[Flatout]

I made a little video to clarify my point.

[Silent Service]

I made a little video to clarify my point.

Thanks for the video.

This is definitely perplexing to me since I did the math and it doesn't support what you show in the video.  So, there are one of two options: 1) either I messed up the math (which is totally possible) or 2) whatever tool you're using in that program isn't displaying what you think its displaying.  I'll leave the math here, if you can figure out what I did wrong then I'll have learned something.

Initial assumptions
Initial velocity is due east (velocity only in the west-east direction)
No change in altitude (R remains constant)

Math sources
https://en.wikipedia.org/wiki/Circle_of_a_sphere
https://en.wikipedia.org/wiki/Arc_(geometry)
https://en.wikipedia.org/wiki/Chord_(geometry)
(I also used some basic calculus so if you don't know how to do calculus I can walk you through it)

Variables
R = distance from us to center of the earth
d = radius of the circle that bisects the earth at our given latitude
Theta = longitude degrees
Phi = latitude degrees
WE = west-east distance
NS = north-south distance
dWE/dt = west-east velocity
dNS/dt = north-south velocity

Okay, so the first thing we need to do is figure out the radius of the circle that bisects the earth at our latitude.  From the first link we have the radius of a circle that bisects a sphere as:

BC^2 = AB^2 + AC^2

BC = R and AB = d in our example.  We can figure out that AC = R^2*sin(Phi) based on trigonometry.  Plugging in each of our variables we get:

R^2 = d^2 + R^2*sin(Phi)

Now solve for d

d^2 = R^2 - R^2*sin(Phi)
d^2 = R^2(1 - sin(Phi))
d = R*SQRT(1 - sin(Phi))

Ok, from here we can calculate distance traveled in the west-east direction and distance traveled in the north-south direction in terms of our spherical coordinates.  For both directions we'll use the equation for an arc.

WE = d*Theta (theta in radians)
WE = R*SQRT(1 - sin(Phi))*Theta

NS = R*Phi (phi in radians)

Lets look at the north south direction first.  Our initial condition is that our initial velocity is in the west east direction only.  Therefore, dNS/dt = 0.  We also know that R is constant (and non-zero.)  If we take the derivative of our NS equation we get:

dNS/dt = R*dPhi/dt

Since dNS/dt equals zero and R does not equal zero then dPhi/dt equals zero.  Since dPhi/dt equals zero then integrating will give us that Phi is a constant.

Now going back to our west east direction if phi is a constant than SQRT(1-sin(Phi)) is also a constant.  We'll call this constant C.  This simplifies our equation to:

WE = R*C*Theta (theta in radians)

therefore, our velocity in the west east direction is:

dWE/dt = R*C*dTheta/dt

Since spherical coordinates defines Theta and Phi as being orthogonal to each other we know that our initial velocity, which only has an west-east component, will result only in a change in Theta and not a change in Phi.  This results in only longitude changing but latitude remaining constant.

Like I said, maybe my math is off.  Alternatively, maybe you're using the computer program incorrectly or you have a concept error somewhere.  I'll try to figure stuff out in the program and you figure out if I messed up in the math.

And if we do find out my math is wrong I'd love to calculate how much change in latitude you see for every one degree of longitude.  That would be a fantastic experiment that people could test to give evidence for flat/round earth (of course you'd have to account for things like wind.)

[Roundy]

I don't know if this helps (I only had time to glance at your post) but it seems like you think that any latitude line bisects the Earth in the RE model. That is incorrect; it's true about longitude lines but not latitude. Only one line of latitude actually bisects the RE Earth, geometrically speaking: the Equator! I hope that helps clear things up.

Honestly I think your biggest problem is that you're overcomplicating things.

[Silent Service]

I don't know if this helps (I only had time to glance at your post) but it seems like you think that any latitude line bisects the Earth in the RE model. That is incorrect; it's true about longitude lines but not latitude. Only one line of latitude actually bisects the RE Earth, geometrically speaking: the Equator! I hope that helps clear things up.

Sorry, I may have misspoke or been unclear.  I didn't mean bisect in the sense that it cuts through the center of the earth.  I mean take a slice of the sphere that is parallel to the equator at a constant latitude.  The equations will calculate that circle for any latitude.  Does that clear things up?

[Roundy]

Not really. I don't have any interest in checking your math, but if it shows that a circle drawn anywhere on a sphere but the exact middle doesn't curve, it's obviously wrong. Indeed the very definition of "great circle" is basically that it is a straight line drawn across the surface of a sphere, and that only happens at the exact center (the Equator in the case of Round Earth).

[Roundy]

Here's a link explaining great circles. Hope it helps. https://en.m.wikipedia.org/wiki/Great_circle

[Flatout]

Silent, I understand the math but fail to understand how it applies.

[Flatout]

I think you should play with a globe.

Round 2😁

[Flatout]

Maybe this will help

[Rounder]

So let's say I'm facing east on a flat earth.  North is 90 degrees to the left of me.  However, if I walk/run/drive/fly/etc in a straight line after a few hundred miles I will no longer be going due east.  In fact, the longer I go in that straight line the more south I will actually travel because north is always defined as the direction between an observer and the north pole.  In order to keep going due east I would routinely have to turn left.  However, we know this doesn't happen in real life.  So what explains this on a flat earth?

Getting back to the original question: I've highlighted the problem in your proposed experiment.  How do you propose to know this?  Suppose we fly, since any surface route introduces obstacles to the straight line path that far exceed our ability to determine straightness.  So, flight.  Against what standard are you determining the straightness of your flight?  Feels straight?  Air is rough, and moving, you can't use your "hands off the wheel" method.  Visual?  That only gets you a few miles before you have to pick new objects to use as reference.  Compass?  Well now you're turning left in either model.  More so on the flat, but still a little bit on the round.

I agree with you about the earth's shape, but this is not the way to demonstrate it.