# Hippopede

In geometry, a **hippopede** (from Ancient Greek ἱπποπέδη, "horse fetter") is a plane curve determined by an equation of the form

- ,

where it is assumed that *c* > 0 and *c* > *d* since the remaining cases either reduce to a single point or can be put into the given form with a rotation. Hippopedes are bicircular rational algebraic curves of degree 4 and symmetric with respect to both the *x* and *y* axes.

## Special casesEdit

When *d* > 0 the curve has an oval form and is often known as an **oval of Booth**, and when *d* < 0 the curve resembles a sideways figure eight, or lemniscate, and is often known as a **lemniscate of Booth**, after 19th-century mathematician James Booth who studied them. Hippopedes were also investigated by Proclus (for whom they are sometimes called **Hippopedes of Proclus**) and Eudoxus. For *d* = −*c*, the hippopede corresponds to the lemniscate of Bernoulli.

## Definition as spiric sectionsEdit

Hippopedes can be defined as the curve formed by the intersection of a torus and a plane, where the plane is parallel to the axis of the torus and tangent to it on the interior circle. Thus it is a spiric section which in turn is a type of toric section.

If a circle with radius *a* is rotated about an axis at distance *b* from its center, then the equation of the resulting hippopede in polar coordinates

or in Cartesian coordinates

- .

Note that when *a* > *b* the torus intersects itself, so it does not resemble the usual picture of a torus.

## See alsoEdit

## ReferencesEdit

- Lawrence JD. (1972)
*Catalog of Special Plane Curves*, Dover. Pp. 145–146. - Booth J.
*A Treatise on Some New Geometrical Methods*, Longmans, Green, Reader, and Dyer, London, Vol. I (1873) and Vol. II (1877). - Weisstein, Eric W. "Hippopede".
*MathWorld*. - "Hippopede" at 2dcurves.com
- "Courbes de Booth" at Encyclopédie des Formes Mathématiques Remarquables