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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 ...
Various pseudospheres – surfaces with constant negative Gaussian curvature – can be embedded in 3-D space under the standard Euclidean metric, and so can be made into tangible models. Of these, the tractoid (or pseudosphere) is the best known; using the tractoid as a model of the hyperbolic plane is analogous to using a cone or cylinder as ...
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 ...
From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (), and a surface of positive Gaussian curvature ().In higher dimensions, a manifold may have different curvatures in different directions, described by the Riemann curvature tensor.