Difference between revisions of "Hyperboloid"

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The construction of a 1 sheeted hyperboloid can be approximated by having two  circles connected by many straight beams.  This structure is used most recognizably in nuclear cooling towers.
 
The construction of a 1 sheeted hyperboloid can be approximated by having two  circles connected by many straight beams.  This structure is used most recognizably in nuclear cooling towers.
  
[[Image:Hyperboloid of one sheet.jpg|300px]][[Image:Cooling-tower-01.jpg|300px]]
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[[Image:Hyperboloid of one sheet.jpg|200px]][[Image:Cooling-tower-01.jpg|200px]]
  
  

Revision as of 13:39, 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.pngHyperboloid 1-Sheet.png

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

Coshx.png Hyperboloid 2-Sheet.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

Other Uses

The construction of a 1 sheeted hyperboloid can be approximated by having two circles connected by many straight beams. This structure is used most recognizably in nuclear cooling towers.

Hyperboloid of one sheet.jpgCooling-tower-01.jpg





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