# Cosmology and Spherical Coordinates

My question refers to page 10 of this document. Specifically, when using spherical polar coordinates in cosmology, why does the author of this work choose the origin of the coordinate system to be at the north pole, as opposed to the centre of the sphere? Is this more useful to cosmologists, if so: why?

• Minor comment to the post (v2): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. – Qmechanic Apr 12 at 23:07
• You can edit the question and do that if you so wish. – wrb98 Apr 13 at 21:59

Let's drop one dimension to make the illustration. Now our job is to construct coordinates on the surface of a sphere. The angles $$\theta,\phi$$ of the spherical polar coordinate system could be used, but they also include an initial step in which an arbitrary place on the sphere's surface is taken to act as a pole for the definition of $$\theta$$, and an arbitrary 'zero' direction is required for the definition of $$\phi$$. So they do not avoid this issue of picking a place to call 'north pole'.
• I was looking back at this yesterday, and was wondering if you can come up with an analog for the surface of the hyperboloid $x^2 + y^2 - z^2 = R^2$? That is, a justified parameterisation with the reference point being on the surface as opposed to at the origin (which does not lie on the surface in question). If you can find one, I would very much appreciate it. – wrb98 Apr 23 at 1:22
• Should be $-R^2$ there. – wrb98 Apr 23 at 1:29