Gaussian Curvature — Definition, Formula & Examples

Gaussian curvature is a measure of how a surface curves at a given point, defined as the product of the two principal curvatures. A positive value means the surface curves like a sphere, a negative value means it curves like a saddle, and zero means the surface is flat in at least one direction.

For a regular surface $S$ in $\mathbb{R}^3$ , the Gaussian curvature at a point $p$ is $K = \kappa_1 \kappa_2$ , where $\kappa_1$ and $\kappa_2$ are the principal curvatures at $p$ . Equivalently, $K = \det(dN_p)$ , where $dN_p$ is the differential of the Gauss map. By Gauss's Theorema Egregium, $K$ depends only on the first fundamental form and is therefore an intrinsic invariant of the surface.

Key Formula

K = \kappa_1 \, \kappa_2

Where:

$K$ = Gaussian curvature at the point
$\kappa_1$ = First principal curvature (maximum normal curvature)
$\kappa_2$ = Second principal curvature (minimum normal curvature)

How It Works

At each point on a surface, there are two directions along which the normal curvature reaches its maximum and minimum values — these are the principal curvatures

\kappa_1

and

\kappa_2

. Multiplying them gives the Gaussian curvature

K

. If both principal curvatures bend the same way (both positive or both negative),

K > 0

and the surface is locally dome-shaped. If they bend in opposite directions,

K < 0

and the surface is saddle-shaped. When

K = 0

, the surface can be unrolled flat without stretching, like a cylinder.

Worked Example

Problem: Find the Gaussian curvature at any point on a sphere of radius 3.

Step 1: On a sphere of radius

r

, every normal cross-section is a circle of radius

r

. Both principal curvatures equal

1/r

\kappa_1 = \kappa_2 = \frac{1}{3}

Step 2: Multiply the principal curvatures to get the Gaussian curvature.

K = \kappa_1 \, \kappa_2 = \frac{1}{3} \cdot \frac{1}{3} = \frac{1}{9}

Answer: The Gaussian curvature at every point on a sphere of radius 3 is

K = \frac{1}{9}

Why It Matters

Gaussian curvature is central to differential geometry and general relativity, where it generalizes to Riemann curvature on higher-dimensional manifolds. Engineers use it to classify surface patches in CAD modeling — regions with

K > 0

K < 0

, or

K = 0

require different fabrication techniques like stamping, stretching, or bending.

Common Mistakes

Mistake: Confusing Gaussian curvature with mean curvature, which is

H = \frac{\kappa_1 + \kappa_2}{2}

Correction: Gaussian curvature is the product

\kappa_1 \kappa_2

, not the average. A cylinder has

H \neq 0

but

K = 0

, which shows these two measures capture different geometric information.