Hippopede

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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 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.
Hippopedes with b=1, a=0.1, 0.2, 0.5, 1.0, 1.5, and 2.0.
r^2 = 4 b (a- b \sin^{2} \theta)\,

or in Cartesian coordinates

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

[edit] 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).

[edit] External links


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