I once asked John Davis a related question, as to whether the sum of the angles in triangles on the flat earth would add to 180 degrees, he said that they wouldn't because the surface was non-euclidean, so the (semi) official answer is that the surface of the earth is non euclidean but flat. ( if you can figure that out, let me know )

In my opinion all those flight times and distances are consistent with a non euclidean flat earth Now cue the argument about metrics. My money is on Minkowski. I think the flat earthers will disagree.

The entire basis of it is that you can orbit around Earth with Earth remaining the same distance from you, and as your orbit is by definition a straight line (or geodesic) through space-time, it must mean the surface of Earth is a straight line as well and thus Earth must be flat.

But there are a few massive problems with this.

The first, and most important, is that this reasoning only works in Euclidean spaces, not non-Euclidean spaces like space-time.

Another is that not all geodesics through space-time remain the same distance from Earth, not even all those which have some point travelling perpendicular to a line connecting Earth to the trajectory (which by definition would include all lines except those going directly through the centre of Earth).

Some will be hyperbolic trajectories, where Earth gets closer and closer, and then starts moving further away, which would indicate that Earth is convex. Others are sub-orbital elliptical trajectories, colliding with the surface of Earth which would indicate Earth is concave. Others are elliptical orbits where the distance to Earth varies periodically, which would indicate an undulating Earth.

Another issue is that it is ignoring all the other points on Earth's surface. Similar to being able to measure the distance to the point below, you can also measure the distance to other points on the surface of Earth, and that indicates Earth is a sphere.

But the simplest analysis is with the equivalence principle which standing on Earth in the presence of gravity is indistinguishable from being on an object accelerating upwards at g. This shows that Earth's surface is not a space-time geodesic (or flat), and instead is constantly curving up (or out), constantly "expanding" as the time axis tries to crush it.

The other option is a completely different route where Earth is merely in non-Euclidean space (without trying to appeal to space-time).

The problem with this is how you measure the curvature of space and why light bends.

Unless you can make it distinguishable from a RE and light going straight (at least through space-time), then it is just a RE in disguise to pretend it is flat.

It also creates the issue of all the FE claims of missing curvature would work equally well against this model (i.e. not at all).

Without light bending, the simplest way to measure if space is Euclidean or not is with light, and there are several ways.

One is to shine 2 beams of light parallel to one another (preferably in a vacuum), and see how the distance between them changes (and as a bonus, the distance from the starting point to them, which is not necessarily measured along the line the light traveled but can be along another straight line)), doing so in multiple directions.

In Euclidean space, they remain the same distance apart.

In spherical space, they first converge (get closer together), then cross, then diverge (get further apart), reaching a maximum (which should be the initial separation distance) before starting to converge again, crossing and diverging and reaching a maximum at the starting point.

In hyperbolic space, they diverge, with the rate of the change in rate of divergence continually decreasing until eventually they reach a point where they act like 2 lines which started at an angle in Euclidean space.

In other spaces you can have some combination of the above.

In cylindrical spaces they remain the same distances apart, yet can loop around, with beams fired at an angle crossing one another multiple times (depending on the starting directions).

But if Earth's surface is going to be flat, then a beam of light fired parallel to Earth's surface should remain the same distance above.

Another option is to measure the apparent intensity of a light source with changing distance.

The 1/r^2 law is based upon Euclidean geometry where as the light propagates outwards, the same energy is spread over the surface of a sphere.

In hyperbolic geometry, due to the divergence stated above, the surface area of the sphere grows faster than r^2, and thus the apparent intensity decreases faster.

In spherical geometry, due to the convergence/divergence above, the intensity becomes periodic. It start decreasing as you move away from the source, but with a decrease slower than 1/r^2, reaching a minimum and then starts increasing, reaching a maximum (which is the initial intensity), and then starts decreasing again, reaching a minimum, then growing in intensity to original maximum at the starting point.

This can also be done in 2D with a 1/r law to compare with, but requires a 2D light source.