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Wednesday, November 1, 2017

Flat Earth Follies: Isle of Man, seen from the Great Orme

Flat Earth Claim:  Isle of Man, seen from the Great Orme

Ok... and?  Is that supposed to be impossible?  How do you know?  Did you dishonestly use the 8" × d² formula which ignores observer height and height of the distance object?

What is the elevation of the observer?  What is the vertical field of view of that camera and lens?

Where exactly was your image taken from?

I'm going to use this higher resolution image because it has the EXIF data intact and I know the source which tells me the location of the photographer.

Image Credit: (post)

EXIF data:

NIKON D80 (Sensor: 23.6 mm × 15.8 mm Nikon DX format - APS-C crop ~1.53x)
135mm F5.6 1/100 ISO400 (35mm focal length: 202)
3753 x 2501
2010:11:23 14:32:46

Given these stats we also know we have a 10° x 6.61° Field of View and each pixel is ~0.00259°.

And from the blog post we're told "This picture was taken from the Great Orme, by the Rest & Be Thankful Cafe" -- so we have a location:

Latitude: 53.341525°
Longitude:  -3.869581°

Let's see - the Rest & Be Thankful is about 400 feet over the water which matches this image very well.

There are several peaks on the Isle of Man, Snaefell being the highest.

Snaefell 2,034' (Heading 339°)
Beinn-y-Phott 1,791'
North Barrule 1,842'

This point is 68.2 miles from Snaefell peak.

We can use PeakFinder to help identify the terrain:

By the way -- there is a whole bunch more of the island off to the left but you can't see it in this image because it falls below the horizon and then out of the frame.

And we can overlay this image in Google Earth Pro -- I've drawn lines from the observer to Greeba, Snaefell, North Barrule, and North end of the island mountain range.

All of these give us a very good match with the observation.

Now let's compute the angular size of Snaefell, which is 2,034 feet high, at our distance of 360,084 feet.  Angular size formula is:

α = 2*arctan(g/2/r)
α = 2*arctan(2034/2/360084) = 5.649 mrad or 0.3226°

And now we can estimate the number of pixels by dividing this by the angular size of each pixel.

0.3236°/0.00259° ~125 pixels (estimated) -- which means Snaefell should be this tall if we could see the whole mountain -- but we're missing about 60 pixels or 48% of the mountain.

Using the FEI Horizon Calculator with the Standard Atmospheric Refraction of 15% (link) we can see that we expect only the bottom 837.1 feet of this mountain to be missing at this distance - or 41.2%.   So this is probably closer to 10% refraction.

This span also measures just about 12 miles wide at the distance of Snaefell, which gives you just about 10 degrees wide using our angular size formula.  We cannot see the exact extents but we can estimate the distance using Google Maps and using existing features to estimate the portions on the side.

This is HOW perspective works.  There isn't an ADDITIONAL perspective factor at play here.  The angles made by line-of-sight are what cause perspective in the first place.

From another one:

I'm glad you took into account the observer height but you missed Refraction.

In other words, the ray curvature is about 15% that of the Earth
from Calculating Ray Bending 

Tangent drop according to 8" ESTIMATE formula for 63-6 miles:  2166 feet -- we agree on this

However, using the more accurate formula and accounting for refraction we find:

Amount hidden, or obscured, at 63 miles from 26 feet, given standard refraction (15%): 1773.3 feet

So that actually leaves about 261 feet of the tallest peak above the Horizon (12.8%).

If the Earth was Flat you would see much more of those distant peaks than you do, they extend well below the horizon.  Fly a drone up as a high as you can and overlay two images taken from the same position with one at high altitude and one down lower, lining up the peaks.

I've explained why you only see the peaks but Flat Earth can't -- on a Flat Earth we should see all the way down to the shoreline to the limit our of resolution.  The resolution cannot leave the top half in scale at dozens of pixels and magically squish the bottom dozens of pixels down into 1 pixel.  That isn't how perspective works.

If you calculate the angular size of Snaefell at 63 miles you can calculate the angular resolution here very easily just as I did in the first example.

Don't just TELL ME that we're seeing "the whole mountain", DEMONSTRATE IT.

And look at this next one, you can just barely see the peaks with big gaps in between.  This is clearly NOT the whole mountain, this is about 65 miles from Blackpool.   Behaving exactly as we expect on a Globe, shrinking FASTER with distance than angular size alone -- 3% further but much more than 3% shorter.

Isle of Man from Blackpool (~65 miles)

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