Paraboloid

(Redirected from Hyperbolic paraboloid)
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HyperbolicParaboloid.png
Hyperbolic paraboloid
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ParaboloidOfRevolution.png
Paraboloid of revolution

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

[itex]

\left( \frac{x}{a} \right) ^2 + \left( \frac{y}{b} \right) ^2 + 2z = 0 [itex] (elliptic paraboloid),

or

[itex]

\left( \frac{x}{a} \right) ^2 - \left( \frac{y}{b} \right) ^2 + 2z = 0 [itex] (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 is the shape of the parabolic reflectors used in mirrors, antenna dishes, and the like. It is also called a circular paraboloid.

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.

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