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For example, a sphere of radius r has Gaussian curvature 1 / r 2 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 property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it ...
It is a singular space (the equator is a singularity), but away from the singularities, it has constant negative Gaussian curvature and therefore is locally isometric to a hyperbolic plane. The name "pseudosphere" comes about because it has a two-dimensional surface of constant negative Gaussian curvature, just as a sphere has a surface with ...
The curvature radius is r = a cot x / y . A great implication that the tractrix had was the study of its surface of revolution about its asymptote: the pseudosphere. Studied by Eugenio Beltrami in 1868, [2] as a surface of constant negative Gaussian curvature, the pseudosphere is a local model of hyperbolic geometry.
The Gaussian curvature of the surface is then given by the second order deviation of the metric at the point from the Euclidean metric. In particular the Gaussian curvature is an invariant of the metric, Gauss's celebrated Theorema Egregium. A convenient way to understand the curvature comes from an ordinary differential equation, first ...
Ricci curvature plays an important role in general relativity, where it is the key term in the Einstein field equations. Ricci curvature also appears in the Ricci flow equation, first introduced by Richard S. Hamilton in 1982, where certain one-parameter families of Riemannian metrics are singled out as solutions of a geometrically-defined ...
Whereas the Gaussian curvature of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive. In spite of its positive curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a model for hyperbolic geometry.
The product k 1 k 2 of the two principal curvatures is the Gaussian curvature, K, and the average (k 1 + k 2)/2 is the mean curvature, H. If at least one of the principal curvatures is zero at every point, then the Gaussian curvature will be 0 and the surface is a developable surface. For a minimal surface, the mean curvature is zero at every ...