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These surfaces all have constant Gaussian curvature of 1, but, for either have a boundary or a singular point. do Carmo also gives three different examples of surface with constant negative Gaussian curvature, one of which is pseudosphere. [4] There are many other possible bounded surfaces with constant Gaussian curvature.
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).
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 ...
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.
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 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 ...
Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the ...
The Uniformization theorem guarantees that every Riemann surface is conformally equivalent to one that has constant Gaussian curvature. In the case of a torus, the constant curvature must be zero. Then one defines the "moduli space" of the torus to contain one point for each conformal equivalence class, with the appropriate topology.