mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 Longitude (, ) is a geographic coordinate that specifies the east- west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lett ...

(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. In geography, Southern latitudes are defined to be negative, and as a result the colatitude ...

(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 In mathematics, Viviani's curve, also known as Viviani's window, is a Lemniscate, figure-eight-shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder (geometry), cylinder that ...

and

spherical spiral In mathematics, a spiral is a curve which emanates from a point, moving further away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects. Two-dimensional A two-dimension ...

s are special cases of Clelia curves. In practice Clelia curves occur as the

ground track A satellite ground track or satellite ground trace is the path on the surface of a planet directly below a satellite's trajectory. It is also known as a suborbital track or subsatellite track, and is the vertical projection of the satellite's ...

satellite A satellite or an artificial satellite is an object, typically a spacecraft, placed into orbit around a celestial body. They have a variety of uses, including communication relay, weather forecasting, navigation ( GPS), broadcasting, scient ...

s in polar circular orbits, i.e., whose traces on the earth include the poles. If the orbit is a

geosynchronous A geosynchronous orbit (sometimes abbreviated GSO) is an Earth-centered orbit with an orbital period that matches Earth's rotation on its axis, 23 hours, 56 minutes, and 4 seconds (one sidereal day). The synchronization of rotation and orbital ...

one, then

c=1

and the trace is a Viviani's curve.

Parametric representation

If the sphere of radius

r

is parametrized in the

spherical coordinate system In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are * the radial distance along the line connecting the point to a fixed point ...

by :

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

where

\theta

and

\varphi

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

linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...

\; \varphi=c\theta

, then using this equation to replace

\varphi

gives 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

rational Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...

(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 plan In architecture and building engineering, a floor plan is a technical drawing to scale, showing a view from above, of the relationships between rooms, spaces, traffic patterns, and other physical features at one level of a structure. Dimensio ...

s 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 be 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