Paper by Erik D. Demaine

Reference:

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 of Geometric Objects in R³, edited by Jorge Calvo, Kenneth Millett, and Eric Rawdon, 2002, pages 287–311, American Mathematical Society. Collection of papers from the Special Session on Physical Knotting and Unknotting at the AMS Spring Western Section Meeting, Las Vegas, Nevada, April 21–22, 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 self-touching or 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 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 [BDD⁺02] 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 [CDR02] is the best possible.

Copyright:

The paper is \copyright American Mathematical Society.

Length:

The paper is 25 pages.

Availability:

The paper is available in PostScript (432k) and PDF (253k).

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Related papers:

InfinitesimallyLocked_CGW2001 (Infinitesimally Locked Linkages with Applications to Locked Trees)

Related webpages:

Carpenter's Rule Theorem