paraboloid n : a surface having parabolic sections parallel to a single coordinate axis and elliptic sections perpendicular to that axis
- Swedish: paraboloid
- In the context of "maths|lang=sv": paraboloid
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.
PropertiesWith 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.
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 tableIf 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