To clarify :
I'm assuming the geodetic coordinate axes at point P on the surface of the earth are :
- The perpendicular to the plane of tangency at P (up or down)
- On the plane of tangency a north vector
- A third vector determined with right hand rule
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)
- A North vector
- An East or West vector
At this point I'm wondering why this second system of axis has a complicated Euclidean norm.
Maybe the North vector remained the same as in the first system of axes, then it means that the system of geocentric axes is not orthogonal (there is a small residual angle)
Is it correct up to this point ?
If yes, the other option would be to modify the North vector, to make it perpendicular to the line from the center of the earth. Then the North vector would go either into or out of Earth, and this is obviously a problem because it means that if you're travelling north or south at constant altitude then you actually have a vertical speed in this system of axes...
Regarding your last line : Up to reading your message I thought that GPS coordinates were actually the coordinates of the point, on the surface, of which we're vertical. But it's quite evident that it wouldn't work that way.