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Flat-Earth: Finding the curvature of the Earth

Mittwoch, 1. Februar 2017 - 22:18 | Autor: wabis | Themen: Wissen, Facts, Animation, Interaktiv | Kommentare(1)
For us living beings on the surface of the earth, the earth looks flat. For this reason the so-called Flat-Earther (FE) claim that the earth is a flat disc rather than a globe.

With the help of a simulation, I show up to what altitudes the earth appears flat, although it actually has a spherical shape. I prove by means of photos, how the simulation shows the actual conditions, by superimposing the simulation results onto photos.

 Deutsche Version: Wie stark ist die Krümmung der Erde?

Simulation of the Eearths Curvature

The blue grid in the following simulation shows the curvature of the earth at a certain altitude as it appears at a certain Angle of view. The Angle of View can also be set as a 35mm focal length.

(Klick: Zoom)
ZoomInformationen zum BildCalculated Values
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ZoomInformationen zum BildOverview of the Simulation

 JavaScript: Simulation of the Eearths Curvature

Simulation Parameters

The simulation can be used to compare a spherical earth with a flat earth. The grid of a flat earth can be displayed by selecting the appropriate Model.

For comparison with a flat-earth model, a red grid can be displayed with the Grid setting Globe+Flat. The red grid shows the projection of the blue grid onto the plane of the flat earth. For low altitudes, the deviations between the blue ball grid and the red flat grid are minimal. So small indeed that by turning off the red grid the curvature can barely be noticed.

Note that the globe grid does not have a constant spacing. Instead, a certain number of grid lines are displayed, adjustable with the parameter Lines. This corresponds to the natural seeing, because on the earth we have no fixed grid which shows the relative distances either. As a result the distance between the grid lines varies with the distance to the globe earth horizon. The actual distance between the lines is displayed at LineSpacing.

HeightRange specifies the range of the Height slider. Log uses a logarithmic scale from 2 m to 1 000 000 km. In the Height input field you can enter values from 0,1 m = 0,0001 km to 1 000 000 km.

View∠ (view angle α) and f (focal length = zoom) are linked together via the formula:

(1)
f = { 43{,}2\ \mathrm{mm} \over 2 \cdot \tan( \alpha / 2 ) }

You can enter a View∠ between 0,1° and 160° or a focal length f between 3,81 mm and 24 800 mm.

Eye-Level shows a line at infinity that is the distance Height obove the flat earth plane. So this line is at eye height of the observer and is called Eye-Level.

FlatHorizon shows a line at the distance of the globe horizon on the flat earth plane.

FE-Equator shows a line at the equator of the flat earth. The equator of the flat earh has a distance of d = R · π / 2 from the north pole, where R is the radius of the globe earth.

AngDiameter (angular diameter) is the angular measurement describing how large the globe earth appears from the distance Height.

Observations

The earth is huge in comparison to us humans, 12 742 000 m compaired to 2 m. So huge, indeed, that we are not able to see their spherical shape with the naked eyes from the surface of the earth. We can measure the distance to the horizon and its lowering due to the spherical shape only with precise technical instruments. Only from high altitudes or from space can we clearly see the ball shape of the earth.

Even at altitudes of several kilometers, such as the cruise altitudes of airliners, the spherical shape can not always be clearly identified. A slight curvature can only be detected on wide angle images. It must be taken into account, however, that wide-angle lenses can distort the scene. On cheap cameras or smartphones the curvature can therefore only be observed to a limited extent.

The visibility of the curvature is therefore dependent on the altitude and the angle of view respective the focal length, i.e. zooming!

The fact that the horizon is lower than Eye-Level can not be recognized by the naked eye, since in nature there is no eye-level line above the horizon. However, this drop can be seen with appropriate instruments such as an Overhead Display of an Aircraft.

Superimpose the Grid onto Photos

In order to prove that the calculated blue grid actually reflects reality correctly, the grid can be matched with a real photo.

To superimpose a grid onto a real photo the right way, the following information is required:

  • Altitude in which the photo of the earth was taken
  • 35mm Focal Length or Angle of View used on taking the image
  • Aspect ratio of the image
  • The lens must not be distorting, fish-eye lenses are excluded or a lens correction must be applied, which removes the distortion.

