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Tuesday, November 28, 2017

Pic Gaspard (3880m)/Grand Ferrand (2758m) from Pic de Finestrelles (2826m) in the Pyrénées

I have some new toys to share in this installment of 'Yes, the Earth really is Curved even though you can see one distant mountain peak from another', so this should be fun!

This time we're going to look at a view of two distant mountains from Pic de Finestrelles (2826m) in the Pyrénées, taken by Marc Bret of Beyond Horizons (see also the Flickr album).

Pic Gaspard (3880m) in the Massif des Écrins range at a distance of 443 km.

Grand Ferrand (2758m) at a distance of 392.48 km.

Our view is right around 42.414466°N, 2.132839°E at about 2826 meters elevation, looking right along the coast.

 Pic Gaspard/Grand Ferrand from Pic de Finestrelles in the Pyrénées, image by Marc Bret
The EXIF metadata shows this image was taken by a Panasonic DMC-FZ72 with a focal length of 215mm -- given the 5.62 crop factor of the 1/2.3" sensor in this camera, this gives you a 35mm equivalent focal length of 1200mm giving a full frame view of 1.644° wide and 1.215° high.
Since 1521 x 1014 isn't the exact same aspect ratio as the size of image this camera shoots it's likely that some small amount of cropping has taken place.

To double check this we can draw lines from our viewpoint to each peak and we find that, at the distance of Ferrand (392.48 km), our lines are about 1.7 km apart. This gives us an angle of approximately 0.24817° -- if we then apply that over the full 1521 pixels of the image we get an estimate of 1.63° wide -- very close to the 1.644° we expect so let's just use 1.644° / 1.215° as our field of view.
Now we're in a very good position to size the mountain and compare it to our photo.
First, we need the Law of Perspective, which says that the angular size (α) of an object whose height perpendicular to our line of sight (g) at distance (r) is:
α = 2*arctan(g/2/r)
Excellent - we just need to know the height and the distance.
Technically, we should find the straight-line distance but at 440 km it doesn't change the answer much.  If you want to find the straight-line distance from a curved distance (o) along the surface of the Earth you could use this formula (where R is Earth's radius of curvature, for 42°N latitude R ~ 6369km):

g = R*2*sin((o/R)/2) = 6369*2*sin((443/6369)/2) = 442.9 km

So let's just keep it simple since this would be about 1/1000th of a degree.
Pic Gaspard (3880m) @ 443 km = 2*arctan(3.880/2/443) = 0.5018° = 461 pixels
Grand Ferrand (2758m) @ 392.48 km = 2*arctan(2.758/2/392.48) = 0.4026° = 375 pixels

Just for scale, one-half a degree is the apparent size of the Moon, so we're talking about a pinkie-at-arms-length size -- but that that is the size those mountains should appear at the distances they are from this observer.
So we can mark those out on the image now as the full height of the mountain at that distance.

As expected, the "sea-level" bottom for the more distant mountain (Gaspard) is lower than the closer mountain (Ferrand) and since it's a larger mountain the total apparent size is larger even though it's slightly further away.

So please do not try to tell me we're seeing the "whole mountain" here.

Now, Walter Bislin has an amazing calculator that will let you place objects in the world and includes a very advanced refraction module (which I left at defaults except to set the 'surveying standard' 7/6th refraction, about 14%.  This is very modest amount of refraction and since the photographer tells you that this is very rare to be able to see this, the actual refraction is likely greater than shown here.

Here is Ferrand and Gaspard on a Globe model vs a Flat Earth model.  Notice how the Globe model correctly shows just the distant peaks with Ferrand peak above Gaspard even though Ferrand is shorter.  When you put these two mountains on a flat plane then Gaspard clearly stands taller.

There is no contest here, the Globe best fits the observations and really ONLY fits the observations.  If we could see the entire mountain, even at this great distance with perspective, they would stand vastly larger than seen here.

 Walter Bislin Horizon Rendering Tool (link)
Another fun tool that comes from the Beyond Horizons page is a 'panorama' rendering tool that will label the mountain peaks.  I've used PeakFinder in the past but this one seems to be more full featured.  Here is our approximate view:

 View of Grand Ferrand from Finestrelles via udeuschle panorama generator
So this view comports with the Globe model extremely well and utterly destroys the Flat Earth model.  The problem is that Flat Earthers do not how to do basic estimations like how big something would be in an image (if you could see the whole thing).

Here is also a video made by Sly Sparkane: