Bullet-nose curve

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.

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.

References

J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 128–130. ISBN 0-486-60288-5.