# 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 Tiong-Seng 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 Second-Order 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 3-Space from 1986 with some basic results.

- A draft chapter on Parallel Drawings in 3-Space from 1987

- Preprint on 3-Space 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 Some of this was superseded by the work of Rybnykov, but the explicit connections to the reciprocals of Rankin is still of interest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 2-Rigidity

- Bases and Circuits for 2-rigidity: 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.

- Draft chapter on Global and Second-Order Rigidity of Tensegrity Frameworks from the 1987 draft book.

## Schlegel Diagrams

- Preprint on Schlegal Diagrams of Simple 4-Polyhedra are all exact projections of convex 4-polytopes, 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 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://www-iri.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 2|V|-3 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 point-hyperplane and spherical frameworks