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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)


 
 
The two arches of the vault opposite, in San Fedele in Milan are two arches of a half-hippopede, as the figure on the right shows (intersection of a half-sphere and a cylinder):

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