Bullet-nose curve

Bullet-nose curve with Template:Math and Template:Math

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

$a^2{y}^2-b^2{x}^2=x^2{y}^2$

The bullet curve has three double points in the real projective plane, at Template:Math and Template:Math, Template:Math and Template:Math, and Template:Math and Template:Math, and is therefore a unicursal (rational) curve of genus zero.

If

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

then

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

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

References

• Template:Cite book