Elliptic Paraboloid — Definition, Formula & Examples

An elliptic paraboloid is a bowl-shaped three-dimensional surface where every cross-section parallel to the base is an ellipse and every cross-section through the central axis is a parabola.

A quadric surface defined by an equation of the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z}{c}$ , where $a$ , $b$ , and $c$ are positive constants. When $a = b$ , the cross-sections are circles, producing a circular paraboloid as a special case.

Key Formula

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

Where:

$a$ = Controls the spread of the surface in the x-direction
$b$ = Controls the spread of the surface in the y-direction
$c$ = Controls the rate at which the surface rises in the z-direction

How It Works

To analyze an elliptic paraboloid, examine its cross-sections (also called traces). Setting

z = k

for a positive constant

k

gives an ellipse

\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{k}{c}

, which grows larger as

k

increases. Setting

y = 0

gives the parabola

z = \frac{c}{a^2}x^2

, and setting

x = 0

gives

z = \frac{c}{b^2}y^2

. The vertex of the surface sits at the origin, and the paraboloid opens upward when

c > 0

. Recognizing these traces is the standard technique for sketching quadric surfaces.

Worked Example

Problem: Describe the cross-sections of the elliptic paraboloid

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

z = 1

and

y = 0

Trace at z = 1: Substitute

z = 1

into the equation to get the horizontal cross-section.

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

Identify the ellipse: This is an ellipse with semi-axis

a = 2

along the

x

-direction and semi-axis

b = 3

along the

y

-direction.

Trace at y = 0: Substitute

y = 0

to get the vertical cross-section in the

xz

-plane.

z = \frac{x^2}{4}

Answer: At

z = 1

, the cross-section is an ellipse

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

. At

y = 0

, the cross-section is an upward-opening parabola

z = \frac{x^2}{4}

Why It Matters

Elliptic paraboloids appear throughout engineering and physics — satellite dishes and parabolic reflectors exploit the surface's focusing property. In multivariable calculus, they serve as a primary example when learning to classify quadric surfaces, find tangent planes, and evaluate surface integrals.

Common Mistakes

Mistake: Confusing an elliptic paraboloid with a hyperbolic paraboloid because both contain the word "paraboloid."

Correction: An elliptic paraboloid has a sum of squared terms (

x^2/a^2 + y^2/b^2

), producing elliptical cross-sections that all open the same way. A hyperbolic paraboloid has a difference (

x^2/a^2 - y^2/b^2

), creating a saddle shape.