Total Central Curvature of Curves
by Thomas Banchoff

Consider a space curve conatined in a sphere. The total central curvature is calculated by taking the arithmetic mean of the total absolute curvatures of the curves obtained by central projection from all the points on the bounding sphere.

The aim of this paper is to prove that this definition is equivalent to the classical total absolute curvature of the original space curve. This result is then generalized to curves in n-space. As a corollary, it is shown that a curve on S^3 in E^4 with total absolute curvature < 4 in E^4 can be unknotted in S^3.