Clélie

In mathematics, a Clélie or Clelia curve is a curve on a sphere with the property:[1]

  • If the surface of a sphere is described as usual by the longitude (angle ) and the colatitude (angle ) then
.
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).

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 polar orbits of satellites with circular orbits, whose traces on the earth include the poles. If the orbit is a geosynchronous one, then and the trace is a Viviani's curve.