Bullet-nose curve

In mathematics, a bullet-nose curve is a unicursal quartic curve with three inflection points, given by the equation

${\displaystyle a^{2}y^{2}-b^{2}x^{2}=x^{2}y^{2}\,}$^[1]

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

${\displaystyle f(z)=\sum _{n=0}^{\infty }{2n \choose n}z^{2n+1}=z+2z^{3}+6z^{5}+20z^{7}+\cdots }$

then

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

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

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