The same issue applied to the early explorers going to the North and South Poles.

Dozens of men (and horses) would carry food for the few men that would eventually make the final push for the Pole. The food would be dropped off along the way and the the extra men would turn back way short of the Pole.

Of course, the horses were also "dropped off" along the way and supplied some of the food for the return trip.

The problem is that there is no such thing as "a map." There are dozens of ways to project a sphere onto a flat surface, none of them good for all purposes. Conclusions such as these may be correct for one projection but not for another. If you are assuming a Mercator projection, say so, though its distortion at high northern and southern latitudes makes it useless for flight paths that go into those regions or that one is comparing with other paths that go into those regions.

Aside from that, though, your earlier point about the lack of refueling stops anywhere near the middle of several long-distance great-circle routes is valid. Breaking up a long trip to refuel, thus carrying less fuel, makes sense only if such stops exist. On most polar routes they don't, though IIRC some flights from the US east coast to Asia refueled in Anchorage in the days before non-stop flights of that distance were practical. ANC may not have been in the ideal location for that purpose, but it was close enough.

Pan Am went to great lengths to build up the infrastructure for its TPAC Clipper services...including setting up the then-desolate Wake Island among other Pacific stations for refueling and resupply. It's unlikely we'll ever see an airline take, or need to take, such an initiative to build a few fuel stops from the ground up like that. But such will be necessary for various space travel needs in the future. A brief article on the early Pan Am setup days:

https://www.clipperflyingboats.com/t...irline-service

Of course the world's top air forces do have the flexibility to place an aircraft refueling point at lots of locations as needed for a given mission, thanks to air-to-air refueling! And no need to land and taxi in for the fill-up! ;)

Cargo carriers do this, like FedEx/UPS. They'll make a technical stop in Alaska before continuing on to Asia. The slight loss in time is made up with fuel savings.

They don't do it for fuel savings. They do it because the cargo load is too heavy to make it non-stop.

For the method in which the cargo is carried, it represents a fuel savings, ie 1 heavy trip with a fuel stop, lighter trip with no stop.

How is it a fuel savings when you left cargo behind that will have to be flown on an additional flight? The will certainly burn more fuel.

In terms of fuel per weight*mile, isn't a heavier plane with a fuel stop cheaper?

whoops, should have said

not sure when the energy equation will ever make sense, for there to be electric-assisted commercial passenger aircraft... just like cars. fixed weight. on-demand power delivery. limited capacity, but enough to assist on take-off phase, and then "sip" on jet fuel for the cruise....

No. Flying non-stop will use less fuel. You also save time on the aircraft which reduces maintenance and crew costs.

your course assumptions are further complicated by wind velocities. the last couple times on the western route, we flew from middle of southern europe to iAd, we used the great southern trade wind route. south of great Britain and Gurrnsey,

This is a really really good video that explains it well. It's mind blowing how much fuel it takes to move fuel.