Bullet-nose curve

Short description: Plane curve of the form a²y² – b²x² = x²y²

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

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

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

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

then

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

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

References

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 128–130. ISBN 0-486-60288-5. https://archive.org/details/catalogofspecial00lawr/page/128.