A Generalization of Cauchy’s Arm Lemma with an Application to Curve Development

Cauchy’s arm lemma says that if n− 2 consecutive angles of a convex polygon are opened but not beyond π, 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). We generalize this lemma to permit opening of the angles beyond π, as far reflex as they were originally convex. The conclusion remains the same: e cannot shorten. We apply this to prove that the “slice” curve that is the intersection of a plane with a convex polyhedron develops without self-intersection.