Gaussian Curvature - Informal Definition

Informal Definition

At any point on a surface we can find a normal vector which is at right angles to the surface. The intersection of a plane containing the normal with the surface will form a curve called a normal section and the curvature of this curve is the normal curvature. For most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures, call these κ₁, κ₂. The Gaussian curvature is the product of the two principal curvatures Κ = κ₁ κ₂.

The sign of the Gaussian curvature can be used to characterise the surface.

• If both principal curvatures are the same sign: κ₁κ₂ > 0, then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points the surface will be dome like, locally lying on one side of its tangent plane. All sectional curvatures will have the same sign.
• If the principal curvatures have different signs: κ₁κ₂ < 0, then the Gaussian curvature is negative and the surface is said to have a hyperbolic point. At such points the surface will be saddle shaped and intersect its tangent plane in two curves. For two directions the sectional curvatures will be zero giving the asymptotic directions.
• If one of the principal curvature is zero: κ₁κ₂ = 0, the Gaussian curvature is zero and the surface is said to have a parabolic point.

Most surfaces will contain regions of positive Gaussian curvature (elliptical points) and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a parabolic line.