Tom, I've never posted in direct reply to the "experimental proof" before, but since nobody including youself has responded to this, I am going to quote myself here... Read the whole thing, including the edit, before commenting please.
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I hate to use an analogy here but I can't think of a better way to do it... to simplify this experiment take a cylinder (or globe if you want, that's fine too) and place upon it a couple of Styrofoam blocks or shapes of some kind.. relatively thin is fine and glue them down good and tight. place them relatively close even.. the more so the better to prove the point. Farther away and it will be more of an effect, but this shows the minor differences you were pointing out also.
Now stab a pencil or needle or what have you into each block wherever you choose, string a thread tight between them, then cut the cylinder along the backside, and lay it out flat. You will notice that the thread now droops. This would be an example of a northern discrepancy between known current distance between two points and if you were to flatten out the earth.
For the south pole, all you have to do is reverse the concept, cut the cylinder from the front and the thread either springs tight trying to keep it together or snaps.
Hope that helps, if not let me know... I'll try to do better.
Take care,
John
Edit: Also, in case the last part doesn't seem feasible, take a circle or disk, wrap it into even a gentle cone and repeat the same experiment, attempting to flatten the cone back out into a disk... the string will tighten out or snap sue to the distortion.
Before anyone says this proves nothing but experimental shapes, these are only experiments to prove distance variances - take into account we ALL know how far it is from say Tokyo to LA, or in common discussion here; say San Francisco, CA to Sydney, Australia - 7408 miles (11922 km) (6438 nautical miles) [a long haul I make at least once a year to see my brother... ugh!]
If you try to alter the map (as in move and/or twist continents around) to make the distance work... then you are off on all your other distances to elsewhere in the world. try to adjust that also... then you just start making it worse and worse.
Take care,
John