en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q4929992

2013-03-02T12:04:23Z

Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q4929992

← Older revision		Revision as of 14:04, 2 March 2013
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	~~Hi there~~. ~~Allow me begin~~ by ~~introducing~~ the ~~author~~, ~~her title~~ is ~~Myrtle Cleary~~. ~~Years in~~ the ~~past we moved~~ to ~~Puerto Rico~~ and ~~my family loves it~~. ~~One~~ of the ~~issues he enjoys most~~ is ~~ice skating but he is struggling~~ to ~~discover time for~~ it. ~~Supervising~~ is ~~my profession~~.<br><br>~~Feel free to surf to my website~~ - [http://www.~~1a-pornotube~~.com/~~user~~/~~WCrompton std testing~~ at ~~home~~]		[[Image:PedalCurve1.gif\|500px\|right\|thumb\|Hippopede (red) given as the [[pedal curve]] of an [[ellipse]] (black). The equation of the hippopede is 4''x''<sup>2</sup>+y<sup>2</sup>=(''x''<sup>2</sup>+y<sup>2</sup>)<sup>2</sup>.]]
			In [[geometry]], a '''hippopede''' (from ἱπποπέδη meaning "horse fetter" in ancient Greek) 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 ''c''>0 and ''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 [[circular algebraic curve\|bicircular]] [[Algebraic curve\|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 ''d''<0 the curve resembles a sideways figure eight, or lemniscate, and is often known as a '''lemniscate of Booth''', after James Booth (1810–1878) who studied them. Hippopedes were also investigated by [[Proclus]] (for whom they are sometimes called '''Hippopedes of Proclus''') and [[Eudoxus of Cnidus\|Eudoxus]]. For ''d'' = −''c'', the hippopede corresponds to the [[lemniscate of Bernoulli]].
			{{-}}

			==Definition as spiric sections==
			[[Image:Hippopede02.svg\|right\|thumb\|350px\|Hippopedes with ''a'' = 1, ''b'' = 0.1, 0.2, 0.5, 1.0, 1.5, and 2.0.]]
			[[Image:Hippopede01.svg\|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 coordinate]]s

			:<math>
			r^2 = 4 b (a- b \sin^{2} \theta)\,
			</math>

			or in [[Cartesian coordinate]]s

			:<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. 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==
			*[http://curvebank.calstatela.edu/hippopede/hippopede.htm "The Hippopede of Proclus" at The National Curve Bank]

			[[Category:Algebraic curves]]
			[[Category:Spiric sections]]

en>RDBury: rm link per WP:ELNO, see Wikipedia talk:WikiProject Mathematics#More links to digi-area.com

2012-02-06T14:37:59Z

rm link per WP:ELNO, see Wikipedia talk:WikiProject Mathematics#More links to digi-area.com

New page

Hi there. Allow me begin by introducing the author, her title is Myrtle Cleary. Years in the past we moved to Puerto Rico and my family loves it. One of the issues he enjoys most is ice skating but he is struggling to discover time for it. Supervising is my profession.<br><br>Feel free to surf to my website - [http://www.1a-pornotube.com/user/WCrompton std testing at home]

Cochleoid - Revision history

en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q4929992

en>RDBury: rm link per WP:ELNO, see Wikipedia talk:WikiProject Mathematics#More links to digi-area.com