I'm assuming the geodetic coordinate axes at point P on the surface of the earth are ...
The curvilinear geodetic coordinates are simply longitude, geodetic latitude, and height. Your description is for an ENU or
NED frame.
Whereas the geocentric axes at the same point would be
- A "vertical" on a line from center of the earth to P (up or down) ...
Yes but this radius vector is not normal to the surface at P. Any other point Q along the surface normal to P has a radius vector with a different direction (i.e. different geocentric latitude). Contrast this with the fact that the
geodetic latitudes of P and Q are the same. In other words, varying height has the major advantage of leaving geodetic latitude unvaried, which is a property of orthogonal coordinates.
At this point I'm wondering why this second system of axis has a complicated Euclidean norm.
Additional terms are needed to account for the change in geocentric latitude with changing height.
... I thought that GPS coordinates were actually the coordinates of the point, on the surface, of which we're vertical.
That is correct. WGS 84 coordinates are geodetic and therefore orthogonal.