Spherical coordinates

[Qacer]

Hey all,

I'm brushing up on my vector stuff. In reference to spherical coordinates, I'm trying to understand why the differential length for the zenith (z) is R dz and the azimuth (b) is R sin (z) db.

Thanks!

 

[RossGr]

It is all about arc length. The zenith coordinates describe great circles through the poles. So the r coordinate determines the radius of the circle, therefore r dz is an element of arc length. For the azimuth coordinate the zenith angel determines the radius of the circle described.

Look at a globe, the zenith angle follows the lines of longitude. No matter which line of longitude you follow your distance from distance from the center of earth does not vary, so the distance you travel along the line is given by r dz. (This is the simple arc length formula.)

Now consider a line of latitude. If you travel 1 deg along the equator you will have traveled further then if you travel 1 deg at 45N. The radius of a circle of latitude is given by r sin(z). So the arc length is found by R sin(z) db where db is the change in angle along the circle of latitude

 

[CycloWizard]

Originally posted by: RossGr
It is all about arc length. The zenith coordinates describe great circles through the poles. So the r coordinate determines the radius of the circle, therefore r dz is an element of arc length. For the azimuth coordinate the zenith angel determines the radius of the circle described.

Look at a globe, the zenith angle follows the lines of longitude. No matter which line of longitude you follow your distance from distance from the center of earth does not vary, so the distance you travel along the line is given by r dz. (This is the simple arc length formula.)

Now consider a line of latitude. If you travel 1 deg along the equator you will have traveled further then if you travel 1 deg at 45N. The radius of a circle of latitude is given by r sin(z). So the arc length is found by R sin(z) db where db is the change in angle along the circle of latitude

:beer: Wish you would have taught my Calc III class. I think that's by far the best explanation of that I've ever heard.

 

[Qacer]

Thanks for the tip, RossGr!