Gaussian curvature

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In differential geometry, the Gaussian curvature or Gauss curvature (symbol Κ, named after Carl Friedrich Gauss) of a smooth surface in three-dimensional space at a point is the product of the two principal curvatures, κ1 and κ2, at the given point: K = κ 1 κ 2 . {\displaystyle K=\kappa _{1}\kappa _{2}.} For example, a sphere of radius r has Gaussian curvature ⁠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, meaning that it could in principle be measured by a 2-dimensional being living entirely within the surface, because it depends only on distances that are measured "within" or along the surface, not on the...

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