Round Earth is a better model than Flat Earth

Not that I probably need this discussion but there is a sizeable group of people who maintain the Earth is flat. And this is important for the next blog on the nature of Society. Well flat earth is fine, its certainly a good approximation for the local geometry but is the entire earth flat is the question. And if it is the concept of "entire earth" becomes interesting because what is the boundary like. Does the water fall off the earth in a massive waterfall, or is the earth an infinite sheet. Boundary conditions are almost always more important than the main model.

So here is a picture I took in 2019 of turbine in the Rampion Offshore Wind Farm on the South Coast of Britain. It is perfect for measuring the curvature of the Earth because there are platforms near the base.

Now lets look at the theory predictions:

Flat Earth predicts that as things disappear into the distance they get smaller simply by perspective.

Round Earth predicts not only that things will get smaller with perspective, but that they will drop over a horizon too.

Regarding the platforms:

Flat Earth predicts that the turbines will just get smaller with distance, so the ratio of the distance below the platform and above will remain the same. 

Round Earth predicts that the turbines will fall below an horizon and so the ratio of distance below and above the platform will also reduce.

If we measure the turbines from the picture and scale our turbines to be the same we get this:

 Clearly the distance below the platform is reducing. This suggest an horizon below which the turbines are falling.

Let now try a Round Earth model. Very simply an "horizon" is the tangent to a great circle on the Earth sphere. we can convert to 2D now. Lets project everything onto the line that runs through the horizon so that distances are relative (what we see with perspective). This is the also the ry axis of the model. I've labelled 3 points (w1,w2,w3) as windmill base, platform and height respectively. As the windmill moves farther away on the circle I've drawn an example of how the points move. We then project these back onto ry from the perspective of the viewer (the star) who is 'a' metres from the horizon line. That is just scale it according to how far away the point is. Earth Radius is just R. The angle 't' that a windmill on the surface of the earth subtends to the horizon line is simply distance from horizon/R. I have two vectors rx and ry for the x and y of the model. This doesn't need dot or cross products as rx and ry are orthogonal and I've chosen ry for the projection to make it ultra simple.

Here are results for an observer 10km from the horizon. You can work out how far you are from the horizon from your elevation (D). And can work out the physical drop like this:

D = R(1-cos(a/R))

But this is not projected. For projected results you get this. Yellow is the apparent/relative top of an 80m wind turbine as it moves away from the horizon. The y-axis is just for relative measure. Magenta the platform and blue the base.

What is interesting is that in the picture the turbines appear to be dropping over the horizon at a constant rate. If you work it out the turbine platforms lose 2m as they move each step away. They have already lost 10m by the 1st turbine. Can work this out cos the platform should be 20% of the overall height so you can see how much is missing. And a turbine stands 80m out of the water. While the tops of the turbines appear to drop very unlinearly due to perspective the platform appears much more linear and by the time it is approaching disappearing over the horizon at 14km its looks almost linear.

The Rampion Offshore Wind Farm is massive. It is over 10km from the coast and is almost 10km wide. I don't have any data however on which part I photographed. I need to go back and get an accurate photo.  

 In the photo it looks like the next turbine will have its platform on the horizon so from the model can say that it is 14km from the horizon so that is 24km from land.

This model looks much more promising than the Flat Earth model which suggests the turbines would just get smaller as they went into the distance and we would be able to see their bases in the water as far out as we could see. Clearly in this photo you cannot see the bases in the water: they all appear to rise from the horizon. And the platforms drop to meet the horizon so they are clearly "over the horizon."

There may however be other ingenious models that I have not considered to explain this behaviour. Perhaps a new Copernican revolution is yet to happen. Certainly physicists talk of the world being in 11 dimensions not 3 so we don't really understand the full geometry of the Earth. Plus it is fractal so has no clear surface. But for the purposes of simple observations like this the Round Earth does seem to be a satisfactory model.