The types of the development of a space curve at a small scale

In the year 1879, Christian Wiener commissioned eight models illustrating space curves. Space curves are maps \[f : \mathrm{I} \to \mathbb{R}, s \mapsto (f_1(s),f_2(s),f_3(s)), \] where $\mathrm{I}$ is a real interval, and $f$ has derivatives of all orders.

Space curves are characterized by their curvature and their torsion. The curvature gives a measure of how much the curve deviates from being a straight line, the torsion determines how much the behavior of the curve differs from a plane (twisting).

The shape of the curve can be described by a point progressing along the curve and either maintaining its direction of motion (+) or reversing it ($-$), analogously for the turning angles of tangent and osculating plane. There are precisely eight combinations, each of them illustrated by a model. Furthermore, the projections onto the tangential plane ($z=0$), the rectifying plane ($y=0$) and the normal plane ($x=0$) are displayed.

Text written by: Martin Müller and Anne Hein

Translated by: Lea Renner