hippopede


last updated: 20050423 
Hippopede means literally 'foot of a horse'; the curve is also known as the horse
fetter.
Because of the work done by J. Booth (18101878), the curve is also named the curve
of Booth.
The curve has three forms:
For a < 1, the curve is the oval of Booth.
For a > 1, the curve is the lemniscate of
Booth.
When the parameter a has the value 2, the curve is the lemniscate
of Bernoulli.
Proclus (75 BC) and Eudoxus were the first to investigate the curve.
After Proclus the curve has been named hippopede of Proclus.
In Cartesian coordinates the curve has the form: (x^{2} + y^{2})^{2}
= b x^{2} + y^{2} with b = 1  a.
And we see that the curve is a bicircular
(rational) quartic.
The hippopede can be seen as:
 the cissoid of two
circles of equal size
 a Watt's curve for which the length
of the rod and the distance between the circles are equal
 a special case of the spiric section for which
the plane is tangent to the interior of the torus
