# Dictionary Definition

paraboloid n : a surface having parabolic
sections parallel to a single coordinate axis and elliptic sections
perpendicular to that axis

# User Contributed Dictionary

## English

### Noun

- A surface having a parabolic cross section parallel to an axis, and circular or elliptical cross section perpendicular to the axis; especially the surface of revolution of a parabola

### Derived terms

#### Translations

- Swedish: paraboloid

#### See also

# Extensive Definition

In mathematics, a paraboloid is
a quadric
surface of special kind. There are two kinds of paraboloids:
elliptic and hyperbolic. The elliptic paraboloid is shaped like an
oval cup and can have a maximum or minimum point. In a
suitable coordinate system, it can be represented by the
equation

z = \frac + \frac.

This is an elliptical paraboloid which opens
upward.

The hyperbolic paraboloid is a doubly ruled
surface shaped like a saddle. In a
suitable coordinate system, it can be represented by the
equation

z = \frac - \frac.

This is a hyperbolic paraboloid that opens up
along the x-axis and down along the y-axis.

## Properties

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; and is also the shape of the surface of a rotating liquid, a principle used in liquid mirror telescopes. 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.

The hyperbolic paraboloid is a ruled
surface: it contains two families of mutually skew lines.
The lines in each family are parallel to a common plane, but not to
each other. The Pringles potato
chip gives a good physical approximation to the shape of a
hyperbolic paraboloid.

## Curvature

The elliptic paraboloid, parametrized simply as

- \vec \sigma(u,v) = \left(u, v, + \right)

- K(u,v) =

- H(u,v) =

The hyperbolic paraboloid, when parametrized as

- \vec \sigma (u,v) = \left(u, v, - \right)

- K(u,v) =

- H(u,v) = .

## Multiplication table

If the hyperbolic paraboloid- z = -

- z = (x^2 + y^2) \left( - \right) + x y \left(+\right)

- z = x y .

- z = .

- \ z = x y

The two paraboloidal \mathbb^2 \rarr \mathbb
functions

- z_1 (x,y) =

- \ z_2 (x,y) = x y

- f(z) = z^2 = f(x + i y) = z_1 (x,y) + i z_2 (x,y)

paraboloid in Arabic: سطح مكافئ

paraboloid in Catalan: Paraboloide

paraboloid in Czech: Paraboloid

paraboloid in Danish: Hyperbolsk
paraboloide

paraboloid in German: Paraboloid

paraboloid in Spanish: Paraboloide

paraboloid in French: Paraboloïde

paraboloid in Italian: Paraboloide

paraboloid in Dutch: Paraboloïde

paraboloid in Polish: Paraboloida

paraboloid in Portuguese: Parabolóide

paraboloid in Russian: Параболоид

paraboloid in Finnish: Paraboloidi

paraboloid in Swedish:
Paraboloid