Difference between revisions of "Hyperboloid"

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[[Image:Sinhx.png|300px]][[Image:HyperboloidOfOneSheet.png|300px]]
 
[[Image:Sinhx.png|300px]][[Image:HyperboloidOfOneSheet.png|300px]]
  
Hyperboloids can be '''hyperboloids of one sheet''' or '''hyperboloids of two sheets'''.
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[[Image:Coshx.png|300px]] [[Image:HyperboloidOfTwoSheets.png|300px]]
[[Image:HyperboloidOfTwoSheets.png|300px]]
 
  
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They are represented by the following equations.
 
<math>{x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over c^2}=1 \,</math> &nbsp;'''hyperboloid of one sheet,'''
 
<math>{x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over c^2}=1 \,</math> &nbsp;'''hyperboloid of one sheet,'''
  
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<math>- {x^2 \over a^2} - {y^2 \over a^2} + {z^2 \over c^2}=1 \,</math> &nbsp;'''hyperboloid of two sheets.'''
 
<math>- {x^2 \over a^2} - {y^2 \over a^2} + {z^2 \over c^2}=1 \,</math> &nbsp;'''hyperboloid of two sheets.'''
  
These curves are graphed using the following <balloon title="Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters" style="color:green"> parametric equations: </balloon>
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These curves can also be defined using the following <balloon title="Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters" style="color:green"> parametric equations: </balloon>
  
 
<math>x(u,v)=a(\sqrt{1+u^2})cos{v}\,</math>
 
<math>x(u,v)=a(\sqrt{1+u^2})cos{v}\,</math>

Revision as of 13:02, 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

Coshx.png HyperboloidOfTwoSheets.png


They are represented by the following equations. {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 can also be defined 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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