We have a sphere with $r=1$, and we want the coordinates of $C$. $A$ is the north pole, and $AB$ is our prime meridian. See picture:

enter image description here

I'm familiar with an $(x,y,z)$ coordinate system, but not so much with polar coordinates. My booklet says that we can find the coordinates of $C$ using:

$$ x = \sin(b) \cdot \cos(A)$$ $$y = \sin(b) \cdot \sin(A)$$ $$z = \cos(b)$$

But it doesn't explain how, and I can't seem to figure it out myself (as this is my introduction to spherical geometry/trigonometry). Can anybody explain how this makes sense?


1 Answer 1


LettThe arc length be $l$ and so $$l=r\theta=\theta \implies b=\angle AOC$$ Then $$z=OC\cdot \cos \angle AOC=1\cos b$$ and $$x=OC\cdot \sin \angle AOC \cos \angle COB=1\sin b \cos a$$ enter image description here

  • $\begingroup$ I still don't understand how to get the expression for the other 2, this one I get now. $\endgroup$
    – Phaptitude
    Feb 16, 2014 at 16:54
  • $\begingroup$ Is it clear now @Phaptitude $\endgroup$
    – Semsem
    Feb 16, 2014 at 17:21
  • $\begingroup$ I have just added a picture @Phaptitude $\endgroup$
    – Semsem
    Feb 16, 2014 at 22:32
  • $\begingroup$ Sorry for my inactiviy, but yes, thanks a lot for this answer, I fully understand it now. $\endgroup$
    – Phaptitude
    Feb 17, 2014 at 15:38

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