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If you drew a straight line on a flat chart or map, took the course ateach place where they crossed a line of longitude, you would come up with a series of constantly changing headings. That is what the flight aware "course" or route of flight reflects. Take the place where the aircraft crosses each 10 degrees of longitude on a flat chart, plot those points on a globe and you will see that it is actually a straight line. This aircraft shows a course initially headed north and eventually finishes south but in truth IS flying a straight line...Show more →
Aviation sectional charts employ a Lambert Conformal Conical Projection. According to the Wikipedia article, "Pilots favor these charts because a straight line drawn on a Lambert conformal conic projection approximates a great-circle route between endpoints as long as distances are not great." Which is what Mike is getting at, I'm guessing.
But over shorter distances, a great circle route is negligibly shorter than a route flown at a constant heading. Over longer distances and especially for roughly east-west routes between points at at higher latitudes, I don't know of any flat map projection that allows you to draw a straight line to plot a great circle, at least not in the general case.
As for that plot of the ground track of an orbiting satellite: that's a whole 'nother discussion having to do with the fact that the Earth rotates underneath an object in orbit. If the satellite is placed into an orbit inclined w/r/t to the Equator, the ground track depicted on a Mercator projection looks like a sine wave.