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In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes.A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.
A hyperboloid of one sheet is a doubly ruled surface: it can be generated by either of two families of straight lines. Four images of hyperboloid towers. The Shukhov Tower in Polibino , the world's first hyperboloid structure , a water tower by Vladimir Shukhov at the All-Russian Exposition in Nizhny Novgorod , Russia
A hyperboloid of two sheets. Sphere; Spheroid. Oblate spheroid; Prolate spheroid; Ellipsoid; Cone (geometry) Hyperboloid of one sheet; Hyperboloid of two sheets; Hyperbolic paraboloid (a ruled surface) Paraboloid
For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle. A surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces.
For example, a hyperboloid of one sheet is a quadric surface in ruled by two different families of lines, one line of each passing through each point of the surface; each family corresponds under the Plücker map to a conic section within the Klein quadric in .
Hyperboloid structures are architectural structures designed using a hyperboloid in one sheet. Often these are tall structures, such as towers, where the hyperboloid geometry's structural strength is used to support an object high above the ground. Hyperboloid geometry is often used for decorative effect as well as structural economy.
A string model of a portion of a regulus and its opposite to show the rules on a hyperboloid of one sheet. In three-dimensional space, a regulus R is a set of skew lines, every point of which is on a transversal which intersects an element of R only once, and such that every point on a transversal lies on a line of R.
For positive ν, the half-hyperboloid is above the x-y plane (i.e., has positive z) whereas for negative ν, the half-hyperboloid is below the x-y plane (i.e., has negative z). Geometrically, the angle ν corresponds to the angle of the asymptotes of the hyperbola. The foci of all the hyperbolae are likewise located on the x-axis at ±a.