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HIPPOPEDE (OF EUDOXUS)

From the Greek "Hippos" horse and "pede" fetter
(a greek horse fetter had this eight-like shape).
Eudoxus of Cnidus (406 - 355 BC): Greek philosopher, mathematician and astronomer. Other names: horse fetter, spherical lemniscate. |

Cartesian equation:
(R is the radius of the sphere and a the distance from the
center
O of the sphere to the axis of the cylinder, with 0 <
a <
R).
Cartesian parametrization: . Rational biquadratic (quartic of the first kind). |

The hippopede of Eudoxus is the intersection between a
sphere and a tangent cylinder of revolution; therefore, it is, at the same
time, a spherical and a cylindrical
curve.

When a = R/2, we get the Viviani
curve.
But in fact, careful consideration of the Cartesian parametrization shows that all hippopedes are images of one another by a scaling in one direction. In particular, all hippopedes are images of the Viviani curve by a scaling in one direction. |

The projections on the planes *xOy*, *xOz* and
*yOz*
are a circle, an arc of a parabola and a lemniscate
of Gerono with equation:,
respectively.

The hippopede has the following dynamic construction,
that corresponds to the historical definition of Eudoxus:

The hippopede is the locus of a point on a great circle
forming an angle α with respect to the equatorial plane and turning
at constant speed around the axis of the poles while the point itself travels
on the great circle at the same speed in the opposite direction. The radius
of the cylinder in which the hippopede is inscribed is then equal to .
With this construction, we get the parametrization: |

The hippopede is therefore a special case of satellite
curve, that corresponds to geosynchronous satellites (with a revolution
period equal to one day).
Here is for example the projection on a planisphere of the visible trajectory of the satellite Syncom 2. (thanks to Thierry Pre) |

Do not mistake for the hippopede of Proclus, also called Booth's curve.

Compare to the bicylindrical
curve obtained with two tangent cylinders.

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© Robert FERRÉOL , Jacques MANDONNET 2018