# Bicorn

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

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

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

## History

In 1864, James Joseph Sylvester studied the curve

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

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 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

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

This curve, a limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at x = ± i and z=1.

The parametric equations of a bicorn curve are:

${\displaystyle x=a\sin(\theta )}$ and ${\displaystyle y=a{\frac {\cos ^{2}(\theta )\left(2+\cos(\theta )\right)}{3+\sin ^{2}(\theta )}}}$ with ${\displaystyle -\pi \leq \theta \leq \pi }$