Bicorn

For the hat, see Bicorne. For the mythical beast, see Bicorn and Chichevache.

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

Bicorn

$$y^{2}\left(a^{2}-x^{2}\right)=\left(x^{2}+2ay-a^{2}\right)^{2}.$$

It has two cusps and is symmetric about the y-axis.

History

In 1864, James Joseph Sylvester studied the curve

$$y^{4}-xy^{3}-8xy^{2}+36x^{2}y+16x^{2}-27x^{3}=0$$

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

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

$$\left(x^{2}-2az+a^{2}z^{2}\right)^{2}=x^{2}+a^{2}z^{2}.$$

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

$$x=a\sin(\theta )$$

and

$$y=a{\frac {\cos ^{2}(\theta )\left(2+\cos(\theta )\right)}{3+\sin ^{2}(\theta )}}$$

with $-\pi \leq \theta \leq \pi$.

See also

• List of curves

External links

• Weisstein, Eric W. "Bicorn". MathWorld.