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1982 (Publication #37)

Frenet Frames and Theorem of Jacobi and Milnor for Space Polygons

Jugoslavenska Akademija Znanosti I Umjetnosti , iz Rada 396 (1982), pp. 101-108.

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ABSTRACT

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.