How Helical Can a Closed, Twisted Space Curve Be?

Abstract

Ix'(s)| = 1, for all s. For this parametrization we call x' the (unit) tangent vector field along the curve. Using our observation about helices as motivation, one calls a regular space curve x: -R > *R a generalized helix if its tangent vector field x' makes a constant angle 0 with a fixed unit vector u, where 0 < 0 < 7-/2. Such curves can never be closed; this means that x cannot be a periodic function of its arc length parameter. This is so since the tangent vector field always has a positive component in the direction of u. The question we ask is how close to a generalized helix can a closed space curve be. To make this question more precise we need to consider the differential geometry of space curves. In particular, we first introduce the curvature and torsion of a space curve and then use them to describe what it means for a space curve to be a generalized helix.