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 and 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 cases.
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.
Definition as spiric sections.
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.