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 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 letter l ...

(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 Guido Grandi Dom Guido Grandi, O.S.B. Cam. (1 October 1671 – 4 July 1742) was an Italian monk, priest, philosopher, theologian, mathematician, and engineer. Life Grandi was born on 1 October 1671 in Cremona, Italy and christened Luigi ...

after Clelia Borromeo.McTutor Archive
/ref>

Viviani's curve In mathematics, Viviani's curve, also known as Viviani's window, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder that is tangent to the sphere and pa ...

and

spherical spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:polar orbit A polar orbit is one in which a satellite passes above or nearly above both poles of the body being orbited (usually a planet such as the Earth, but possibly another body such as the Moon or Sun) on each revolution. It has an inclination of about ...

s of

satellite A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioisotope ...

s with circular orbits, 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 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

rational Rationality is the quality of being guided by or based on reasons. 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 abili ...

(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 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