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Bullet-nose curve

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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 {\displaystyle 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 ) = ∑ n = 0 ∞ ( 2 n n ) z 2 n + 1 = z + 2 z 3 + 6 z 5 + 20 z 7 + ⋯ {\displaystyle f(z)=\sum _{n=0}^{\infty }{2n \choose n}z^{2n+1}=z+2z^{3}+6z^{5}+20z^{7}+\cdots }

then

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

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

References

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