pseudosphere

[′süd·ə‚sfir]

(mathematics)

The pseudospherical surface generated by revolving a tractrix about its asymptote.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Pseudosphere

the surface of constant negative curvature formed by the rotation of a tractrix about its asymptote (see Figure 1). Its name emphasizes both its similarity to and difference from a sphere, which is an example of a surface with constant positive curvature. The pseudosphere is of particular interest because figures drawn on smooth parts of this surface

obey the laws of Lobachevskian non-Euclidean geometry. This fact, established in 1868 by E. Beltrami, was of great importance in the dispute over the reality of Lobachevskian geometry.

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References in periodicals archive

Caption: Figure 3: Upper half of a tractricoid (pseudosphere).

Generalized [alpha]-Attractor Models from Elementary Hyperbolic Surfaces

The origin of [S.sub.(2)] is O = (1, 0, 0, 0, 0, 0) and this space is identified with the intersection of the pseudosphere [[summation].sub.(2)] determined by [I.sub.(2)] with a quadratic cone P known as Plucker or Grassmann relation (invariant under the group action); these constraints are given by [66]

Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups

Then, by using a perturbation method, we were able to analytically calculate the shape of the buckled membrane and show that it is a pseudosphere.

Defects in nematic membranes can buckle into pseudospheres. Physical Review E, v.

Morphology change in nematic membranes induced by defects/Mudanca de morfologia em membranas nematicas induzida por defeitos

In this paper we define nonnull and null pseudospherical Smarandache curves according to the Sabban frame of a spacelike curve lying on pseudosphere in Minkowski 3-space.

The pseudosphere with center at the origin and of radius r = 1 in the Minkowski 3-space [R.sup.3.sub.1] is a quadric defined by

Let [alpha] : I [subset] R [right arrow] [S.sup.2.sub.1] be a curve lying fully in pseudosphere [S.sup.2.sub.1] in [R.sup.3.sub.1].

On pseudospherical Smarandache curves in Minkowski 3-space

Example 7 (the de Sitter pseudosphere is shown in Figure 6).

that is, the de Sitter pseudosphere satisfies condition (3).

On the gauss map of surfaces of revolution with nonlightlike axis in Minkowski 3-space

Then a is congruent to an osculating curve of the second kind with tangential component g([alpha], T) = 0 if and only if [alpha] lies in a pseudosphere [S.sup.3.sub.1](r) in [E.sup.4.sub.1].

Hence a is a null helix lying in the pseudosphere. According to Theorem 4.4, [alpha] is an osculating curve of the second kind.

Some characterizations of null osculating curves in the Minkowski space-time/Nullkooldumiskoverate iseloomustusest Minkowski aegruumis

It is then clear that all complexified Heisenberg (k, n - k) pseudospheres

On Propagation of Sphericity of Real Analytic Hypersurfaces across Levi Degenerate Loci