It is the purpose of this paper
to show that a regular C8 space curve Γ in a Euclidean 3-space, whose curvature κ≠0,
can be bent into a piecewise helix (i.e., a curve that is a helix but for a finite number
of corners) in such a way that the piecewise helix remains within a tubular region
about C of arbitrarily small preassigned radius. Moreover, we shall show that the
bending can be carried out in such a way that either (a) the piecewise helix is
circular or (b) the piecewise helix has the same curvature as Γ at corresponding
points except possibly at corners, of (c) if the torsion of Γ is nowhere zero, then the
piecewise helix has the same torsion as Γ at corresponding points except possibly at
corners.
