Hippopede#Special cases
{{short description|1=Plane curves of the form (x² + y²)² = cx² + dy²}}
Image:PedalCurve1.gif of an ellipse (black). The equation of this hippopede is: ]]
In geometry, a hippopede ({{ety|grc|ἱπποπέδη (hippopédē)|horse fetter}}) is a plane curve determined by an equation of the form
:
where it is assumed that {{math|c > 0}} and {{math|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 {{mvar|x}} and {{mvar|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.
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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
:
r^2 = 4 b (a - b \sin^{2}\! \theta)
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
References
- Lawrence JD. (1972) Catalog of Special Plane Curves, Dover Publications. 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}}
- [http://www.2dcurves.com/quartic/quartich.html "Hippopede" at 2dcurves.com]
- [http://www.mathcurve.com/courbes2d/booth/booth.shtml "Courbes de Booth" at Encyclopédie des Formes Mathématiques Remarquables]
External links
- [https://web.archive.org/web/20090318143501/http://curvebank.calstatela.edu/hippopede/hippopede.htm "The Hippopede of Proclus" at The National Curve Bank]