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In differential geometry, the Gaussian curvature or Gauss curvature ''Κ'' of a surface at a point is the product of the principal curvatures, ''κ''1 and ''κ''2, at the given point: : For example, a sphere of radius ''r'' has Gaussian curvature ''1/r2'' 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 hyperboloid or the inside of a torus. Gaussian curvature is an ''intrinsic'' measure of curvature, depending only on distances that are measured on the surface, not on the way it is isometrically embedded in any space. This is the content of the Theorema egregium. Gaussian curvature is named after Carl Friedrich Gauss who published the Theorema egregium in 1827. ==Informal definition== At any point on a surface we can find a normal vector which is at right angles to the surface; planes containing the normal are called ''normal planes''. 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 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. For two directions the sectional curvatures will be zero giving the asymptotic directions. *If one of the principal curvatures 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gaussian curvature」の詳細全文を読む スポンサード リンク
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