Bicorn

File:Bicorn.svg

Bicorn

In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation $$y^2(a^2-x^2)={(x^2+2ay-a^2)}^2.$$ It has two cusps and is symmetric about the y-axis.

History

In 1864, James Joseph Sylvester studied the curve $$y^4-x{y}^3-8x{y}^2+36x^{2}y+16x^2-27x^3=0$$ in connection with the classification of quintic equations; he named the curve a bicorn because it has two cusps. This curve was further studied by Arthur Cayley in 1867.

Properties

File:Bicorn-inf.jpg

A transformed bicorn with a = 1

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 and perform an imaginary rotation on $x$ by substituting $ix/z$ for $x$ and $1/z$ for $y$ in the bicorn curve, we obtain $${(x^2-2az+a^2{z}^2)}^2=x^2+a^2{z}^2.$$ This curve, a limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at $x=\pm i$ and $z=1$.

The parametric equations of a bicorn curve are $$\begin{array}{rl}x& =a\sin \theta \\ y& =a\frac{(2+\cos \theta ){\cos }^{}\theta }{3+{\sin }^2\theta }\end{array}$$ with $-\pi \le \theta \le \pi .$

See also

• List of curves