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.

If

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

This article is issued from Wikipedia - version of the Thursday, March 14, 2013. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.