Subject: What would the horizon look like on a very large planet?

What would the horizon look like on a very large planet?
I've heard that on the Earth the horizon is a little over
20 miles away when viewed from sea level by an observer
of average height.  This led me to wonder whether the
horizon would seem farther away if the observer were on
a much larger planet (assuming no atmospheric haze or 
variation in surface elevation), and just what this
"farther away" effect would consist of.

Presumably, the horizon would just occupy a greater proportion
of the sky/earth view; there would appear to be less sky, and
the horizon would appear "higher" than it would on Earth.
In fact, one could measure the angle formed by the observer's 
line of sight and the ground directly below his or her eyes 
(the "eye angle"); the angle would logically increase as the 
diameter of the planet increased.

So if the planet were very large, the observer would look
almost straight ahead.  Our only experience with looking
almost straight ahead and seeing some sort of "horizon"
would probably occur if we were standing in a depression
such as a crater, a valley, or just a wide hole.  So it
would seem that if we stood on this large planet and looked
at the very distant horizon it would look as if we were
standing in a bowl; and the larger the planet the deeper
the bowl. [On an infinite plain, how deep would the bowl
be?]

On the other hand, maybe the increase in the eye angle would be
so small that the observer wouldn't perceive a difference.
After all, although the angle would increase as the 
diameter of the planet increased, the *rate* of increase would
diminish.  Perhaps beyond a certain diameter the eye angle
change would simply be imperceptible; the horizon would always
appear to be about as far away as it is on the Earth. [On an
infinite plain would this still be true?]

Sorry about the long preamble.  My question is which of these
two guesses is most likely true?  Would an infinite plain make
a different answer altogether?

Re:What would the horizon look like on a very large planet?