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topology > differential topology > Gaussian curvature

geometry > differential geometry > differential geometry of surfaces > Gaussian curvature

Preferred term

Gaussian curvature

Definition(s)

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, $κ 1$ and $κ 2$ , at the given point:
${\\displaystyle K=\\kappa _{1}\\kappa _{2}.}$
The Gaussian radius of curvature is the reciprocal of $Κ$ .
For example, a sphere of radius $r$ has Gaussian curvature $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/r2$ everywhere, and a flat plane and a cylinder have Gaussian curvature zero 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 “within” or along the surface, not on the way it is isometrically embedded in Euclidean space. This is the content of the Theorema egregium.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Gaussian_curvature)

Broader concept(s)

Synonym(s)

Gauss curvature

In other languages

courbure de Gauss

French
courbure totale

URI

http://data.loterre.fr/ark:/67375/PSR-FKRLMSC9-6

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