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Frenet Frames and Theorems of Jacobi and Milnor for Space Polygons

by Thomas F. Banchoff

This article defines polygonal analogues of normal, binormal, and
tangemt indicatrix curves for smooth space curves. Additionally,
previous theorems relating length of those curves and the enclosed areas
and generalized so that they encompass the polygonal case. An analogue
of Jacobi's theorem (which states that every space curve with
a simple normal indicatrix separates the sphere into two regions
of equal area) is proven, and a polygonal analogue of Milnor's theorem
regarding total absolute torsion of closed space curves is proven.
Finally, the notion of a Frenet frame and tangent developable for
space polygons is established.