Why Is \(\phi\) Defined Differently in Spherical Coordinates?

AI Thread Summary
The discussion centers on the definition of the angle \(\phi\) in spherical coordinates, which is measured from the +z axis to a position vector projected onto the yz plane. The question arises as to why this convention differs from polar and cylindrical coordinates, where angles are measured from the +x axis. It is noted that there is no universal agreement on the conventions used for these angles, allowing for flexibility in choice. However, practical applications, such as calculating intersections between a sphere and a cone, demonstrate the utility of the established definition of \(\phi\). Understanding these conventions is crucial for simplifying complex problems in three-dimensional geometry.
bomba923
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Just curious, why is \phi calculated as the angle between the +z axis and a position vector of a point of a function, as projected onto the yz plane? Why this convention?

In polar & cylindrical, \theta is calculated from the +x axis to the +y axis (counterclockwise) for position vectors.

*So, why not extrapolate alphabetically, to have the \phi be the angle between the +y axis and the position vector of a point as projected onto the yz plane? :smile:
 
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Use whatever convention you like, there is no general agreement on this issue.
 
arildno said:
Use whatever convention you like, there is no general agreement on this issue.

But nowhere have I found \phi calculated as the angle between the +y axis and the position vector of a point, projected onto the yz plane...
 
You will see why once you start working with problems involving these. For example, take the intersection between a sphere and a cone (I posted about this before). This can be computed by a double integral, but if you use a triple integral and spherical coordinates, it becomes much simpler (thanks to HallsofIvy on this).
 

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