Paraboloid

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

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

$\left( \frac{x}{a} \right) ^2 - \left( \frac{y}{b} \right) ^2 + 2z = 0$ (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.

See also: ellipsoid, hyperboloid.

Categories: Surfaces