples. It was written prior to our reconstructing a proof of Maxwell's theorem. It was based on discussions with Henry Crapo and Janos Baracs.
Links to Short Movies of Rigidity Examples
Movies to accompany Rigidity of frameworks on expanding spheres by Nixon, Schulze, Shinichi and Whiteley, 2018
Movies for finite flexes
Links for other resources
All issues of Structural Topology (which published a number of the early articles in this generation of work on rigidity) are accessible at the Barcelona web site: http://wwwiri.upc.es/people/ros/StructuralTopology/
Some other articles, including recent articles and draft introductory chapters for the never completed book on Rigidity mentioned above (51 on his list) are downloadable at: http://www.math.cornell.edu/~connelly/
Algorithms actually used focus on the 'pebble game' in different variations. See the flexweb implementation for 2V3 at http://flexweb.asu.edu/software/pebble_game/2D_interactive/ See also the interactive Java program for general counts at http://www.cs.umass.edu/~alee/bodyBar/
The continuing work of Bernd Schulze, including his Ph.D Thesis at York University: http://www.maths.lancs.ac.uk/~schulzeb/#pub
The Thesis of Elissa Ross at York University on Rigidity of Periodic Frameworks as Graphs on the Torus: http://www.math.yorku.ca/~ejross/RossThesis.pdf
The continuing work of Adnan Sljoka, including his Ph.D. thesis at York University: http://www.math.yorku.ca/~adnanslj/
The Thesis of Yaser Eftekhari at York University, 2017 on Geometry of pointhyperplane and spherical frameworks