Curvatura Gaussiana

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In differential geometry, the Gaussian curvature or Gauss curvature Κ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, κ1 and κ2, at the given point:

https://en.wikipedia.org/wiki/Gaussian_curvature

Concepte de curvatura principal

Podem definir una

Curvatura de Línia. Aleshores bla bla.

Així doncs la curvatura Gaussiana és simplement el producte de les dues curvatures principals

La clau està en que aquesta curvatura és intrínseca. A diferència d’una curvatura de línia normal () que només podíem entendre-la dins d’un espai superior (), és a dir una curvatura extrerna. Una curvatura Gaussiana no necessita per ser mesurada.

Relacionat

El tema està bastant relacionat amb el Teorema de Egregium.

Theorema Egregium

Gauss's Theorema Egregium is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface.

https://en.wikipedia.org/wiki/Theorema_Egregium

Relacionat

Geometria Esfèrica