Cylinder between pencils of elliptic and hyperbolic paraboloids Paraboloid

elliptic paraboloid, parabolic cylinder, hyperbolic paraboloid

the pencil

z
=

x

2


+



y

2



b

2




,
 
b
>
0
,


{\displaystyle z=x^{2}+{\frac {y^{2}}{b^{2}}},\ b>0,}

of elliptic paraboloids , pencil

z
=

x

2


−



y

2



b

2




,
 
b
>
0
,


{\displaystyle z=x^{2}-{\frac {y^{2}}{b^{2}}},\ b>0,}

of hyperbolic paraboloids approach same surface

z
=

x

2




{\displaystyle z=x^{2}}

for



b
→
∞


{\displaystyle b\rightarrow \infty }

, parabolic cylinder (see image).