Difference between revisions of "Hyperboloid"

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A hyperboloid is a quadric surface in 3 dimensions that is generated by [[revolution of line|rotating]] a hyperbolic function around an axis.   
 
A hyperboloid is a quadric surface in 3 dimensions that is generated by [[revolution of line|rotating]] a hyperbolic function around an axis.   
  
The hyperbolic function <math> sin{hx} </math> produces the graph on the left, and by rotating it about the y-axis we get a '''hyperboloids of one sheet''.
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The hyperbolic function <math> sin{hx} </math> produces the graph on the left, and by rotating it about the y-axis we get a '''hyperboloids of one sheet'''.
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[[Image:Sinhx.png|300px]][[Image:HyperboloidOfOneSheet.png|300px]]
 
[[Image:Sinhx.png|300px]][[Image:HyperboloidOfOneSheet.png|300px]]
  

Revision as of 12:53, 26 May 2009


Hyperboloid
Hyperboloid.jpg
Field: Calculus
Image Created By: Paul Nylander
Website: Paul Nylander's Site

Hyperboloid

A hyperboloid is a quadric, a type of surface in three dimensions.


Basic Description

A hyperboloid is a quadric surface in 3 dimensions that is generated by rotating a hyperbolic function around an axis.

The hyperbolic function  sin{hx} produces the graph on the left, and by rotating it about the y-axis we get a hyperboloids of one sheet.

Sinhx.pngHyperboloidOfOneSheet.png

Hyperboloids can be hyperboloids of one sheet or hyperboloids of two sheets.

HyperboloidOfTwoSheets.png

{x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over c^2}=1 \,  hyperboloid of one sheet,

or

- {x^2 \over a^2} - {y^2 \over a^2} + {z^2 \over c^2}=1 \,  hyperboloid of two sheets.

These curves are graphed using the following parametric equations:

x(u,v)=a(\sqrt{1+u^2})cos{v}\,

y(u,v)=a(\sqrt{1+u^2})sin{v}\,

z(u,v)=cu\,

==Analyzing a Hyperboloid from 2 Dimensionals==





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