In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation:
(elliptic paraboloid),
or
(hyperbolic paraboloid).
Hyperbolic paraboloid.
There are two kinds of paraboloid: elliptic and hyperbolic. The elliptic paraboloid is shaped like a cup and can have a maximum or minimum point. The hyperbolic paraboloid is shaped like a saddle and can have a critical point called a saddle point. It is a ruled surface.
With a = b an elliptic paraboloid is a paraboloid of revolution: a surface obtained by revolving a parabola around its axis. It the shape used by the parabolic reflectors used in mirrors, antenna dishes, and the like. It is also called a circular paraboloid.
Paraboloid of revolution.
A point light source at the focal point produces a parallel light beam. This also works the other way around: a parallel beam of light incident on the paraboloid is concentrated at the focal point. This applies also for other waves, hence parabolic antennas.
The hyperbolic form of the tower is remarkably similar to that of the pseudosphere used to illustrate explanations of Lobachevskii's disproof of Euclid's parallel postulate.
His design, as well as the full set of supporting calculations analyzing the hyperbolic geometry and sizing the network of members, was completed by February of 1919; however, the 2200 tons of steel required to build the tower to 350m were not available.
Antoni Gaudi used structures in the form of hyperbolicparaboloid (hypar) and hyperboloid of revolution in the Sagrada Familia in 1910 [1].
The hyperbolicparaboloid configuration has been utilized successfully for a number of years in the building industry and has been applied in the shape of the potato chip presently marketed under the trademark Pringle.
The principal object of the invention is to provide a family of structural building components formed from a plurality of integrated hyperbolicparaboloid elements which have a common midpoint and linear edges and allow for the prestressing of reinforcing elements along straight lines.
The hyperbolicparaboloid elements B used in the tetralith A are designed so that from any given point on the surface of the element B two straight lines can, respectively, be projected between opposing linear edges through that point.
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