Clelia curve

Clelia curve for c = 1/4 with direction of passage (the curve is passed through from bottom to top on the coordinate axes, see also the floor plan.)

Clelia curves: Outlines of examples, curve parts on the lower hemisphere are dashed. The lower 4 curves start and end at the poles. The upper curves are periodic due to the choice of (see: rosettes ). ${\displaystyle c}$

A Clelia curve is a curve on a sphere with the property:

• If the spherical surface is described in the usual way by degrees of longitude (angles ) and degrees of latitude (angles ), the following applies ${\displaystyle \varphi }$ ${\displaystyle \theta }$

${\displaystyle \varphi =c\;\theta ,\quad c>0}$.

It was named by the Italian mathematician and Camaldolese Guido Grandi in honor of Countess Clelia Borromeo.

Special cases of Clelia curves appear in geometry as Vivian curves and spherical spirals . They have a practical meaning as ground curves of satellites whose orbits are circular and which fly over the poles. If the satellite orbit is also geosynchronous , then and the ground curve is a Vivian curve. ${\displaystyle c=1}$

Parametric representation

If you describe the spherical surface with the parametric representation

${\displaystyle {\begin{aligned}x&=r\cdot \cos \theta \cdot \cos \varphi \\y&=r\cdot \cos \theta \cdot \sin \varphi \\z&=r\cdot \sin \theta \end{aligned}}}$

and sets , you get a parametric representation of a Clelia curve: ${\displaystyle \;\varphi =c\theta }$

• ${\displaystyle {\begin{aligned}x&=r\cdot \cos \theta \cdot \cos c\theta \\y&=r\cdot \cos \theta \cdot \sin c\theta \\z&=r\cdot \sin \theta .\end{aligned}}}$

Examples

Each Clelia curve hits the two poles at least once.

Spherical spiral: ${\displaystyle \quad c\geq 2\ ,\quad -\pi /2\leq \theta \leq \pi /2}$

A spherical spiral usually begins and ends in one of the poles.

Vivian curve: ${\displaystyle \quad c=1\ ,\quad 0\leq \theta \leq 2\pi }$

Ground curve of a polar orbit (of a satellite): ${\displaystyle \quad c\leq 1\ ,\quad \theta \geq 0}$

In the case , the curve is periodic, if is rational (see rosette). E.g .: the period is the same in the case . If it is not rational, the curve is not periodic. ${\displaystyle \;c\leq 1\;}$ ${\displaystyle c}$ ${\displaystyle \;c=1/n\;}$ ${\displaystyle \;n\cdot 2\pi \;}$ ${\displaystyle c}$

The table only shows the floor plans of the respective curves. The lower four curves are spherical spirals. The upper four curves are periodic ground tracks from polar orbits. In the case , the lower parts of the curve are exactly below the upper parts of the curve. The middle picture shows the floor plan of a Vivian curve. The typical image in the form of an 8 is only obtained when projecting along the axis (elevation). ${\displaystyle \;c=1/3\;}$ ${\displaystyle x}$

Relationship to other curves

• The floor plan of a Clelia curve is a rosette .

Individual evidence

1. ↑ McTutor archive

literature

• HA Pierer : Universal Lexicon of the Present and Past or the latest encyclopedic dictionary of the sciences, arts and crafts. Verlag HA Pierer, 1844, p. 82.

Web links

• Clelia. At: Mathcurve.com. (English).