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Two surfaces which both have constant positive Gaussian curvature but with either an open boundary or singular points. Minding's theorem (1839) states that all surfaces with the same constant curvature K are locally isometric. A consequence of Minding's theorem is that any surface whose curvature is identically zero can be constructed by ...
If a surface has constant Gaussian curvature, it is called a surface of constant curvature. [52] The unit sphere in E 3 has constant Gaussian curvature +1. The Euclidean plane and the cylinder both have constant Gaussian curvature 0. A unit pseudosphere has constant Gaussian curvature -1 (apart from its equator, that is singular).
This is the original form of the sine-Gordon equation, as it was considered in the 19th century in the course of investigation of surfaces of constant Gaussian curvature K = −1, also called pseudospherical surfaces. Consider an arbitrary pseudospherical surface. Across every point on the surface there are two asymptotic curves.
In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature. [1] [2] This includes minimal surfaces as a subset, but typically they are treated as special case. Note that these surfaces are generally different from constant Gaussian curvature surfaces, with the important exception of the sphere.
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 ...
A sphere of radius R has constant Gaussian curvature which is equal to 1/R 2. At the same time, a plane has zero Gaussian curvature. As a corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling. Conversely, the surface of a sphere cannot be unfolded onto a flat plane without distorting the distances.
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 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.