Editing Gaussian curvature (section)

==Informal definition==
[[File:Minimal surface curvature planes-en.svg|thumb|300px|right|[[Saddle surface]] with normal planes in directions of principal curvatures]]

At any point on a surface, we can find a [[Normal (geometry)|normal vector]] that is at right angles to the surface; planes containing the normal vector are called ''[[normal plane (geometry)|normal plane]]s''. The intersection of a normal plane and 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 “smooth” surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called the [[principal curvature]]s, call these {{math|''κ''<sub>1</sub>}}, {{math|''κ''<sub>2</sub>}}. The '''Gaussian curvature''' is the product of the two principal curvatures {{math|''Κ'' {{=}} ''κ''<sub>1</sub>''κ''<sub>2</sub>}}.

The sign of the Gaussian curvature can be used to characterise the surface.
*If both principal curvatures are of the same sign: {{math|''κ''<sub>1</sub>''κ''<sub>2</sub> > 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: {{math|''κ''<sub>1</sub>''κ''<sub>2</sub> < 0}}, then the Gaussian curvature is negative and the surface is said to have a hyperbolic or [[saddle point]]. At such points, the surface will be saddle shaped. Because one principal curvature is negative, one is positive, and the normal curvature varies continuously if you rotate a plane orthogonal to the surface around the normal to the surface in two directions, the normal curvatures will be zero giving the [[asymptotic curve]]s for that point.
*If one of the principal curvatures is zero: {{math|''κ''<sub>1</sub>''κ''<sub>2</sub> {{=}} 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]].