Hippopede

In geometry, a hippopede (from Ancient Greek ἱπποπέδη (hippopédē) '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.

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 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.

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.