## Robert Connelly, Erik D. Demaine, and Günter Rote:

# Infinitesimally locked self-touching linkages with applications to locked
trees

*in: "Physical Knots: Knotting, Linking, and Folding Geometric Objects in
***R**^{3}." Editors: Jorge Alberto Calvo, Kenneth C. Millett, and
Eric J. Rawdon. Contemporary Mathematics 304, American Mathematical Society
2002, pp. 287-311,
### Abstract

We propose a new algorithmic approach for analyzing whether planar linkages
are locked in many cases of interest. The idea is to examine *self-touching*
or *degenerate* frameworks in which multiple edges coincide geometrically.
We show how to study whether such frameworks are locked using techniques
from rigidity theory, in particular first-order rigidity and equilibrium
stresses. Then we show how to relate locked self-touching frameworks to
locked frameworks that closely approximate the self-touching frameworks.

Our motivation is that most existing approaches to locked linkages
are based on approximations to self-touching frameworks. In particular,
we show that a previously proposed locked tree in the plane can be easily
proved locked using our techniques, instead of the tedious arguments required
by standard analysis. We also present a new locked tree in the plane
with only one degree-3 vertex and all other vertices degree 1 or 2. This
tree can also be easily proved locked with our methods, and implies that
the result about opening polygonal arcs and cycles (Connelly,
Demaine, and Rote 2002) is the best possible.
Last update: May 5, 2003.