Paper by Erik D. Demaine

Reference:

Robert Connelly, Erik D. Demaine, and Günter Rote, “Infinitesimally Locked Linkages with Applications to Locked Trees”, in Abstracts from the 11th Annual Fall Workshop on Computational Geometry, New York, New York, November 2–3, 2001.

Abstract:

Recently there has been much interest in linkages (bar-and-joint frameworks) that are locked or stuck in the sense that they cannot be moved into some other configuration while preserving the bar lengths and not crossing any bars. We propose a new algorithmic approach for analyzing whether planar linkages are locked in many cases of interest. The idea is to examine degenerate frameworks in which multiple edges converge to geometrically overlapping configurations. 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 degenerate frameworks to locked frameworks that closely approximate the degenerate frameworks. Our motivation is that most existing approaches to locked linkages are based on approximations to degenerate frameworks. In particular, we show that a previously proposed locked tree in the plane [1] 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 [2] is the best possible.

Length:

The abstract is 2 pages.

Availability:

The abstract is available in PostScript (115k) and gzipped PostScript (42k).

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InfinitesimallyLocked_LasVegas (Infinitesimally Locked Self-Touching Linkages with Applications to Locked Trees)