{{Short description|Class of quartic plane curves}} [[Image:PedalCurve1.gif|500px|right|thumb|Hippopede (red) given as the pedal curve of an ellipse (black). The equation of this hippopede is: <math>4x^2 + y^2 = (x^2 + y^2)^2</math>]]
In geometry, a '''hippopede''' ({{ety|grc|''ἱπποπέδη'' (hippopédē)|horse fetter}}) is a plane curve determined by an equation of the form :<math>(x^2+y^2)^2=cx^2+dy^2,</math> 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. {{-}}
==Definition as spiric sections== right|thumb|350px|Hippopedes with ''a'' = 1, ''b'' = 0.1, 0.2, 0.5, 1.0, 1.5, and 2.0. right|thumb|350px|Hippopedes with ''b'' = 1, ''a'' = 0.1, 0.2, 0.5, 1.0, 1.5, and 2.0. 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
:<math> r^2 = 4 b (a - b \sin^{2}\! \theta) </math>
or in Cartesian coordinates
:<math>(x^2+y^2)^2+4b(b-a)(x^2+y^2)=4b^2x^2</math>.
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 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]
Category:Quartic curves Category:Spiric sections