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2013-09-12T14:34:28Z

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:''For the hat, see [[Bicorne]].''
:''For the mythical beast, see [[Bicorn (legendary creature)]].''

[[Image:Bicorn.svg|thumb|right|226px|Bicorn]]
In [[geometry]], the '''bicorn''', also known as a '''cocked hat curve''' due to its resemblance to a [[bicorne]], is a [[Rational curve|rational]] [[quartic curve]] defined by the equation
:<math>y^2(a^2-x^2)=(x^2+2ay-a^2)^2.</math>
It has two [[cusp (singularity)|cusp]]s and is symmetric about the y-axis.

==History==
In 1864, [[James Joseph Sylvester]] studied the curve
:<math>y^4-xy^3-8xy^2+36x^2y+16x^2-27x^3=0</math>
in connection with the classification of [[quintic equation]]s; he named the curve a bicorn because it has two cusps. This curve was further studied by [[Arthur Cayley]] in 1867.

==Properties==
The bicorn is a [[algebraic curve|plane algebraic curve]] of degree four and [[geometric genus|genus]] zero. It has two cusp singularities in the real plane, and a double point in the [[complex projective plane]] at x=0, z=0 . If we move x=0 and z=0 to the origin substituting and perform an imaginary rotation on x bu substituting ix/z for x and 1/z for y in the bicorn curve, we obtain
:<math>(x^2-2az+a^2z^2)^2 = x^2+a^2z^2.\,</math>
This curve, a [[limaçon]], has an ordinary double point at the origin, and two nodes in the complex plane, at x = ± i and z=1.
[[Image:Bicorn-inf.jpg|thumb|A transformed bicorn with ''a'' = 1]].

The parametric equations of a bicorn curve are:

<math>x = a \sin(\theta)</math> and
<math>y = \frac{\cos^2(\theta) \left(2+\cos(\theta)\right)}{3+\sin^2(\theta)}</math> with <math>-\pi\le\theta\le\pi</math>

==See also==
* [[List of curves]]

==References==
* {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=147–149 }}
* [http://www-history.mcs.st-andrews.ac.uk/history/Curves/Bicorn.html "Bicorn" at The MacTutor History of Mathematics archive]
* {{MathWorld|title=Bicorn|urlname=Bicorn}}
* [http://www.mathcurve.com/courbes2d/bicorne/bicorne.shtml "Bicorne" at Encyclopédie des Formes Mathématiques Remarquables]
* ''The Collected Mathematical Papers of James Joseph Sylvester. Vol. II'' Cambridge (1908) p. 468 ([http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;cc=umhistmath;idno=aas8085.0002.001;view=toc online])

[[Category:Curves]]
[[Category:Algebraic curves]]

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