to 3Dgeek
Notice how all the examples you gave involve explaining "why crows don't fly straight sometimes".
I meant give me an example in math or any technology where the earth is not involved.
My hypothesis is that the only time the GREAT CIRCLE comes into play with scientists is when things look perfect on a flat map and awkward on a globe map. That's when they drag out the great circle; but nowhere else in science or mathematics.
Yea, give me non-earth examples.
Sure: Take a nice large ball...a basketball, or a bowling ball will do nicely. Mark two points on it...put them a good distance apart. Now, take a length of thread, tape one end to the first mark and the other end to the second mark. If the thread is tight, it will follow a "Great Circle" path across the ball...because that is the shortest path (if it wasn't the shortest path, then you'd be able to pull the thread tighter).
Do this with a globe (a three-dimensional map of the Round Earth) - and look where the thread lies...notice the path it takes across the oceans and continents. Stretch your thread between two large cities and compare it to the maps that the airlines put out showing the routes their long distance flights take. Mostly - they follow the same path that your piece of thread takes - because the shortest distance between two points on a sphere is a "great circle". Sometimes airlines cannot take the great circle route because of political or air traffic control considerations - US airlines don't over-fly North Korea these days - but long distance flights over oceans are almost always great circles.
The reason great circle routes confuse people is this:
When you try to flatten a sphere out into a flat paper map - there are GUARANTEED to be some distortions - some parts of the map are stretched and other parts are squished.
When you distort a "great circle" path onto a flat map, the result is some weird curve (which exact shape depends on which "projection" was used to flatten out the map).
So when you look on a flat map, great circle routes LOOK very inefficient. They aren't the shortest paths across the map...but the problem isn't with the route - the problem is with the map itself.
Are "great circles" used in other ways? Presumably. Every time anyone needs the shortest distance across a sphere. It's hard to come up with examples though. We use spheres often enough...we make balls for sports, balls for kids to play with, ball bearings...but I'm having a hard time thinking of other things. Does anyone really care about the distance over a ball bearing or a basketball? I guess not.
Just because you can't come up with another example of something doesn't make it untrue.