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

History
In 1864, James Joseph Sylvester studied the curve $$y^4 - xy^3 - 8xy^2 + 36x^2y+ 16x^2 -27x^3 = 0$$ 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
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 and perform an imaginary rotation on $$x$$ by substituting $$ix/z$$ for $$x$$ and $$1/z$$ for $$y$$ in the bicorn curve, we obtain $$\left(x^2 - 2az + a^2 z^2\right)^2 = x^2 + a^2 z^2.$$ This curve, a limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at $$x= \pm i$$ and $$z=1$$.

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