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
It has two cusps and is symmetric about the y-axis.
History
In 1864, James Joseph Sylvester studied the curvein 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
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 and to the origin and perform an imaginary rotation on by substituting for and for in the bicorn curve, we obtainThis curve, a limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at and.
The parametric equations of a bicorn curve are
with