Understanding Spherical Coordinates and Their Range

[madachi]

Homework Statement

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

x = 3 sin \phi cos \theta
y = 3 sin \phi sin \theta
z = 3cos\phi

where 0 \le \theta \le 2 \pi and 0 \le \phi \le \pi .

I don't understand about the range of \phi.

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

 

[Pengwuino]

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

2) Look at how the coordinate system is defined and notice that you must go beyond \frac{\pi}{2} 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 \theta and \phi switched.

 

[phyzguy]

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).