Hyperbolic Paraboloid — Definition, Formula & Examples

A hyperbolic paraboloid is a saddle-shaped surface that curves upward in one direction and downward in the perpendicular direction. Its cross-sections are parabolas in two directions and hyperbolas in the third.

A hyperbolic paraboloid is a quadric surface defined by an equation of the form $z = \frac{x^2}{a^2} - \frac{y^2}{b^2}$ , where $a$ and $b$ are positive real constants. It is a doubly ruled surface with a saddle point at the origin, and every horizontal cross-section (constant $z$ ) is a hyperbola (or, at $z = 0$ , a pair of intersecting lines).

Key Formula

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

Where:

$a$ = Scaling constant controlling the curvature along the x-axis
$b$ = Scaling constant controlling the curvature along the y-axis
$z$ = Height of the surface at point (x, y)

How It Works

To analyze a hyperbolic paraboloid, set one variable to a constant and examine the resulting cross-section. Setting

z = k

for

k > 0

gives a hyperbola opening along the

x

-axis; for

k < 0

, the hyperbola opens along the

y

-axis. Setting

y = 0

yields an upward-opening parabola

z = x^2/a^2

, while setting

x = 0

yields a downward-opening parabola

z = -y^2/b^2

. The origin is a saddle point — it is simultaneously a minimum along the

x

-direction and a maximum along the

y

-direction.

Worked Example

Problem: Identify the cross-sections of the surface z = x²/4 − y²/9 at z = 0, z = 1, and x = 0.

At z = 0: Set z = 0 and solve. This gives two intersecting lines through the origin.

0 = \frac{x^2}{4} - \frac{y^2}{9} \implies y = \pm \frac{3}{2}x

At z = 1: Set z = 1. The cross-section is a hyperbola opening along the x-axis.

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

At x = 0: Set x = 0. The cross-section is a downward-opening parabola in the yz-plane.

z = -\frac{y^2}{9}

Answer: At z = 0: two lines y = ±(3/2)x. At z = 1: a hyperbola. At x = 0: a downward-opening parabola.

Why It Matters

Hyperbolic paraboloids appear in structural engineering — thin-shell roofs (like those designed by Félix Candela) exploit the saddle shape for strength with minimal material. In multivariable calculus, they are the standard example of a surface with a saddle point, making them essential for understanding the second derivative test for functions of two variables.

Common Mistakes

Mistake: Confusing a hyperbolic paraboloid with an elliptic paraboloid by overlooking the minus sign.

Correction: An elliptic paraboloid has the form z = x²/a² + y²/b² (both terms positive), which opens in one direction like a bowl. The subtraction in z = x²/a² − y²/b² is what creates the saddle shape.