Vol. 32, No. 1, 1970

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The bending of space curves into piecewise helical curves

James McLean Sloss

Vol. 32 (1970), No. 1, 231–239

It is the purpose of this paper to show that a regular C8 space curve Γ in a Euclidean 3-space, whose curvature κ0, 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.

Mathematical Subject Classification
Primary: 53.01
Received: 17 March 1969
Revised: 13 June 1969
Published: 1 January 1970
James McLean Sloss