Turn or Integral Curvature of a Curve

Abstract

To any curve K in the space E n there can correspond a certain number к(K) , such that 0 ≤ к(K) ≤ ∞, which will be hereafter referred to as a turn or an integral curvature of the given curve K. In a regular case, which is an object of investigation in differential geometry, the turn of a curve is equal to the integral of the curve’s curvature with respect to the arc length. Let us first define the concept of a turn for polygonal lines. In this case it refers, essentially, to elementary geometry. For arbitrary curves a turn is defined by way of approximating a curve with polygonal lines (see below). The fact that in a regular case a turn, in the sense of the definitions given here, is really equal to the integral of the curvature with respect to the arc length will be stated below. There is another possible way of constructing the theory of a curve turn, when the notion of a turn is defined by way of approximating an arbitrary curve by regular curves.