As I've said, it is the only accurate determination of the geometry of Earth's surface, and it's straight and therefore flat.
Yes, you have said that repeatedly, but you are yet to justify it. Until you do, your argument is just a pile of crap.
I see what you mean here, but it doesn't rebut my point here, as I will explain.
Yes it does, as the example I provided clearly does.
Address that example as it uses your method and gets a completely false result.
What I'm claiming here is that if you have a line cross a surface, and you draw a straight line to the surface across the traversal at an equal distance, it will reveal the geometry of the surface. The key word here is "surface", I used that in the definition.
Your example with the wall 1000 km or so north of the equator you are traversing is able to get a parallel straight line geodesic parallel to a curved line, this is because in Non-Euclidean geometry (will represent a sphere here for demonstration), the line traversing the surface is parallel to a line straight across, which also change position in a third dimension relative to the parallel lines, which means this can only be represented in a 3D coordinate system.
No. It can be represented in a 2D non-Euclidean system.
You can also represent it in a 3D Euclidean system noting that it is a surface.
The same applies to space-time, which can be represented as a 4D surface in 5D.
Again, here is a 2D representation, noting that it is non-Euclidean space and thus the line, while it appears curved, is actually straight.

The blue and green lines are straight, the red and purple lines curve.
This is required to define the curved and straight as parallel in this specific case.
No, not just this specific case.
In all cases dealing with non-Euclidean geometry.
This is because this isn't relying on this principle of non-euclidean geometry to determine a flat surface, so it doesn't hold and is irrelevant.
No, you are trying to define a surface in non-Euclidean geometry. You can do the same with the wall example.
Rather than accepting reality of the wall is (a curved line), you can instead try to determine the surface (you can't say you are using to determine a flat surface as that is what you are trying to prove. If you need to start with the asserted that Earth is flat you have made your entire argument useless).
Using your method you will conclude that the red and purple circles are straight. In reality, they are not.
No it's not, since one doesn't rely on 3D non-euclidean geometry to determine this
Neither rely on 3D. Both are 2D problems. (alternatively you can state yours is 4D, mine is 2D)
Yours uses radius and longitude (or fraction around orbit).
Mine uses latitude and longitude, where latitude can be treated as the radius.
Both are allegedly trying to map surfaces in non-euclidean space.
the only reason why the curved and straight line can be parallel is because it is all plotted in an exclusively 3D coordinate system, it can't be represented in a 2D plane, like the surface geodesics can (these lines passing straight through the ground can be represented as 2D and straight parallel lines being the same distance apart holding.
Again, pure bullshit.
I represented it in a 2D plane. Straight "parallel" lines were not the same distance apart.
Instead a straight line remained the same distance from 2 non-straight lines.
It doesn't, there is a clear distinction.
No there isn't.
Again, both work exactly the same.
You have a reference line. This line is straight.
You have some other line which you are trying to determine if it is straight.
You traverse your reference line, measuring the distance to the other line (perpendicular to your reference).
The other line remains the same distance during the entire traversal.
Thus you conclude the other line is straight as well.
This describes both situations.
(Again, I'm ignoring the rest of your bullshit until you deal with this).
You failed to, you went on to an irrelevant 'gotcha' with non-euclidean geometry that misses how the surface is being determined here, I hardly call that 'showing it's not the case'.
No. I went on a highly relevant "gotcha", with non-Euclidean geometry, and attempting to determine a "surface" just like you were, showing that it does not work.
So no, that is showing it is not the case.
For flat-space it is. Not for curved space.
Yes, for flat space it is the case. 2 straight, parallel lines remain equidistant. This is not the case for curved space (or curved space-time).
But as I have said, it can only match the surface geometry it is parallel, an equal distance to match the changes in the surface directly, doing this with Earth (in non-euclidean space-time) reveals it's flat.
Similarly, doing it with a line which remains 10 degrees north of the equator of a sphere in spherical geometry shows that that is straight. But in reality, it is curved.
So no, doing this doesn't reveal anything except your dishonesty and unwillingness to admit error.
This doesn't work.
Exactly, and a straight line has the same vector as the Earth's surface and can be represented in a 2D plane as actually equivalent lines and not the non-euclidean 3D exclusive example you brought up which isn't relevant.
Is your space-time Euclidean or non-Euclidean.
