hyperboloid

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Hyperboloid of one sheet

Hyperboloid of two sheets

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

  (hyperboloid of one sheet),

or

  (hyperboloid of two sheets).

These are also called elliptical hyperboloids. If and only if a = b, it is a hyperboloid of revolution, and is also called a circular hyperboloid.

Properties

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

An elliptic hyperboloid of one sheet. The wires are straight lines. For any point on the surface, there are two straight lines lying entirely on the surface which pass through the point. This illustrates the doubly ruled nature of this surface.

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.

Whereas the intrinsic curvature of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive, yet the geometry of both is hyperbolic, demonstrating the need to define hyperbolic geometry in terms of parallel lines rather than curvature.

Degenerate

A degenerate hyperboloid is of the form

if a = b then this will give a cone, if not then it gives an elliptical cone.

Hyperboloid structures

One-sheeted hyperboloids are used in construction, with the structures called hyperboloid structures. A notable (and recognizable) use of these is in the cooling towers utilized by some power stations (mainly nuclear power plants). Since a hyperboloid is a doubly-ruled surface, it can be built with straight steel beams, and allows minimisation of wind cross-section while retaining structural integrity with minimal material.

See also

• Hyperbola
• Ellipsoid
• Paraboloid / Hyperbolic paraboloid
• Hyperboloid structure
• Ruled surface
• de Sitter space
• Vladimir Shukhov

Wikimedia Commons has media related to: Hyperboloid

External links

• Weisstein, Eric W., "Hyperboloid" from MathWorld.
• Weisstein, Eric W., "Elliptic Hyperboloid" from MathWorld.

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