A Generalization of Cauchy's Arm Lemma
By Joseph O'Rourke
To accompany Folding and Unfolding in Computational Geometry:Part III: Ch.1.2: Cauchy's Rigidity Theorem
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Research supported by National Science Foundation grant CCR-9731804. Views expressed are those of the authors, and do not necessarily reflect those of the NSF.

Introduction

Cauchy's arm lemma says that if n-2 consecutive angles of a convex polygon are opened but not beyond pi, keeping all but one edge length fixed and permitting that "missing" edge e to vary in length, then e lengthens (or retains its original length).

The generalization of this lemma permits opening of the angles beyond pi, as far reflex as they were originally convex. The conclusion remains the same: e cannot shorten.

This theorem can be derived from Chern's proof1 of a theorem of Axel Schur2, employing differential geometry, or, independently, by induction [O'R00].

Illustration of Theorem

The theorem is illustrated by a Java applet launched by clicking the button at the bottom of this page. The user is presented with a canvas on which to enter a clockwise convex chain:



The turn angle ranges are displayed in green, and the "forbidden shoulder circle" is drawn in blue. The user may then turn any joint of the chain by double-clicking on it and dragging the next link, which rotates the (red) subchain beyond that link as rigid unit:


The theorem says that no reconfiguration within the allowable turn angle ranges will permit the hand to enter the forbidden circle:


A corollary says that every joint has a correspoding forbidden circle, which can be viewed by selecting:View all concentric circles


Finally, the complete reachability of a chain can be viewed by clicking: Show reachability


Note that the forbidden circle lies external to the reachability region.

For more details, see the paper [O'R00].


(Best viewed on a monitor with resolution greater than 800x600.)

Applet Platform/OS/Browser notes

PlatformOS Browser
Internet ExplorerNetscape Communicator
PCWindows 9x, WindowsNT4IE v4: No reported problems at above resolutions.NC v4.7: No reported problems on above resolution.
Red Hat Linux 6.1Not tested.Netscape v4.61 for Linux. No reported problems.
MACMacOS 8.5IE v4: No reported problems at above resolutions.Some Mac versions of Netscape do not support the Java1.2 event model, which is used extensively on the applet. Therefore, IE is recommended for the Mac
SGIIrix 6.xNot tested.Tested on Netscape 4.07. Reported problems are:
  • Initial Java frame appears as a blank blue frame with no components. Frame needs to be moved on the screen in order to bring up the components.
  • Frame may need resizing.
1  S.S. Chern. Curves and surfaces in Euclidean space. In S.S. Chern, editor, Global Differential Geometry, volume 27 of Studies in Mathematics, pages 99-139. Math. Assoc. Amer., 1989
2  A. Schur. Uber die Schwarzche Extremaleigenschaft des Kreises unter den Kurven konstantes Krummung. Math. Ann., 83:143-148, 1921