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^{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}^{\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

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

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