Quadric — Definition, Formula & Examples

A quadric (or quadric surface) is the three-dimensional analogue of a conic section — a surface defined by a second-degree polynomial equation in three variables

x

y

, and

z

A quadric surface is the set of all points $(x, y, z) \in \mathbb{R}^3$ satisfying a general second-degree equation $Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0$ , where the coefficients $A$ through $J$ are real constants and at least one of $A, B, C, D, E, F$ 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 mixed product terms
$G, H, I$ = Coefficients of the linear terms
$J$ = Constant term

How It Works

Just as conic sections (ellipses, parabolas, hyperbolas) arise from second-degree equations in two variables, quadric surfaces arise from second-degree equations in three variables. The six standard non-degenerate quadric surfaces are the ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, elliptic paraboloid, hyperbolic paraboloid, and elliptic cone. You classify a quadric by completing the square in each variable and reducing the equation to standard form. The signs and structure of the resulting terms determine which type of surface you have.

Worked Example

Problem: Identify the quadric surface given by

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

Divide by 36: 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

Recognize the form: This matches

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

with

a = 6

b = 3

c = 2

. All three terms are positive and sum to 1.

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

Quadric surfaces appear throughout multivariable calculus when you evaluate triple integrals, parametrize surfaces, or compute flux. They also model real objects in physics and engineering — satellite dishes are paraboloids, cooling towers are hyperboloids, and planets are roughly ellipsoids.

Common Mistakes

Mistake: Confusing a quadric surface with a conic section.

Correction: Conic sections are curves in 2D defined by second-degree equations in two variables. Quadrics are surfaces in 3D defined by second-degree equations in three variables. A cross-section of a quadric taken by a plane is typically a conic section.