hyperboloid n : a quadric surface generated by rotating a hyperbola around its main axis

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

+ - =1  (hyperboloid of one sheet),

or

- - + =1  (hyperboloid of two sheets).

If, and only if, a = b, it is a hyperboloid of revolution. 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 focal 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.

A degenerate hyperboloid is of the form

+ - =0

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

A notable (and recognizable) use of a hyperboloid structure is in the cooling towers utilized by some power stations. 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
• Hyperboloid structure
• Ruled surface

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