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Fred, you wrote:
"The author states that height can only be known to some few cm or
whatever because of variations in gravity, if I remember correctly.
It would seem that this is due to our tradition of assuming we are on
the surface of a spheroid or ellipsoid when doing navigation."

The assumption they mentioned is the "geoid" which is a complicated surface
of contant gravitational potential. So we're well beyond the ellipsoid
already when it comes to extremely precise measurements.

"Confining ourselves to a surface makes the trig easier, but couldn't
one position oneself with greater accuracy (with feet firmly planted
on earth, not on a boat) using only stars or stars plus the sun,
ignoring the earth's horizon, by measuring star-star distances?  Make
it a true 3-D problem.  Or would uncertainties in the positions of
stars still hamper ones efforts, especially uncertainty in their
distance from us?"

Stars? No. Much too far away. We need much closer objects whose positions
would shift noticeably relative to the distant starry background. So why
don't we launch some fake stars into orbit close to the Earth? And since
timings are easier to measure than angles, we'll put some transmitters on
our fake stars. As I'm sure you've recognized, this is just the GPS system.
But it wasn't designed for that next level of accuracy, a couple of orders
of magnitude higher, required by the next generation of atomic clocks.
That's the trouble. Necessity is the mother of invention, and I'm sure
someone will invent a solution, especially if anyone can think of a
practical application of these super-clocks. It may be the case, though,
that the business of ever more accurate time measurement has hit a brick
wall with no means of comparing clocks at different locations which may last
for a few decades conceivably.

 -FER



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