Quadratic Surface — Definition, Formula & Examples

A quadratic surface is a surface in three-dimensional space defined by a second-degree polynomial equation in three variables

x

y

, and

z

. Common examples include ellipsoids, hyperboloids, paraboloids, and cones.

A quadratic surface (or quadric surface) is the set of all points $(x, y, z) \in \mathbb{R}^3$ satisfying an equation of the form $Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0$ , where $A, B, C, D, E, F, G, H, I, J$ are real constants and at least one of $A$ through $F$ is nonzero.

Key Formula

Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0

Where:

$A, B, C$ = Coefficients of the squared terms
$D, E, F$ = Coefficients of the cross-product terms
$G, H, I$ = Coefficients of the linear terms
$J$ = Constant term

How It Works

To identify a quadratic surface, you examine which squared terms and cross terms appear in its equation. By completing the square and rotating axes (to eliminate cross terms), any quadric can be reduced to one of six standard forms: ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, elliptic paraboloid, hyperbolic paraboloid, and elliptic cone. Degenerate cases—such as a single point, a line, or a pair of planes—can also arise. A useful technique is to take cross-sections by setting one variable equal to a constant and recognizing the resulting conic section (ellipse, hyperbola, or parabola) in the remaining two variables.

Worked Example

Problem: Identify the quadratic surface given by

x^2 + 4y^2 + 9z^2 = 36

Step 1: Divide every term by 36 to write the equation in standard form.

\frac{x^2}{36} + \frac{y^2}{9} + \frac{z^2}{4} = 1

Step 2: Recognize this as the standard form of an ellipsoid

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

with

a = 6

b = 3

c = 2

a = 6,\; b = 3,\; c = 2

Step 3: Cross-sections confirm this: setting

z = 0

gives the ellipse

\frac{x^2}{36} + \frac{y^2}{9} = 1

, and every other horizontal slice is also an ellipse (or a single point at

z = \pm 2

Answer: The surface is an ellipsoid centered at the origin with semi-axes of length 6, 3, and 2 along the

x

y

-, and

z

-axes respectively.

Why It Matters

Quadratic surfaces appear throughout multivariable calculus when evaluating triple integrals, optimizing functions of several variables, and parametrizing surfaces. In physics and engineering, they model satellite dish shapes (paraboloids), cooling tower profiles (hyperboloids), and antenna reflectors.

Common Mistakes

Mistake: Confusing a cone with a hyperboloid of one sheet because both have hyperbolic cross-sections.

Correction: Check whether the equation has the form

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

(cone, equals zero on the right after rearranging) versus

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

(hyperboloid of one sheet). A cone passes through the origin; a hyperboloid of one sheet does not.