Bicorn

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

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] $${\displaystyle y^{2}\left(a^{2}-x^{2}\right)=\left(x^{2}+2ay-a^{2}\right)^{2}.}$$ It has two cusps and is symmetric about the y-axis.^[2]

Bicorn

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.^[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 ${\displaystyle (x=0,z=0)}$. If we move ${\displaystyle x=0}$ and ${\displaystyle z=0}$ to the origin and perform an imaginary rotation on ${\displaystyle x}$ by substituting ${\displaystyle ix/z}$ for ${\displaystyle x}$ and ${\displaystyle 1/z}$ for ${\displaystyle y}$ in the bicorn curve, we obtain $${\displaystyle \left(x^{2}-2az+a^{2}z^{2}\right)^{2}=x^{2}+a^{2}z^{2}.}$$ This curve, a limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at ${\displaystyle x=\pm i}$ and ${\displaystyle z=1}$.^[4]

The parametric equations of a bicorn curve are $${\displaystyle {\begin{aligned}x&=a\sin \theta \\y&=a\,{\frac {(2+\cos \theta )\cos ^{2}\theta }{3+\sin ^{2}\theta }}\end{aligned}}}$$ with ${\displaystyle -\pi \leq \theta \leq \pi .}$

See also

• List of curves