In
geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a
bicorne, is a
rationalquartic curve defined by the equation[1]
It has two
cusps and is symmetric about the y-axis.[2]
History
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]
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 . 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]