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

[math]\displaystyle{ a^2y^2-b^2x^2=x^2y^2 \, }[/math]

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

[math]\displaystyle{ f(z) = \sum_{n=0}^{\infty} {2n \choose n} z^{2n+1} = z+2z^3+6z^5+20z^7+\cdots }[/math]

then

[math]\displaystyle{ y = f\left(\frac{x}{2a}\right)\pm 2b\ }[/math]

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.