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[1]

y

2

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

[2] In 1864, James Joseph Sylvester studied the curve

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.

[3] 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

If we move

to the origin and perform an imaginary rotation on

by substituting

in the bicorn curve, we obtain

This curve, a limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at

[4] The parametric equations of a bicorn curve are

= a sin ⁡ θ

( 2 + cos ⁡ θ )

cos

⁡ θ

sin

⁡ θ

{\displaystyle {\begin{aligned}x&=a\sin \theta \\y&=a\,{\frac {(2+\cos \theta )\cos ^{2}\theta }{3+\sin ^{2}\theta }}\end{aligned}}}

− π ≤ θ ≤ π .