## Gaussian curvature |

In **Gaussian curvature** or **Gauss curvature** *Κ* of a *κ*_{1} and *κ*_{2}, at the given point:

For example, a sphere of radius *r* has Gaussian curvature *1/r ^{2}* everywhere, and a flat plane and a cylinder have Gaussian curvature 0 everywhere. The Gaussian curvature can also be negative, as in the case of a

Gaussian curvature is an *intrinsic* measure of

Gaussian curvature is named after

- informal definition
- relation to geometries
- further informal discussion
- alternative definitions
- total curvature
- important theorems
- surfaces of constant curvature
- alternative formulas
- see also
- references
- books
- external links

At any point on a surface, we can find a * normal planes*. The intersection of a normal plane and the surface will form a curve called a

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

- If both principal curvatures are of the same sign:
*κ*_{1}*κ*_{2}> 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:
*κ*_{1}*κ*_{2}< 0, then the Gaussian curvature is negative and the surface is said to have a hyperbolic orsaddle 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 theasymptotic curves for that point. - If one of the principal curvatures is zero:
*κ*_{1}*κ*_{2}= 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.

Other Languages

català: Curvatura gaussiana

Deutsch: Gaußsche Krümmung

eesti: Täiskõverus

español: Curvatura de Gauss

français: Courbure de Gauss

한국어: 가우스 곡률

Bahasa Indonesia: Lengkungan Gauss

italiano: Curvatura gaussiana

Nederlands: Gaussiaanse kromming

日本語: ガウス曲率

polski: Krzywizna Gaussa

português: Curvatura gaussiana

русский: Кривизна Гаусса

slovenščina: Gaussova ukrivljenost

svenska: Gausskrökning

Türkçe: Gauss eğriliği

українська: Кривина Гауса

Tiếng Việt: Độ cong Gauss

中文: 高斯曲率