# Resources in Rigidity Theory

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.

## Parallel Drawing in 3D

A set of preprint papers with connections to parallel drawings of configurations in dimension 2 and in higher dimensions:

• 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 Some of this was superseded by the work of Rybnykov, but the explicit connections to the reciprocals of Rankin is still of interest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 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

## Global Rigidity

There are a few older preprints on global rigidity which may be of continuing interest.

## 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, Shin-ichi and Whiteley, 2018

• A movie of a single path on one variable radius circle
• A movie of a single path linking 2 variable radii circles
• A movie of a cycle fixing one radius while extended path allows second radius to change

#### Movies for finite flexes

• An animation of Bottema's Mechanism with K44 in the plane with C2V symmetry,
• An animation of Bottema's Mechanism with K44 in the Minkowski Plane with C2V symmetry.
• An animation of a Desargue's Mechanism in Minkowski Space

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