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
\varphi
\theta
\varphi=c \theta, c>0
The curve was named by Luigi Guido Grandi after Clelia Borromeo.[2] [3]
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. If the orbit is a geosynchronous one, then
c=1
If the sphere of radius
r
\begin{align} x&=r ⋅ \cos\theta ⋅ \cos\varphi\\ y&=r ⋅ \cos\theta ⋅ \sin\varphi\\ z&=r ⋅ \sin\theta \end{align}
\theta
\varphi
\varphi=c\theta
\varphi
\begin{align} x&=r ⋅ \cos\theta ⋅ \cosc\theta\\ y&=r ⋅ \cos\theta ⋅ \sinc\theta\\ z&=r ⋅ \sin\theta. \end{align}
Any Clelia curve meets the poles at least once.
Spherical spirals:
c\ge2 , -\pi/2\le\theta\le\pi/2
A spherical spiral usually starts at the south pole and ends at the north pole (or vice versa).
Viviani's curve:
c=1 , 0\le\theta\le2\pi
Trace of a polar orbit of a satellite:
c\le1 , \theta\ge0
In case of
c\le1
c
c=1/n
n ⋅ 2\pi
c
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
c=1/3