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The Gaussian curvature is the product of the two principal curvatures Κ = κ 1 κ 2. The sign of the Gaussian curvature can be used to characterise the surface. If both principal curvatures are of the same sign: κ 1 κ 2 > 0, then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points, the surface ...
Formally, Gaussian curvature only depends on the Riemannian metric of the surface. This is Gauss's celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking. An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r.
In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.. A pseudosphere of radius R is a surface in having curvature −1/R 2 at each point. Its name comes from the analogy with the sphere of radius R, which is a surface of curvature 1/R 2.
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 ...
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 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 ...
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.
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 ...