## Bicorn

In geometry, the bicorn, also known as a "cocked hat curve" due to its resemblance to a bicorne, is a rational quartic curve.
The quartic curves (from the equation $\displaystyle y^4-xy^3-8xy^2+36x^2y+16x^2-27x^3=0$) were studied by James Joseph Sylvester in 1864 and by Arthur Cayley in 1867. The particular bicorn given by Sylvester and Cayley is a different quartic from the one given here but this one, with a simpler formula, has essentially the same shape. The bicorn is a plane algebraic curve of degree four and 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 $\displaystyle \frac{ix}{z}$ for x and $\displaystyle \frac{1}{z}$ for y in the bicorn curve, we obtain the cartesian equation. The area enclosed by the curve is $\displaystyle A=\frac{1}{3}\left ( 16\sqrt{3}-27 \right )\pi a^2$. The curve has cusps at $\displaystyle \left ( \pm a,0 \right )$. There does not seem to be a simple closed form for the arc length of the curve, but its numerical value is approximately given by $\displaystyle 5.0565300a$. Interactive Graphic: |
Equations:Cartesian Equation:
$\displaystyle y^2\left ( a^-x^2 \right )=\left ( x^2+2ay-a^2 \right )^2$
Parametric Equations:
$\displaystyle x=a \sin t$
$\displaystyle y=\frac{a \cos^2 t \left ( 2+\cos t \right )}{3+\sin^2 t}$ Graph: |

**References:**

"Bicorn." Bicorn. N.p., Jan. 1997. Web. 02 Nov. 2013.

"Bicorn." Wikipedia. Wikimedia Foundation, 11 June 2013. Web. 2 Nov. 2013.

Weisstein, Eric W. "Bicorn." MathWorld. Wolfram, n.d. Web. 2 Nov. 2013.