Hippopede

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

Hippopedes with a=1, b=0.1, 0.2, 0.5, 1.0, 1.5, and 2.0.

Hippopedes with b=1, a=0.1, 0.2, 0.5, 1.0, 1.5, and 2.0.

$r^{2}=4b(a-b\sin ^{2}\theta )\,$

or in Cartesian coordinates

$\left(x^{2}+y^{2}\right)^{2}+4b(b-a)(x^{2}+y^{2})=4b^{2}x^{2}.$

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 $b=2a$, the hippopede corresponds to the lemniscate of Bernoulli.

References

• Lawrence JD. (1972) Catalog of Special Plane Curves, Dover. Pp.145-146.

• 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