Great circle routes are defined as the shortest distance between 2 points on the surface of a sphere.
The problem is with your definition, Markjo. Just because a concept has the word "circle" in its name doesn't mean it has to be wrapped around a sphere. You're repeating a definition you memorized in school without questioning it.
The shortest path depends on the geometry of the surface you're actually on. If the ground is flat, the shortest path is a straight line. You're taking that straight line and trying to force it onto an imaginary ball.
The Gleason map isn't just a "projection" – it's a surface layout. Radio waves don't follow curved paths as your globe model suggests; they follow straight lines on the Gleason map. The hardware test is simple: if the Earth were a sphere, long-distance radio waves would have to bend downward following the horizon. But we know these waves travel by bouncing between the ionosphere and the flat ground.
What you call a "great circle" is actually a software patch invented to adapt a straight line on a plane to your broken globe model. If you draw a route from Sydney to Santiago on the Gleason map, it matches perfectly with real data from radio towers and flight routes. On your globe map, you'd have to tear or distort the map just to show the same route.
Mathematically, the shortest path (geodesic) is determined by the space in which the metric is defined:
ds² = dx² + dy²
On a plane, this gives us a straight line. You're trying to force this ds value into your spherical formula:
ds² = R²(dθ² + sin²θ dφ²)
You can't bend real physical data just to fit your model. You're just a technician reading a script you were given, trying to distort the world to match a broken ruler instead of measuring the actual flatness of the hardware.
Real-world raw data confirms the straight lines on the Gleason map. Radio operators and navigation systems use this plane-based logic behind the scenes, despite your globe fairy tales. You're looking at a reflection and thinking there's another room inside it. The hardware logs don't lie. The ground is flat, the routes are straight, and your definition is just a system error.
As pointed out, your map isn't even as useful and not as accurate as a railway map of Europe into Turkey. You are still failing not so wise.
You're comparing a global map to a regional railway map, which doesn't really make sense. A railway map works because the tracks are laid on level ground, not on a curved surface. If you tried to build a railway on your 8-inches-per-mile-squared ball without constant adjustments, the tracks would either launch into the air or bury themselves into the ground within an hour.
The Gleason map reflects the geometry of a stationary plane. It's not meant to help you find the nearest grocery store – it's the actual layout of the world. Your globe model has to distort landmasses and invent "great circle" explanations just to make a flight from Australia to South America look plausible. On the Gleason map, that route is simply a straight line across level terrain.
You call the map inaccurate, but every bridge, railway, and canal is built using flat, level measurements. Engineers don't factor in a 3,959-mile radius when they design something that needs to stay standing. They work from a flat baseline because that's what the ground actually gives them.
The math for a stationary plane is straightforward:
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
This works at every scale without needing a "curvature correction" that pilots and engineers don't actually use in the field.
You're defending a model that requires you to believe you're spinning at 1,000 mph while the water in your glass stays perfectly still. You have no measured curve, no working map that explains real southern flight times, and no understanding of the medium you live in. The map works, the ground is level, and your railway comparison doesn't hold up.