Жалпы қисықтығы тұрақты геометриялық бет (Жалпы қисықтығы тұрақты геометриялық бет)

Geometric surface
In geometry, a pseudosphere is a surface with constant negative Gaussian curvature. A pseudosphere of radius R is a surface in having curvature −1/R2 at each point. Its name comes from the analogy with the sphere of radius R, which is a surface of curvature 1/R2. The term was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry. TOC

The same surface can be also described as the result of revolving a tractrix about its asymptote. For this reason the pseudosphere is also called tractroid. As an example, the (half) pseudosphere (with radius 1) is the surface of revolution of the tractrix parametrized by

It is a singular space (the equator is a singularity), but away from the singularities, it has constant negative Gaussian curvature and therefore is locally isometric to a hyperbolic plane. The name "pseudosphere" comes about because it has a two dimensional surface of constant negative Gaussian curvature, just as a sphere has a surface with constant positive Gaussian curvature. Just as the sphere has at every point a positively curved geometry of a dome the whole pseudosphere has at every point the negatively curved geometry of a saddle. As early as 1693 Christiaan Huygens found that the volume and the surface area of the pseudosphere are finite, despite the infinite extent of the shape along the axis of rotation. For a given edge radius R, the area is 4πR^(2) just as it is for the sphere, while the volume is and therefore half that of a sphere of that radius. The pseudosphere is an important geometric precursor to mathematical fabric arts and pedagogy.

In some sources that use the hyperboloid model of the hyperbolic plane, the hyperboloid is referred to as a pseudosphere. This usage of the word is because the hyperboloid can be thought of as a sphere of imaginary radius, embedded in a Minkowski space.

A pseudospherical surface is a generalization of the pseudosphere. A surface which is piecewise smoothly immersed in with constant negative curvature is a pseudospherical surface. The tractroid is the simplest example. Other examples include the Dini's surfaces, breather surfaces, and the Kuen surface.