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.

• Booth J. A Treatise on Some New Geometrical Methods, Longmans, Green, Reader, and Dyer, London, Vol. I (1873) and Vol. II (1877).

External links

• MathWorld description
• 2Dcurves.com description