1. Special cases

2. Definition as spiric sections

3. See also

4. References

5. External links

In geometry, a hippopede (from Ancient Greek ἱπποπέδη, "horse fetter") is a plane curve determined by an equation of the form

,

where it is assumed that {{nowrap|c > 0}} and {{nowrap|c > d}} 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 x and y axes.

Special cases

When d>0 the curve has an oval form and is often known as an oval of Booth, and when {{nowrap|d < 0}} 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 (for whom they are sometimes called Hippopedes of Proclus) and Eudoxus. For {{nowrap|1=d = −c}}, the hippopede corresponds to the lemniscate of Bernoulli.

Definition as spiric sections

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.

See also

• List of curves

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).
• {{MathWorld|title=Hippopede|urlname=Hippopede}}
• "Hippopede" at 2dcurves.com
• "Courbes de Booth" at Encyclopédie des Formes Mathématiques Remarquables

External links

• "The Hippopede of Proclus" at The National Curve Bank

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