Paper by Erik D. Demaine

Reference:
Erik D. Demaine and Sarah Eisenstat, “Expansive Motions for d-Dimensional Open Chains”, in Proceedings of the 23rd Canadian Conference on Computational Geometry (CCCG 2011), Toronto, Ontario, Canada, August 10–12, 2011, to appear.

Abstract:
We consider the problem of straightening chains in d ≥ 3 dimensions, possibly embedded into higher dimensions, using expansive motions. For any d ≥ 3, we show that there is an open chain in d dimensions that is not straight and not self-touching yet has no expansive motion. Furthermore, for any Δ > 0 and d ≥ 3, we show that there is an open chain in d dimensions that cannot be straightened using expansive motions when embedded into ℝd × [−Δ, Δ] (a bounded extra dimension). On the positive side, we prove that any open chain in d ≥ 2 dimensions can be straightened using an expansive motion when embedded into ℝd + 1 (a full extra dimension).

Length:
The paper is 6 pages.

Availability:
The paper is available in PDF (308k).
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Last updated March 21, 2017 by Erik Demaine.