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Spherical coordinates

  1. Jun 4, 2010 #1
    1. The problem statement, all variables and given/known data

    I am confused about spherical coordinates stuff. For example, we can parametrize a sphere of radius 3 by

    [itex] x = 3 sin \phi cos \theta [/itex]
    [itex] y = 3 sin \phi sin \theta [/itex]
    [itex] z = 3cos\phi [/itex]

    where [itex] 0 \le \theta \le 2 \pi [/itex] and [itex] 0 \le \phi \le \pi [/itex] .

    I don't understand about the range of [itex] \phi [/itex].

    1) Why is [itex] 0 \le \phi \le \pi [/itex] ?
    2) If we only want the lower hemisphere, why is the range now [itex] \frac{\pi}{2} \le \phi \le \pi [/itex] ?
    3) What about the range of [itex] \phi [/itex] if we want the upper hemisphere?

    Is there any place where I get to see the diagram so I can get the picture better?

    Thanks!
     
  2. jcsd
  3. Jun 4, 2010 #2

    Pengwuino

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    Gold Member

    1) I assume you mean why doesn't [tex]\phi[/tex] go to [tex]2\pi[/tex]? By allowing [tex]\theta[/tex] to go to [tex]2\pi[/tex], you cover the entire sphere. Any point in the space that you feel you could get by extending [tex]\phi[/tex] to [tex]2\pi[/tex] is covered by simply moving to [tex]\theta > \pi[/tex].

    2) Look at how the coordinate system is defined and notice that you must go beyond [tex]\frac{\pi}{2}[/tex] to be in the lower hemisphere. As for 3), same idea, just look at a diagram as to how the coordinate system is defined to see why certain ranges are why they are.

    http://upload.wikimedia.org/wikiped...nates.svg/429px-Spherical_Coordinates.svg.png

    This is a diagram of how spherical coordinates are typically defined. NOTE: Your definition of the coordinates have [tex]\theta[/tex] and [tex]\phi[/tex] switched.
     
  4. Jun 4, 2010 #3

    phyzguy

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    Science Advisor

    It's just like latitude and longitude on the Earth. Longitude (normally called phi, but what you called theta) runs from 0 to 360 degrees (2 pi), but latitude (normally called theta but what you called phi) only needs to run from -90 degrees to +90 degrees (total range of 180 degrees, or pi) to cover the sphere. The only difference is that on the Earth we define the equator as 0 degrees, the north pole as +90 degrees and the south pole as -90 degrees, whereas in physics, we usually define the north pole as 0 degrees, the equator as 90 degrees (pi/2), and the south pole as 180 degrees (pi).
     
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