Hippopede
Image:PedalCurve1.gif|500px|right|thumb|Hippopede given as the pedal curve of an ellipse. The equation of this hippopede is:
In geometry, a hippopede 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 and 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 and Eudoxus. For, the hippopede corresponds to the lemniscate of Bernoulli.Definition as spiric sections
[Image:Hippopede02.svg|right|thumb|350px|Hippopedes with a = 1, b = 0.1, 0.2, 0.5, 1.0, 1.5, and 2.0.][Image:Hippopede01.svg|right|thumb|350px|Hippopedes with b = 1, a = 0.1, 0.2, 0.5, 1.0, 1.5, and 2.0.]
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.