Hyperboloid of TWO Sheets
a.k.a “Two Opposing Facing Bowls”
𝑥2 𝑦2 𝑧2
− 2− 2+ 2=1
𝑎
𝑏
𝑐
The Shape Looks like two opposite facing
bowls, giving it the nickname, “two opposite
facing bowls”
Example:
𝑥 2 (𝑦 − 2)2 (𝑧 + 1)2
−
−
=1
4
25
9
Some Features that tell you it’s a
Hyperboloid of Two Sheets:
This Equation represents a Hyperboloid of
Two Sheets with a center of (0,2,-1) and the
“bowls” will be flaring out in the X direction.
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

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In the Equation:
o The variables all equal 1
o All three Variables are squared
o 2 of the variables are negative
Horizontal Traces are ellipses when
|z|>c are ellipses
Vertical traces are Hyperbolas
The XY Plane is an Axis of Symmetry
for the graphs of Hyperboloids of
TWO sheets
Hollow
The Graph looks like two opposing
bowls
Trace for x=0
−
(𝑦−2)2
25
−
(𝑧+1)2
9
=1
This trace is impossible because 2
negative numbers can not add up to a positive
1
Trace for y=0
𝑥2
4
−
(𝑧+1)2
9
=1
This trace results in hyperbolas
Trace for z=0
𝑥2
4
−
(𝑦−2)2
25
=1
This trace results in hyperbolas as well
Underview:
Overview: