Bullet-nose curve



In mathematics, a bullet-nose curve is a unicursal quartic curve with three inflection points, given by the equation
 * $$a^2y^2-b^2x^2=x^2y^2 \,$$

The bullet curve has three double points in the real projective plane, at $a = 1$ and $b = 1$, $x = 0$ and $y = 0$, and $x = 0$ and $z = 0$, and is therefore a unicursal (rational) curve of genus zero.

If
 * $$f(z) = \sum_{n=0}^{\infty} {2n \choose n} z^{2n+1} = z+2z^3+6z^5+20z^7+\cdots$$

then
 * $$y = f\left(\frac{x}{2a}\right)\pm 2b\ $$

are the two branches of the bullet curve at the origin.