Clélie

If the sphere is parametrized by

${\displaystyle {\begin{aligned}x&=r\cdot \cos \theta \cdot \cos \varphi \\y&=r\cdot \cos \theta \cdot \sin \varphi \\z&=r\cdot \sin \theta \end{aligned}}}$

and the angles are linearly connected by ${\displaystyle \;\varphi =c\theta }$, then one gets a parametric representation of a Clelia curve:

${\displaystyle {\begin{aligned}x&=r\cdot \cos \theta \cdot \cos c\theta \\y&=r\cdot \cos \theta \cdot \sin c\theta \\z&=r\cdot \sin \theta .\end{aligned}}}$

Any Clelia curve meets the poles at least once.

Spherical spirals: ${\displaystyle \quad c\geq 2\ ,\quad -\pi /2\leq \theta \leq \pi /2}$

A spherical spiral usually starts at the south pole and ends at the north pole (or vice versa).

Viviani's curve: ${\displaystyle \quad c=1\ ,\quad 0\leq \theta \leq 2\pi }$

Trace of a polar orbit of a satellite: ${\displaystyle \quad c\leq 1\ ,\quad \theta \geq 0}$

In case of ${\displaystyle \;c\leq 1\;}$ the curve is periodic, if ${\displaystyle c}$ is rational (see rose). For example: In case of ${\displaystyle \;c=1/n\;}$ the period is ${\displaystyle \;n\cdot 2\pi \;}$. If ${\displaystyle c}$ is a non rational number, the curve is not periodic.

The table (second diagram) shows the floor plans of Clelia curves. The lower four curves are spherical spirals. The upper four are polar orbits. In case of ${\displaystyle \;c=1/3\;}$ the lower arcs are hidden exactly by the upper arcs. The picture in the middle (circle) shows the floor plan of a Viviani's curve. The typical 8-shaped appearance can only achieved by the projection along the x-axis.

• H. A. Pierer: Universal-Lexikon der Gegenwart und Vergangenheit oder neuestes encyclopädisches Wörterbuch der Wissenschaften, Künste und Gewerbe. Verlag H. A. Pierer, 1844, p. 82.

• Clelia., Mathcurve.com..