Difference between revisions of "Spherical coordinates"

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'''Spherical coordinates''' are a way to describe the location of a point in three-dimensional space based on "r", the distance from the origin (where x=y=z=0); "θ", the angle between the point and the x-z plane (in the positive x direction); and Ψ, the angle between the point and the z axis (in the positive z direction).  Thus, each point is described by: (r,θ,Ψ).
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'''Spherical coordinates''' are a way to describe the location of a point in three-dimensional space based on <math>r</math>, the distance from the origin (where <math>x=y=z=0</math>); <math>\theta</math>, the angle between the point and the x-z plane (in the positive x direction); and <math>\phi</math>, the angle between the point and the z axis (in the positive z direction).  Thus, each point is described by: <math>(r, \theta, \phi)</math>.
  
In a sense, cylindrical coordinates are coordinates on a sphere just like [[polar coordinates]] are coordinates on a circle.
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In a sense, spherical coordinates are coordinates on a [[sphere]] just like [[polar coordinates]] are coordinates on a [[circle]].
  
 
The equations converting the parameters are as follows:
 
The equations converting the parameters are as follows:
  
 
:<math>r^2 = x^2 + y^2 + z^2</math>
 
:<math>r^2 = x^2 + y^2 + z^2</math>
:x=r*sin(θ)*cos(Ψ)
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:<math>x = r \sin{\theta} \cos{\phi}</math>
:y=r*cos(θ)*sin(Ψ)
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:<math>y = r \cos{\theta} \sin{\phi}
:z=r*cos(Ψ)
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:<math>z = r \cos{\phi}</math>
  
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The Jacobian in spherical polar coordinates is <math>r^2 \sin{\theta}</math> so that <math>\text{d}x \, \text{d}y \, \text{d}z = r^2 \sin{\theta} \, \text{d}r \, \text{d} \theta \, \text{d} \phi</math>.
 
[[Category:Mathematics]]
 
[[Category:Mathematics]]
 
[[Category:Geometry]]
 
[[Category:Geometry]]

Latest revision as of 14:13, 14 December 2016

Spherical coordinates are a way to describe the location of a point in three-dimensional space based on , the distance from the origin (where ); , the angle between the point and the x-z plane (in the positive x direction); and , the angle between the point and the z axis (in the positive z direction). Thus, each point is described by: .

In a sense, spherical coordinates are coordinates on a sphere just like polar coordinates are coordinates on a circle.

The equations converting the parameters are as follows:

The Jacobian in spherical polar coordinates is so that .