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Gaussian curvature

From Biocrawler, the free encyclopedia.

Gaussian curvature of a point on a surface is the product of the principal curvatures, k₁ and k₂ of the given point.

Symbolically, the Gaussian curvature K is defined as

K = k 1 k 2

.

It is also given by

$K = \frac{\langle (\nabla_2 \nabla_1 - \nabla_1 \nabla_2)\mathbf{e}_1, \mathbf{e}_2\rangle}{\det g}$

where $\nabla_i = \nabla_{{\mathbf e}_i}$ is the covariant derivative and g is the metric tensor.

At a point p on a regular surface in $\mathbb{R}^3$ , the Gaussian curvature is also given by

$K(\mathbf{p}) = \det(S(\mathbf{p}))$

where S is the shape operator.

Retrieved from "http://www.biocrawler.com/encyclopedia/Gaussian_curvature"

Categories: Differential geometry

Wikipedia (http://en.wikipedia.org/wiki/Main_Page) Gaussian_curvature (http://en.wikipedia.org/wiki/Gaussian_curvature) version history (http://en.wikipedia.org/w/index.php?title=Gaussian_curvature&action=history) GNU Free Documentation Lizenz (http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License) CC-by-sa (http://creativecommons.org/licenses/by-sa/2.5/)

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