Originally Posted by microburst8265
Hey,
the physical shape of the earth is probably best resembled as a geoid (equipotential surface approximating mean sea level)...
For Nav Systems an ellipsoid model is used (e.g. WGS84), since the assumption of earth being a sphere is much too inaccuarte to navigate precisely. The shortest distance between two points on the suface of the ellipsoid is called geodatic line. (The same as the great circle on a sphere)
Spherical geometry on an ellipsoid is really not something you want to get involved with
Trust me.... ;-)
Er, is there such a thing as a unique "geodetic" shortest distance on an ellipsoid?
Consider this:
If you look at a pair of points both exactly at the equator and exactly opposite, then the shortest distance is the meridian through a pole. Either pole, those distances are equal. The arcs of equator either side are appreciably longer, though they also are "great circle". One would expect that the other "great circles" would be longer than the pair of meridians, too.
Now look at the pair of points still both exactly at the equator but not precisely opposite - such that the shorter equator arc between them is still longer than the pair of meridians to a pole.
Obviously there is a path which is shorter than the great circle, yet makes an angle at a pole. Therefore, you should also have a non-angled continuous line which does not follow either the great circle at equator (having the equatorial bulge) or meridians with an angle at a pole, but smoothly passes through higher latitudes. Actually a pair of such lines, on either hemisphere.
Now consider a pair of points which are also slightly off the equator, in the same hemisphere. Obviously, now, the continuous line across one hemisphere would be longer than the line across the other hemisphere - but still shorther than the paths in between, spending too much time on the equatorial bulge...
What can be said generally about "shortest" or "straight" lines on Earth?