The standard projection model for the atom probe is an emitter geometry that is based upon a revolution of a conic section, such as a sphere, hyperboloid or paraboloid.

Out of this work came another of Wren's important mathematical results, namely that the hyperboloid of revolution is a ruled surface.

A hyperboloid of revolution of one sheet can be obtained by revolving a hyperbola around its semi-minor axis.

A hyperboloid of one sheet is a doubly ruled surface ; if it is a hyperboloid of revolution, it can also be obtained by revolving a line about a skew line.

If one rotates a line L around another line L ' skew but not perpendicular to it, the surface of revolution swept out by L is a hyperboloid of one sheet.

He then supposed this cylindrical column of water to be divided into two parts, the first, which he called the " cataract ," being an hyperboloid generated by the revolution of an hyperbola of the fifth degree around the axis of the cylinder which should pass through the orifice, and the second the remainder of the water in the cylindrical vessel.

: ( hyperboloid of two sheets ).

Alternatively, a hyperboloid of two sheets of axis AB is obtained as the set of points P such that AP − BP is a constant, AP being the distance between A and P. Points A and B are then called the foci of the hyperboloid.

In spite of its positive curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a model for hyperbolic geometry.

( Note that if one replaces with in the above definition, one obtains a hyperboloid of two sheets.

A hyperboloid of one sheet is a doubly ruled surface: it can be generated by either of two families of straight lines.

In particular, when and move with constant speed along two skew lines, the surface is a hyperbolic paraboloid, or a piece of an hyperboloid of one sheet.

Like the hyperboloid of one sheet, the hyperbolic paraboloid has two families of skew lines ; in each of the two families the lines are parallel to a common plane although not to each other.

Depending on context, it can refer to either a theoretical surface of constant negative curvature, to a tractricoid, or to a hyperboloid.

* Hyperbolic spaces: By the hyperboloid model, an n dimensional hyperbolic space can be identified with the subset of ( n + 1 )- dimensional Minkowski space

The proof makes use of the property that for every conic section we can find a one-sheet hyperboloid which passes through the conic.

Doubly ruled surfaces are the inspiration for curved hyperboloid structures that can be built with a latticework of straight elements, namely:

Cooling towers vary in size from small roof-top units to very large hyperboloid structures ( as in the adjacent image ) that can be up to 200 metres tall and 100 metres in diameter, or rectangular structures ( as in Image 3 ) that can be over 40 metres tall and 80 metres long.

The hyperbolic paraboloid ( not to be confused with a hyperboloid ) is a doubly ruled surface shaped like a saddle.

* Hyperbolic paraboloid ( not to be confused with hyperboloid )

A tradition each year during the holiday season is for the Planetarium's unique hyperboloid structure to be wrapped with a holiday ribbon.

Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities.

A singular monument of industrial architecture is a 128-meter-high open-work hyperboloid tower built on the bank of the Oka near Dzerzhinsk as part of a powerline river crossing by the eminent engineer and scientist Vladimir Shukhov in 1929.

In mathematics, a hyperboloid is a quadric – a type of surface in three dimensions – described by the equation

Natural draft wet cooling towers at many nuclear power plants and large fossil fuel-fired power plants use large hyperboloid chimney-like structures ( as seen in the image at the left ) that release the waste heat to the ambient atmosphere by the evaporation of water.

De Sitter space is the submanifold described by the hyperboloid of one sheet

Krasnodar is home to the steel lattice hyperboloid tower built by the Russian engineer and scientist Vladimir Grigorievich Shukhov in 1928 ; it is located near Krasnodar Circus.

The unique architectural construction — the steel lattice hyperboloid tower built by the Great Russian engineer and scientist Vladimir Grigorievich Shukhov in 1929 — is located near the town of Dzerzhinsk on the left bank of the Oka River.

The world's first hyperboloid structure by Vladimir Shukhov

The hyperboloid tower was built and patented in 1896 by the famous Russian engineer and scientist Vladimir Shukhov.

The hyperboloid structures were subsequently built by other architects, such as Antoni Gaudí, Le Corbusier, and Oscar Niemeyer.

The unique architectural construction – the 128 m steel lattice hyperboloid tower built by the Soviet engineer and scientist Vladimir Grigorievich Shukhov in 1929 is located near the town of Dzerzhinsk on the left bank of the Oka River.

A hyperboloid cooling tower was patented by Frederik van Iterson and Gerard Kuypers in 1918.

The James S. McDonnell Planetarium, thin-shell structure | thin-shell and hyperboloid structure by Gyo Obata, one component of the St. Louis Science Center campus

Recounting lectures of Weierstrass, he there introduced the hyperboloid model described by Weierstrass coordinates.

The James S. McDonnell Planetarium, built in 1963 and featuring a thin-shell structure | thin-shell and hyperboloid structure by Gyo Obata.

In particular the hyperboloid model was identified with velocities by Minkowski ( 1908 ).

A fiber bundle | fibration of projective space by skew lines on nested hyperboloid s.

Access stairs at the base of a massive hyperboloid cooling tower give a sense of its scale ( UK )

The primary reflector is a paraboloid, while the shape of the convex secondary reflector is a hyperboloid.

The geometrical condition for radiating a collimated, plane wave beam is that the feed antenna is located at the far focus of the hyperboloid, while the focus of the primary reflector coincides with the near focus of the hyperboloid.

The world's first radio receiver of engineer Alexander Popov and the world's first hyperboloid tower and lattice shells-coverings of engineer Vladimir Shukhov were demonstrated at the All-Russia industrial and art exhibition in Nizhny Novgorod in 1896.

An elliptic hyperboloid of one sheet.

Whereas the Gaussian curvature of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive.

However, the mechanical induced-draft or forced-draft wet cooling towers in many large thermal power plants, nuclear power plants, fossil-fired power plants, petroleum refineries, petrochemical plants, geothermal, biomass and waste-to-energy plants use fans to provide air movement upward through downcoming water, and are not hyperboloid chimney-like structures.

Unlike optical telescopes which possess simple aluminized parabolic surfaces ( mirrors ), X-ray telescopes generally use a Wolter telescope consisting of nested cylindrical paraboloid and hyperboloid surfaces coated with iridium or gold.