Response from Prof. B.

Yes, when you are in a Euclidean space of any dimension then any point other than the origin lies on a unique ray through the origin. This ray pierces through the unit sphere at a point, and this point, together with the distance from the origin, determine the "polar" coordinates of the point in question. You are then left with the problem of identifying the point on the sphere, and that can be done in several ways. The easiest perhaps is to proceed by induction.

A point on a circle is identified by its angular coordinate, written (cos(t), sin(t)). A point on the two-dimensional sphere in three-space is identified by a point on the equator (longitude) and a "height" above the equator (latitude) so you get something like cos(s) times (cos(t), sin(t), 0) plus sin(s) times (0,0,1), or (cos(s)cos(t), cos(s)sin(t), sin(t)).

In the next dimension, you will get something like cos(w) ((cos(s)cos(t), cos(s)sin(t), sin(t),0) + (sin(w)(0,0,0,1) = (cos(w)cos(s)cos(t), cos(w)cos(s)sin(t), cos(w)sin(t), sin(w)). The pattern is now clear.