Bullet-nose curve
Plane curve of the form a²y² – b²x² = x²y²
Bullet-nose curve with a = 1b = 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 = 0y = 0x = 0z = 0y = 0z = 0genus 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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