Hyperbolic Cylinder — Definition, Formula & Examples

A hyperbolic cylinder is a three-dimensional surface generated by sliding a hyperbola along a straight line perpendicular to the plane of the hyperbola. Its equation in standard form involves only two of the three coordinate variables, meaning the surface extends infinitely along the missing variable's axis.

A hyperbolic cylinder is a quadric surface in $\mathbb{R}^3$ whose standard equation has the form $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ , where $a, b > 0$ . The surface consists of all points $(x, y, z)$ satisfying this equation for every value of $z$ , making it a ruled surface generated by lines parallel to the $z$ -axis through a hyperbola in the $xy$ -plane.

Key Formula

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

Where:

$a$ = Distance from the center to each vertex of the hyperbola along the x-axis
$b$ = Parameter controlling the spread of the hyperbola along the y-axis
$z$ = Free variable (absent from the equation); the surface extends infinitely along the z-axis

How It Works

To identify a hyperbolic cylinder, look for a quadric equation that is missing one variable entirely and has the difference-of-squares structure of a hyperbola. The missing variable tells you the axis along which the surface extends without bound. Every cross-section perpendicular to that axis is the same hyperbola. The surface has two separate "sheets," just like the original hyperbola has two branches.

Worked Example

Problem: Describe the surface defined by the equation

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

in three-dimensional space.

Identify the type: The equation involves only x and y, with a subtraction between squared terms. This is a hyperbolic cylinder extending along the z-axis.

a^2 = 9,\quad b^2 = 4 \implies a = 3,\; b = 2

Describe cross-sections: Every horizontal slice at a fixed z-value gives the same hyperbola in the xy-plane with vertices at

(\pm 3, 0)

and asymptotes

y = \pm \frac{2}{3}x

\frac{x^2}{9} - \frac{y^2}{4} = 1 \quad \text{for all } z

Sketch the surface: The surface consists of two infinite sheets: one where

x \geq 3

and one where

x \leq -3

, each extending without bound in the z-direction.

Answer: The equation represents a hyperbolic cylinder with vertices at

(\pm 3, 0, z)

for all

z

, extending infinitely along the

z

-axis.

Why It Matters

Hyperbolic cylinders appear when classifying quadric surfaces in multivariable calculus and are essential for setting up triple integrals over regions bounded by such surfaces. Engineers encounter them in structural analysis where hyperbolic cross-sections provide specific load-bearing properties.

Common Mistakes

Mistake: Assuming a "cylinder" must have a circular cross-section.

Correction: In analytic geometry, a cylinder is any surface generated by translating a plane curve along a line. The cross-section can be a circle, ellipse, parabola, or hyperbola.