# hippopede

Hippopede (red) given as the pedal curve of an ellipse (black).

The hippopede is a quartic curve described by the equation

(*x *^{2} + *y *^{2})^{ 2} + 4*b*(*b* – *a*)(*x *^{2} + *y *^{2}) – 4*b *^{2}*x *^{2} = 0

where *a* and *b* are positive constants. 'Hippopede' means
literally "foot of a horse." It is often known as the **hippopede
of Proclus**, after Proclus who was
the first to study it, together with Eudoxus (who used it in his theory of how the planets move), and also the **horse
fetter** and the **curve of Booth** because of work
done on it by J. Booth (1810–1878).

Any hippopede is the intersection of a torus (donut) with one of its tangent planes –
that is, a plane parallel to its axis of rotational symmetry. The curve
takes any of a variety of forms depending on where the donut is sliced.
It may be a simple oval, an indented oval or **elliptical lemniscate of Booth** (0 < *b* < *a*),
two isolated circles, or a figure-eight curve
or **hyperbolic lemniscate of Booth** (0 < *a* < *b*).