I don't agree pythagorean distances are the only distance metric. In our discussion we're more interested in arclength along a greatcircle distances.

But fine, let's go with straight distances through the earth.

It isn’t the only distance metric,

* that’s the whole point*. But there are defined metrics for Euclidean and non-Euclidean spaces. You can’t just make up your own based on how you want it to turn out and expect your model to reflect reality.

That’s the metric for non-Euclidean spaces. It takes arclengths, angles and great circles into account.

I can’t help you with the math, but if you aren’t using that formula for your metric, your model doesn’t reflect reality.

These are not arbitrary formulas. This is the way you're supposed to do coord transforms

Some coordinate systems are inherently Euclidean (they only have two coordinates) and some non-Euclidean (three coordinates) The metric is baked into whatever coordinate system you are using. If you transform from a spherical coordinate system to another spherical coordinate system, there is no distortion because they both have the same metric. When you transform from a coordinate system that is inherently Euclidean to one that is non-Euclidean, the transformed system becomes non-Euclidean and there is distortion.

If you are measuring the triangle on a sphere you get the measurements on the right, if measuring on flat space, you get the measurements on the left.

Distortions are the result of using a non-Euclidean metric in a Euclidean space. If there is a triangle that in reality, on a sphere earth, looks like the one on the right, it will look like the one on the left on a flat space. It won’t be accurate. Your model is inherently distorted because you are using celestial coordinates, which have three coordinates, and projecting them onto a Euclidean space, which is measured in only two coordinates.

The geometric properties of the space depends on the metric chosen, and by using a different metric we can construct interesting non-Euclidean geometries such as those used in the theory of general relativity.

http://wiki.gis.com/wiki/index.php/Metric_spaceYou can apply any random metric to any shape, using any coordinate system and mathematically “change” the underlying geometry but if you are mixing and matching Euclidean and non-Euclidean spaces and metrics, your are always going to have distortion from an actual real, physical structure.

How will you tell?

Both models are indistinguishable. You called them a single model.

There is no measurement or observation of reality that can tell them apart. They both represent is equally well.

Your logic is backwards. You are starting with the assumption that the earth doesn’t have real, physically defined geometry to begin with and the geometry isn’t defined until some arbitrary, random metric is applied to it. We can't know the correct geometry because we don't know the "correct" metric.

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*But we do know the correct geometry of the earth*. It has intrinsic curvature, which by definition, can be measured by the “inhabitants”. You keep saying that there is no test or observation we can use, but that is just wrong. There are lots of them.

10.2 Parallel Transport in a Curved Space, D=2

Consider the scenario shown in figure 12. It starts out exactly the same as the previous scenario. However, in this case we suppose that the arrows exist in a two-dimensional space, namely a sphere, i.e. the surface of the earth.

The yellow arrows along the equator are exactly the same here as in the previous scenario. Even though the arrows are the same, we have to describe them differently. We say they all point north along the earth’s surface. They point toward the earth’s geographic pole, not toward the celestial north pole, because the latter does not exist in the two-dimensional space we are using.

As we move northward along the leg of the triangle that goes through North America, the arrows in figure 11 continue to point north toward the geographic north pole. Relative to the arrows in figure 12, these arrows must pitch down so that they remain within the two-dimensional space. They are confined to be everywhere tangent to the surface of the earth. As we move north, each of the arrows is parallel to the previous arrow, as parallel as it possibly could be.

Let’s be clear: Each new arrow is constructed to be parallel to the previous one, as parallel as it possibly could be. What we mean by “parallel” is discussed in more detail in section 10.3.

After we get to the north pole, we start moving south along a the prime meridian. We move south through Greenwich and keep going until we reach the equator at a point in the Gulf of Guinea. As always, each newly constructed vector is parallel to the previous arrow. All the arrows on this leg point due east.

Finally, we move west along the equator until we reach the starting point. Again each arrow is parallel to the previous one. All the red arrows on this leg point due east.

At this point we see something remarkable: The final arrow is not parallel to the arrow we started with.

From this we learn that in a curved space, there cannot be any global notion of A parallel to B. We must instead settle for a notion of parallel transport along a specified path. That is: the notion of parallelism is path-dependent. It also depends on whether you go around the path clockwise or counter-clockwise.

If you start with a northward-pointing vector in Brazil and parallel-transport it to the Gulf of Guinea, you get a northward-pointing vector. If you start with the same vector and transport it clockwise around two legs of the triangle as shown in figure 12, you get an eastward-pointing vector.

**Creatures who live in the curved space can perceive this in a number of ways. Careful surveying is one way. Gyroscopes provide another way. That is, a gyroscope that is carried all the way around a loop will precess relative to a gyroscope that remains at the starting point.**

https://www.av8n.com/physics/geodesics.htmParallel transport in a flat environment is not path dependent. It would be a simple matter to test and if it is found that parallel transport is path dependent, then we know we live in a curved environment. If we want to accurately model it, we know to use a 3 coordinate system, which will give us the correct metric.