Because this argument seems to be indicating your space-time is flat, going completely against GR, meaning orbits are no longer geodesics, and which would show that Earth is round.
You seem to like going off on tangents to cover up your complete failure.
How about you stop setting up such pathetic strawmen and instead focus on what I have actually said?
As I have been doing.
No you haven't. You have been continually dismissing them as irrelevant without any rational justification.
Only, as I have shown, the paths with the surface can be represented in 2D plane, and when you do that, the lines are not parallel as they are not an equal distant apart
And here you go off on circular reasoning yet again.
That is entirely the point, they are parallel yet don't remain the same distance apart because it is not in flat space.
Until you justify this baseless claim of yours, where you use something which only applies in Euclidean space, the rest of your argument is pure bullshit.
You also need to address where I showed this is not the case.
Only, the moment we transfer this to the plane representation of the line cutting to the surface (the line doesn't have width so it doesn't require 3D to represent), we reveal that the straight line reveals a flat surface. Parallel lines can't converge since they are an equal distance apart constantly by definition
Nope. Exactly the opposite. When we transfer this non-Euclidean space to a Euclidean representation, the straight lines appear to curve and parallel lines can converge or diverge.
They only remain the same distance apart by definition in Euclidean space.
Outside of Euclidean space that axiom does not hold.
Instead it is replaced by 1 of 2 other options for parallel lines.
The first one is the simplest which is simply any 2 lines in a plane which do not intersect.
In this case, spherical geometry has no straight parallel lines; Euclidean geometry has a single straight line passing through a point, which is parallel to another straight line; and hyperbolic geometry has infinitely many.
Perhaps this will help you, this is for hyperbolic space:

The other option is using one of the methods of creating a straight parallel line through a point:
You take a straight line (the reference one). You draw another straight line (the construction line) perpendicular to it (technically at any angle), which passes through the point, and then draw a straight line through the point perpendicular (or at a supplementary angle) to the construction line. This line will be parallel to the first.
The issue is that it relies upon key points on the lines, as if you move along the lines, this supplementarity of the lines doesn't always hold.
their vectors are the same.
No they aren't.
The vectors in non-Euclidean geometries are significantly more complex than that.
Believe it or not, I still fail to see how I'm wrong.
It's quite simple, the red and purple lines remain the same distance from the "straight" green line, yet are not straight.
Thus, the method you are trying to use is leading to a false conclusion. As such, it is flawed (i.e. WRONG).
Is that simple enough?
In order for you to be correct, the red and purple lines need to be straight.
Not in this case, you have to look at why this is the case in the non-euclidean example you used
I already know why. It is because the argument you are using ONLY APPLIES TO EUCLIDEAN SPACE!!!!
It does not apply to non-Euclidean spaces.
Instead of trying to look at why it shouldn't apply, try to figure out when it should and what conditions are required for it.
when you have that and apply it to the surface geodesics, it is revealed that they are not connected and therefore irrelevant.
Again, PURE BULLSHIT!!!
Your geodesics are in non-Euclidean space. As such, you have the same issues. It is connected and thus is relevant.
In my case, Earth is a flat surface since an equidistant line traverses it as straight
And in my case, the red and purple lines (which are not straight) were deemed to be straight as an equidistant line traverses it as straight.
the non-euclidean nature of space-time isn't quite relevant in this case (other than that a straight line meets itself because it's in non-euclidean space)
No, it is relavent as it means the parallel lines do not remain the same distance apart.
since the line has no width
It doesn't matter.
The "width" corresponds to the distance between the line and the surface of Earth.
Then show that this is the case with my specific point here with a line parallel to a surface of earth (You will only need to represent it in 2D since the lines have no width, just distance from the surface and length across) and I will abandon this model.
That would require you to first accept that you are wrong and that your method does not work.
You are basically asking me to show your method is wrong, without using anything other than your method and the thing you are applying it to.
In what world does that sound reasonable?
I'm taking the more honest/rational approach. Using your method in a situation where there is agreement on what the facts are (that the red and purple lines are curved), and thus showing your method does not reach the correct conclusion and thus is wrong.
How is it a moot point in determining if Earth is flat? If I can trace a parallel line to the surface that can be represented in a 2D plane (since the lines have no width of course) and it's straight, then isn't that a way of telling?
No it isn't, because you aren't tracing a parallel line, you are tracing a line which is equidistant, but more importantly, because Earth isn't in flat space-time.