Pseudosphere

related topics
{math, energy, light}
{math, number, function}
{line, north, south}
{@card@, make, design}
{area, part, region}

In geometry, the term pseudosphere is used to describe various surfaces with constant negative gaussian curvature. Depending on context, it can refer to either a theoretical surface of constant negative curvature, to a tractricoid, or to a hyperboloid.

Contents


Theoretical Pseudosphere

In its general interpretation, a pseudosphere of radius R is any surface of curvature −1/R2 (precisely, a complete, simply connected surface of that curvature), by 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[1].

Tractricoid

The term is also used to refer to a certain surface called the tractricoid: the result of revolving a tractrix about its asymptote. As an example, the (half) pseudosphere (with radius 1) is the surface of revolution of the tractrix parametrized by[2]

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 is a two-dimensional surface of constant negative curvature just like a sphere with positive Gauss 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 1639 Christian Huygens found that the volume and the surface area of the pseudosphere are finite,[3] despite the infinite extent of the shape along the axis of rotation. For a given edge radius R, the area is 4πR2 just as it is for the sphere, while the volume is 2/3 πR3 and therefore half that of a sphere of that radius.[4][5]

Full article ▸

related documents
Jerk
Angle excess
Nereid (moon)
433 Eros
Gauss (unit)
Zenith
Solid-state physics
Nichols radiometer
Brightness
Camelopardalis
Umbra
Overwhelmingly Large Telescope
Brightness temperature
Pair production
Field ion microscope
Édouard Roche
Guided ray
Alpha Herculis
Exosphere
Hour angle
Directive gain
Triangulum
Pionium
Hexagon
Andromeda (constellation)
Prandtl number
Wilhelm Wien
Gnomon
Supercluster
Oberon (moon)