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 R3." Editors: Jorge Alberto Calvo, Kenneth C. Millett, and Eric J. Rawdon. Contemporary Mathematics 304, American Mathematical Society 2002, pp. 287-311,


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.
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Last update: May 5, 2003.