Journal for Geometry and Graphics, Vol. 2, No. 1, pp. 17 - 31 (1998)

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

Richard Koch, Christoph Engelhardt

Zentrum Mathematik, Munich University of Technology,
Arcisstr. 21, D-80290 München, Germany,
Hans-Pfann-Str. 74, D-81825 München, Germany,

Abstract: A closed regular curve of class $C^r$ ($r \geq 2$) in the Euclidean 3-space having constant curvature $\kappa_0>0$ is called closed $\kappa_0$-curve. We present various examples of nonplanar closed $\kappa_0$-curves of class $C^2$, which are composed of $n$ arcs of circular helices. The construction of $c$ starts from the spherical image (= tangent indicatrix) $c^\star$ of $c$, which then has to be a closed regular curve of class $C^1$ on the unit sphere $S^2$ consisting of $n$ circular arcs and having the center $O^\star$ of $S^2$ as its center of gravity. The case $c^\star \subseteq S^2 \cap \Pi$ is studied in detail, assuming that $\Pi$ is a cube, or, more generally, a regular polyhedron the edges of which are tangent to $S^2$. In order to describe and to visualize the curves $c^\star$ and $c$, and to derive $c$ from $c^\star$, projection methods of Descriptive Geometry are used.

Keywords: closed (composite) space curve, constant curvature, circular helix, spherical image, tangent indicatrix, center of gravity, regular polyhedron

Classification (MSC2000): 53A04; 51M20, 51N05

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