Difference between revisions of "Hyperboloid"

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A hyperboloid is a quadric surface in 3 dimensions that is generated by [[Volume of Revolution|rotating]] a hyperbolic function around an axis.   
 
A hyperboloid is a quadric surface in 3 dimensions that is generated by [[Volume of Revolution|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 '''hyperboloid 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 ''<balloon title="A hyperboloid that can be represented by a single sheet.">hyperboloid of one sheet</balloon>''.   
  
 
[[Image:Sinhx.png|300px]][[Image:Hyperboloid 1-Sheet.png|300px]]
 
[[Image:Sinhx.png|300px]][[Image:Hyperboloid 1-Sheet.png|300px]]
  
The hyperbolic function <math>coshx\,</math> produces the graph on the left, and by rotating it about the y-axis we get a '''hyperboloid of two sheets'''.  
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The hyperbolic function <math>coshx\,</math> produces the graph on the left, and by rotating it about the y-axis we get a ''<balloon title="A hyperboloid that must be represented by two sheets.">hyperboloid of two sheets</balloon>''.
  
 
[[Image:Coshx.png|300px]] [[Image:Hyperboloid 2-Sheet.png|300px]]
 
[[Image:Coshx.png|300px]] [[Image:Hyperboloid 2-Sheet.png|300px]]
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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>
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These curves can also be defined using the following '''[[Parametric Equations|<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. Click for the Helper Page on parametric equations." 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>
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The design has become the standard for nuclear cooling towers because of its minimum usage of material, structural strength, and improved convective air flow.
 
The design has become the standard for nuclear cooling towers because of its minimum usage of material, structural strength, and improved convective air flow.
  
 
 
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|SiteURL=http://www.bugman123.com/
 
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Latest revision as of 09:48, 10 June 2011

Inprogress.png
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 hyperboloid 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 hyperboloid of two sheets.

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\,

The cross sections of a hyperboloid are hyperbolas when taken with respect to the x-z or y-z planes and circles when taken with respect to the x-y plane. This site provides good interactive animations that help illustrate this.

Cooling-tower-01.jpg

Engineering Uses

The construction of a 1 sheeted hyperboloid can be approximated by having two circles connected by many straight beams.

The design has become the standard for nuclear cooling towers because of its minimum usage of material, structural strength, and improved convective air flow.





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