Two-Sheeted Hyperboloid — Definition, Formula & Examples

A two-sheeted hyperboloid is a 3D surface consisting of two separate bowl-shaped pieces that open in opposite directions along a shared axis. It is formed by the set of all points in space satisfying a specific quadratic equation with two negative terms and one positive term.

A hyperboloid of two sheets is a quadric surface in $\mathbb{R}^3$ defined by an equation of the form $\frac{z^2}{c^2} - \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ , where $a$ , $b$ , and $c$ are positive constants. The surface consists of two connected components (sheets), separated by a gap of $2c$ along the $z$ -axis.

Key Formula

\frac{z^2}{c^2} - \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

Where:

$a$ = Semi-axis length controlling spread in the x-direction
$b$ = Semi-axis length controlling spread in the y-direction
$c$ = Distance from the origin to the vertex of each sheet along the z-axis

How It Works

The two sheets appear because the equation requires

\frac{z^2}{c^2} \geq 1

, which means

z \geq c

z \leq -c

. No part of the surface exists between

z = -c

and

z = c

. Cross-sections parallel to the

xy

-plane (fixing

z = k

with

|k| > c

) are ellipses that grow larger as

|k|

increases. Cross-sections containing the

z

-axis are hyperbolas. The axis along which the two sheets open corresponds to the variable with the positive coefficient.

Worked Example

Problem: Identify the surface given by

z^2 - \frac{x^2}{4} - \frac{y^2}{9} = 1

and find the cross-section at

z = 2

Step 1: Rewrite the equation in standard form to identify the surface type.

\frac{z^2}{1} - \frac{x^2}{4} - \frac{y^2}{9} = 1

Step 2: The positive term is

z^2

and both

x^2

and

y^2

terms are negative, so this is a two-sheeted hyperboloid opening along the z-axis with

a = 2

b = 3

c = 1

. The sheets exist where

|z| \geq 1

Step 3: Substitute

z = 2

to find the cross-section.

4 - \frac{x^2}{4} - \frac{y^2}{9} = 1 \;\Rightarrow\; \frac{x^2}{4} + \frac{y^2}{9} = 3 \;\Rightarrow\; \frac{x^2}{12} + \frac{y^2}{27} = 1

Answer: The surface is a hyperboloid of two sheets opening along the z-axis. At

z = 2

, the cross-section is an ellipse with semi-axes

2\sqrt{3}

and

3\sqrt{3}

Why It Matters

Two-sheeted hyperboloids arise in physics and engineering when modeling satellite dish reflectors and certain gravitational equipotential surfaces. Recognizing and classifying quadric surfaces is a core skill in multivariable calculus and analytic geometry courses.

Common Mistakes

Mistake: Confusing a two-sheeted hyperboloid with a one-sheeted hyperboloid.

Correction: In the standard equation, a two-sheeted hyperboloid has two negative terms and one positive term (e.g.,

z^2 - x^2 - y^2 = 1

), producing two separate pieces. A one-sheeted hyperboloid has two positive terms and one negative term (e.g.,

x^2 + y^2 - z^2 = 1

), producing a single connected surface.