You just couldn't have picked a a worse example to prove RE

Did you know the haversine formula works on AE maps unmodified?

spherical distances are calculated using the haversine formula, which takes as input lat/long

lat/long is an invariant for the AE projection and so the globe formula will still work for the AE map unmodified.

The distance is the same number the globe physics model says it is and thus the same as my flat model says. (because the models are equivalent)

True, Haversine is independent of projections. But where I'm kinda hung up is still on a few things mentioned by others and maybe some not mentioned:

- Isn't it a little strange that the AE projection you're basing this all on is just that, a projection...of a globe?

- Isn't Haversine based on spherical geometry, specifically a great circle on a ball?

- In geodesy, Haversine may be used for the quick and dirty shorter calculations, but predominantly Vincenty is used because it's more accurate, based upon an ellipsoid rather than a sphere. That would seem to alter the look of the AE projection to a degree

- You mentioned something about a depiction of reality and not a map. I get that, but what's a "flat earth ruler" and where can I find one because it doesn't seem to exist in said reality

All of this gets wonky pretty fast when you start mixing projections with reality (navigation) with useable maps (navigation). For instance, the AE projection is a projection from a globe. As mentioned before, because projections are required for a globe onto a flat surface, distortions arise. For a flat earth depiction on a flat surface, no distortion is required, it's one-to-one. And it would actually resemble reality. And as I think AATW mentioned, I too have yet to see a non-distorted depiction of a flat earth on a flat surface.

Then there's the reality of navigation. How does that work exactly? Ex., QANTAS JND>SYD flight, the distance may be able to be calculated correctly on the AE, but from a reality/visual perspective, the route makes zero sense.

And of course, little things like how have people been able to circumnavigate Antarctica when it's a ring walling in the earth.

Then there's the whole bendy light rabbit hole which, well, probably needs a whole separate discussion around that.

Curious about your thoughts and thanks for pulling all this stuff together, makes for an interesting examination.