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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 ...
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.
It determines whether a surface is locally convex (when it is positive) or locally saddle-shaped (when it is negative). 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 ...
In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface of constant negative gaussian curvature immersed in .This theorem answers the question for the negative case of which surfaces in can be obtained by isometrically immersing complete manifolds with constant curvature.
In the case of two-dimensional manifolds, negativity of the Ricci curvature is synonymous with negativity of the Gaussian curvature, which has very clear topological implications. There are very few two-dimensional manifolds which fail to admit Riemannian metrics of negative Gaussian curvature.
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 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.