Clélie

In mathematics, a Clélie or Clelia curve is a curve on a sphere with the property:[1] The curve was named by Luigi Guido Grandi after Clelia Borromeo.

[2][3][4] Viviani's curve and spherical spirals are special cases of Clelia curves.

In practice Clelia curves occur as the ground track of satellites in polar circular orbits, i.e., whose traces on the earth include the poles.

and the trace is a Viviani's curve.

is parametrized in the spherical coordinate system by where

are angles, the longitude and latitude (respectively) of a point on the sphere and these two angles are connected by a linear equation

gives a parametric representation of a Clelia curve: Any Clelia curve meets the poles at least once.

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

Trace of a polar orbit of a satellite:

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 picture in the middle (circle) shows the floor plan of a Viviani's curve.

The typical 8-shaped appearance can only be achieved by the projection along the x-axis.

Clelia curve for c=1/4 with an orientation (arrows) (At the coordinate axes the curve runs upwards, see the corresponding floorplan below, too)
Clelia curves: floor plans of examples, arcs on the lower half of the sphere are dotted. The last four curves (spherical spirals) start at the south pole and end at the northpole. The upper four curves are due to the choice of parameter periodic (see: rose ).