## A Generalization Of Cauchy's Arm Lemma

Cauchy's arm lemma says that if n-2
consecutive angles of a convex polygon are opened but not beyond , 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). A generalization of this
lemma, developed by Prof. O'Rourke,^{1}
permits opening of the angles beyond , as far reflex as they were
originally convex. The conclusion remains the same: e cannot shorten.

I first created a Mathematica® program that
simulated the above situation, a chain with all edge-lengths fixed,
leaving the start and end points of the chain to determine the
length-varying edge. The program generates random points and takes their
convex hull to form a random chain, and then generates random angles
(within the constraints) for each vertex (chain joint). We ran the
program for several million iterations, searching for counterexamples to
what was then a conjecture. For every random open chain the result was
the same: e did not shorten (see Fig. 1). Later Prof. O'Rourke proved the
theorem by induction,^{1} and also
discovered it could be derived from Chern's proof^{2} of a theorem of Axel Schur,^{3} employing differential geometry.

Satisfied with the results produced by the
Mathematica® program, we wanted to create a more interactive method of
illustrating the theorem. We decided that a Java applet would provide the
appropriate interactivity. Using Java 1.2's MVC (Model View Control), I
developed an applet
that would allow user input of the chain. Once the chain is drawn,
the applet displays the angle constraints for each vertex (joint) and the
"forbidden circle" (see Fig. 2), the area where the chain hand cannot
penetrate (i.e., penetration would mean shortening of the length-varying
edge e). The user can then change the angles by double-clicking on a
vertex (joint) and dragging the chain from that vertex onward (see Fig.
3). The applet also illustrates other geometric aspects of the theorem,
including the "reachability region" composed of the union of many circle
arcs (see Fig. 4). *(Supported by Schultz grant and NSF.)*

*Fig. 1: Unfolding of polygon created by
Mathematica®* |
*Fig. 2: Convex chain with forbidden circle and angle
constraints.* |

*Fig. 3: Open chain from Java applet* |
*Fig. 4: The "reachability region."* |

^{1}J. O'Rourke, "On the Development of the
Intersection of a Plane with a Polytope." Smith Tech. Rep. 068, Jun 2000.
LANL arXive cs.CG/0006035. Submitted for publication.

^{2}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

^{3}A. Schur. Uber die Schwarzche
Extremaleigenschaft des Kreises unter den Kurven konstantes Krummung.
*Math. Ann.*, 83:143-148, 1921