Bicorn

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

History

In 1864, James Joseph Sylvester studied the curve $$y^4-x{y}^3-8x{y}^2+36x^{2}y+16x^2-27x^3=0$$ 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.

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

The parametric equations of a bicorn curve are $$x=a\sin (\theta )$$ and $$y=a\frac{{\cos }^2(\theta )(2+\cos (\theta ))}{3+{\sin }^2(\theta )}$$ with $-\pi \le \theta \le \pi$.

See also

• List of curves

External links

• Weisstein, Eric W.. "Bicorn". http://mathworld.wolfram.com/Bicorn.html.