Ingeometry, 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]
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 planeat. If we move
and
to the origin and perform an imaginary rotation on
by substituting
for
and
for
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
and
.[4]
The parametric equations of a bicorn curve are
with