Closed Space Curves of Constant Curvature Consisting of Arcs of Circular Helices

R. Koch ; C. Engelhart

Journal for geometry and graphics, Tome 2 (1998) no. 1, pp. 017-032

Résumé

A closed regular curve of class Cr (r >=2) in the Euclidean 3-space having constant curvature kappa0>0 is called closed kappa0-curve. We present various examples of nonplanar closed kappa0-curves of class C2, which are composed of n arcs of circular helices. The construction of c starts from the spherical image (= tangent indicatrix) c* of c, which then has to be a closed regular curve of class C1 on the unit sphere S2 consisting of n circular arcs and having the center O* of S2 as its center of gravity. The case when c* is a subset of the intersection of S2 and Pi is studied in detail, assuming that Pi is a cube, or, more generally, a regular polyhedron the edges of which are tangent to S2. In order to describe and to visualize the curves c* and c, and to derive c from c*, projection methods of Descriptive Geometry are used.