In geometry, a hippopede is a plane curve determined by an equation of the form
(x2+y2)2=cx2+dy2,
When d > 0 the curve has an oval form and is often known as an oval of Booth, and when 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, the hippopede corresponds to the lemniscate of Bernoulli.
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
r2=4b(a-b\sin2\theta)
or in Cartesian coordinates
(x2+y2)2+4b(b-a)(x2+y2)=4b2x2
Note that when a > b the torus intersects itself, so it does not resemble the usual picture of a torus.