I'm also currently writing a complete 737 suite and without much thought implemented a Mercator view. For normal small legs this is fine, but when legs start to get larger than 80 NM, you will see the aircraft, which follows a perfect great circle route, start to move away from the magenta line and later catches up again.
I will look up the stereographic projection, but I'm still interested to know if the real aircraft draws arcs or straight lines only.
Marty