Paper by Erik D. Demaine
- 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.
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
- The paper is \copyright American Mathematical Society.
- The paper is 25 pages.
- The paper is available in PostScript (432k) and PDF (253k).
- See information on file formats.
- [Google Scholar search]
- Related papers:
- InfinitesimallyLocked_CGW2001 (Infinitesimally Locked Linkages with Applications to Locked Trees)
- Related webpages:
- Carpenter's Rule Theorem
See also other papers by Erik Demaine.
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