Bullet-nose curve

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Bullet-nose curve with a = 1 and b = 1
Bullet-nose curve with a = 1 and b = 1

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 x=0 and y=0, x=0 and z=0, and y=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 = \pm 2b f(\frac{x}{2a})

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