# Cauchy's Arm Lemma

Joseph O'Rourke
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.