Plane sections of a paraboloid Paraboloid

hyperbolic paraboloid hyperbolas , parabolas

as plane sections of hyperbolic paraboloid equation

z
=



x

2



a

2




−



y

2



b

2






{\displaystyle z={\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}}

one gets following cases:

a parabola, if plane parallel z-axis equation



u
x
+
v
y
+
w
=
0
,
 
a
u
≠
±
b
v


{\displaystyle ux+vy+w=0,\ au\neq \pm bv}

,
a line, if plane parallel z-axis equation



u
x
+
v
y
+
w
=
0
,
 
a
u
=
±
b
v


{\displaystyle ux+vy+w=0,\ au=\pm bv}

,
a pair of intersecting lines, if plane tangent plane,
a hyperbola, if plane not parallel z-axis , not tangent plane.

remarks: