mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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

\varphi

) and the

colatitude In a spherical coordinate system, a colatitude is the complementary angle of a given latitude, i.e. the difference between a right angle and the latitude. Here Southern latitudes are defined to be negative, and as a result the colatitude is a non- ...

(angle

\theta

) then :

\varphi=c\;\theta, \quad c>0

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

c=1

and the trace is a Viviani's curve.

Parametric representation

If the sphere is parametrized by :

\begin
x &= r \cdot \cos \theta \cdot \cos \varphi \\
y &= r \cdot \cos \theta \cdot \sin \varphi \\
z &= r \cdot \sin \theta
\end

and the angles are linearly connected by

\; \varphi=c\theta

, then one gets a parametric representation of a Clelia curve: :

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

Examples

Any Clelia curve meets the poles at least once. Spherical spirals:

\quad c \ge 2 \ , \quad -\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:

\quad c=1\ , \quad 0 \le \theta\le 2\pi

Trace of a polar orbit of a satellite:

\quad c\le 1\ ,\quad \theta\ge 0

In case of

\;c\le 1\;

the curve is ''periodic'', if

c

is rational (see rose). For example: In case of

\; c=1/n\;

the period is

\;n\cdot 2\pi\;

. If

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

\;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.

References

* 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.

External links

*
Clelia.
', ''Mathcurve.com.''. {{DEFAULTSORT:Clelies Spherical curves