Clélie

 ( Spherical Curves ) 

In mathematics, a Clélie or Clelia curve is a curve on a sphere with the property:.

If the surface of a sphere is described as usual by the longitude (angle $\backslash varphi$) and the colatitude (angle $\backslash theta$) then

:$\backslash varphi=c\backslash ;\backslash theta,\; \backslash quad\; c>0$.

The curve was named by Luigi Guido Grandi after Clelia Borromeo.. McTutor Archive

Viviani's curve and are special cases of Clelia curves. In practice Clelia curves occur as the ground track of in polar orbit circular orbits, i.e., whose traces on the earth include the poles. If the orbit is a geosynchronous one, then $c=1$ and the trace is a Viviani's curve.

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Parametric representation If the sphere of radius $r$ is parametrized in the spherical coordinate system by

$$

\begin{align} x &= r \cdot \cos \theta \cdot \cos \varphi \\ y &= r \cdot \cos \theta \cdot \sin \varphi \\ z &= r \cdot \sin \theta \end{align} where $\backslash theta$ and $\backslash varphi$ are angles, the longitude and latitude (respectively) of a point on the sphere and these two angles are connected by a linear equation $\backslash ;\; \backslash varphi=c\backslash theta$, then using this equation to replace $\backslash varphi$ gives a parametric representation of a Clelia curve:

$$

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

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Examples Any Clelia curve meets the poles at least once.

Spherical spirals: $\backslash quad\; c\; \backslash ge\; 2\; \backslash \; ,\; \backslash quad\; -\backslash pi/2\backslash le\; \backslash theta\backslash le\; \backslash pi/2$

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

Viviani's curve: $\backslash quad\; c=1\backslash \; ,\; \backslash quad\; 0\; \backslash le\; \backslash theta\backslash le\; 2\backslash pi$

Trace of a polar orbit of a satellite: $\backslash quad\; c\backslash le\; 1\backslash \; ,\backslash quad\; \backslash theta\backslash ge\; 0$

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

The table (second diagram) shows the of Clelia curves. The lower four curves are spherical spirals. The upper four are polar orbits. In case of $\backslash ;c=1/3\backslash ;$ 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 be 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.

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External links

• Clelia., Mathcurve.com..

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