A common critique of the Lord of the Rings is that the Great Eagles of Manwë are a deus ex machina; they rescue Frodo and Sam from imminent demise on the slopes of Mount Doom after the ring is destroyed, thus why couldn’t they fly them the entire way there, eliminating the need for a 6 month through-hike of Middle Earth?
We will not comment on the sides of this debate. Instead, we created a Euclidean path outlining the hypothetical least cost path if flying were an option (Figure 5.1).
Figure 5.1 The route from the Shire, to Rivendell, to High Pass, then flying on the backs of the eagles following a Euclidean path to Mordor.
If the Fellowship could call on the Eagles to fly them to Mordor (once they hiked to High Pass) then at an eagle’s flying speed of 50 km/hr, it would take Frodo a mere ten hours to get there and destroy the ring!