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)
If, and only if, a = b, it is a hyperboloid of revolution. A hyperboloid of one sheet can be obtained by revolving a hyperbola around its transversal 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. A and B are then called the foci of the hyperboloid. A hyperboloid of two sheets can be obtained by revolving a hyperbola around its focal axis.
A hyperboloid of one sheet is a 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:
if a = b then this will give a cone, if not then it gives an elliptical cone.
The hyperboloid roofs of the exhibition pavilions of the 1896 All-Russian Industrial and Handicrafts Exposition in Nizhny Novgorod were the first publicly prominent examples of Shukhov’s new system.
A hyperboloid tower in the port of Kobe, Japan.
The world’s first hyperboloid tower is located in Polibino of the Lipetsk region of Russia.
A drawing of the hyperboloid in which some of these lines are explicitly shown.
A drawing of the hyperboloid and one of its tangent planes
Note that the intersection of the hyperboloid and the tangent plane is a reducible plane conic -- accordingly, the union of two lines in the tangent plane.
(hyperboloid of one sheet),
(hyperboloid of two sheets)
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