Help me out here, I (sometimes) do work with GPS data and 3D orientation on an engineering side (i.e. relatively far from the mathematical side of it). Isn't the great circle effect inherent if you are not using a 2D representation of a map? My layman understanding is if you calculate your destination / origin based on sun angles etc, the solution would be inherently represented by great circle route (This coming from a point of poor knowledge on my behalf, not making any statements).
A key distinction here is between a great-circle route and a rhumb line, and it's very easy to conflate them. That's in part because they look pretty similar on non-oceanic scales.
A
rhumb line is the route you take if you follow a constant bearing with respect to (most typically) due north. On a globe, it ultimately tracks out a spiral around the pole.
In contrast, a great-circle route will constantly change the bearing (other than in special cases like travelling directly north or south, or travelling along the equator): for example the route
LHR -> LAX starts off going north-west and ends up south-south-west. This route is directly ‘pointing at’ LA from London, which is obviously true on a globe, but looks wrong on most 2D map projections. That is to say, a great circle is a ‘straight line’ on the
surface of a sphere: if you start off in the right direction and ‘keep on walking straight ahead’, then what you trace out is a great circle (it's initially surprising that this process is
not what you're doing when you follow a rhumb line). This path is also the shortest distance between the start and end points.
The technicality that's special about the Mercator projection is that if you lift a bearing directly from that map, and sail constantly along it, then what you will trace out is the rhumb-line route from A to B, which will get you where you're going, though not quite by the geographically shortest route. If you were to take that bearing from a different map projection, you'd end up in the wrong place.
Going from start- and end-positions as latitudes+longitudes, and calculating the great-circle connecting them, is one of the standard exercises when astronomers or navigators learn their spherical trigonometry.