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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

- $$

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

- $$

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

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.

The elliptic paraboloid, parametrized simply as

- $vec\; sigma(u,v)\; =\; left(u,\; v,\; \{u^2\; over\; a^2\}\; +\; \{v^2\; over\; b^2\}right)$

- $K(u,v)\; =\; \{4\; over\; a^2\; b^2\; left(1\; +\; \{4\; u^2\; over\; a^4\}\; +\; \{4\; v^2\; over\; b^4\}right)^2\}$

- $H(u,v)\; =\; \{a^2\; +\; b^2\; +\; \{4\; u^2\; over\; a^2\}\; +\; \{4\; v^2\; over\; b^2\}\; over\; a^2\; b^2\; left(1\; +\; \{4\; u^2\; over\; a^4\}\; +\; \{4\; v^2\; over\; b^4\}right)^\{3/2\}\}$

The hyperbolic paraboloid, when parametrized as

- $vec\; sigma\; (u,v)\; =\; left(u,\; v,\; \{u^2\; over\; a^2\}\; -\; \{v^2\; over\; b^2\}right)$

- $K(u,v)\; =\; \{-4\; over\; a^2\; b^2\; left(1\; +\; \{4\; u^2\; over\; a^4\}\; +\; \{4\; v^2\; over\; b^4\}right)^2\}$

- $H(u,v)\; =\; \{-a^2\; +\; b^2\; -\; \{4\; u^2\; over\; a^2\}\; +\; \{4\; v^2\; over\; b^2\}\; over\; a^2\; b^2\; left(1\; +\; \{4\; u^2\; over\; a^4\}\; +\; \{4\; v^2\; over\; b^4\}right)^\{3/2\}\}.$

- $z\; =\; \{x^2\; over\; a^2\}\; -\; \{y^2\; over\; b^2\}$

- $z\; =\; \{1over\; 2\}\; (x^2\; +\; y^2)\; left(\{1over\; a^2\}\; -\; \{1over\; b^2\}right)\; +\; x\; y\; left(\{1over\; a^2\}+\{1over\; b^2\}right)$

- $z\; =\; \{2over\; a^2\}\; x\; y$.

- $z\; =\; \{x^2\; -\; y^2\; over\; 2\}$.

- $z\; =\; x\; y$

The two paraboloidal $mathbb\{R\}^2\; rarr\; mathbb\{R\}$ functions

- $z\_1\; (x,y)\; =\; \{x^2\; -\; y^2\; over\; 2\}$

- $z\_2\; (x,y)\; =\; x\; y$

- $f(z)\; =\; \{1over\; 2\}\; z^2\; =\; f(x\; +\; i\; y)\; =\; z\_1\; (x,y)\; +\; i\; z\_2\; (x,y)$

- http://www.exopas.com/beta/faces/pages/gallery/animated.jsp

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Last updated on Wednesday October 01, 2008 at 18:56:58 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday October 01, 2008 at 18:56:58 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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