Hippopede

A hippopede (meaning "horse fetter" in ancient Greek) is plane curve obeying the equation in polar coordinates

$r=1-a\sin ^{2}\theta$

or in Cartesian coordinates

$\left(x^{2}+y^{2}\right)^{2}=y^{2}+(1-a)x^{2}$

The hippopede is a spiric section in which the intersecting plane is tangent to the interior of the torus. It was investigated by Proclus, Eudoxus and, more recently, J. Booth (1810-1878). For $a=2$, the hippopede corresponds to the lemniscate of Bernoulli.

References

• Lawrence JD. (1972) Catalog of Special Plane Curves, Dover.

External link

• Website for 2D curves