Use particularly shootings from high altitudes, for example from an airplane or from space. At lower altitudes, no curvature is clearly visible.

Procedure:

Set the altitude with the blue slider or enter the value in the Height input field. Select the focal length or the corresponding angle of view with one of the black sliders. Select the aspect ratio of the image at AspectRatio. With the green sliders Nick and Roll the viewpoint and the banking angle can be adapted to the photo.

Cut out the area inside the black frame with a program like the Sniping Tool from Windows. Open the photo in any image editing program. Place the cut out area of the grid in a new layer above the photo. Scale the grid plane so that the aspect ratio is maintained and the grid layer becomes the same size as the photo. Set the blending mode of the layer to multiply (or something like that). It may be necessary to move the grid layer slightly and rotate it if the settings of Nick and Roll do not match exactly.

If everything was done correctly, the grid would now match exactly with the image of the earth's surface. The following photographs show how the results can look like:

Pictures taken from the ISS

The International Space Station ISS orbits the earth at an altitude of 400 km. From this altitude, the earth clearly shows itself as a sphere. I now wanted to check if the calculated grids match with photos taken from the ISS. For this I searched original photos, in which data about the camera and lens used is stored in the images in the EXIF-Format. The reason is, in order to get the correct perspective representation, I have to enter the focal length of the used camera in the simulation.

I have found several such images on NASA's website. Below are two such examples with and without superimposed grid:

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ZoomInformationen zum BildTest whether the graphics of the simulation matches a real image taken from the ISS:
Height = 400 km; 35mm focal lengt f = 28 mm; Aspect Ratio 3:2; Line Spacing 48,91 km
Date of recording: 16.09.2016 20:55; Source: NASA; Original image with EXIF Informations
(Klick: Zoom)
ZoomInformationen zum BildTest whether the graphics of the simulation matches a real image taken from the ISS:
Height = 400 km; 35mm focal lengt f = 28 mm; Aspect Ratio 3:2; Line Spacing 48,91 km
Date of recording: 16.09.2016 20:55; Source: NASA; Original image with EXIF Informations
  • Swap images
  • With Grid
  • Without Grid

For the above picture, I used an original photo from NASA. The image was edited according to EXIF data with Photoshop, probably only converted to a JPG. I can't find traces of a composit procedure or any other manipulation and the noise is as expected from a camera with the selected settings.

I have set Height = 400 km and 35mm focal length f = 28 mm in the simulation. With Nick and Roll, I rotated and moved the graphics according to the photo, because the photographer did not aim at the horizon. Then I created a screen copy of the graphics and opened it together with the photo in Photoshop. I have placed the graphics on top of the photo on a new layer and inverted the colors. The graphics and the photo have the same aspect ratio of 3:2. I had to scale the graphics, however, so that it got the same size as the photo. After that, I superimposed the graphics over the photo with blending mode "negative multiply".

And look, the graphics fits exactly to the photo. The lines have a spacing of LineSpacing = 48,91 km. The Gulf of Suez fits exactly between two lines. In Google Earth measured I get about 50 km. So this also fits perfectly.

Below is another picture of the Earth taken from the ISS photographed with the same camera. The superimposed grid of the simulation fits perfectly also here. The faint gray line corresponds to the eye level, i.e. the horizon of a flat earth.

(Klick: Zoom)
ZoomInformationen zum BildAnother picture of the Earth photographed from the ISS. .
Height = 400 km; 35mm focal length f = 28 mm; Aspect Ratio 3:2; Line Spacing 48,91 km
(Klick: Zoom)
ZoomInformationen zum BildAnother picture of the Earth photographed from the ISS. .
Height = 400 km; 35mm focal length f = 28 mm; Aspect Ratio 3:2; Line Spacing 48,91 km
  • Swap Images
  • With Grid
  • Without Grid

Pictures taken from a Rocket Flight

Here are some screen shots taken from the video: GoPro Awards: On a Rocket Launch to Space, which was recorded with a GoPro4 camera with a fish eye lens. I applied the lens correction of Adobe Lightroom to it and after that the images fit perfectly to the calculated grid:

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ZoomInformationen zum BildImage 1
(Klick: Zoom)
ZoomInformationen zum BildImage 2
(Klick: Zoom)
ZoomInformationen zum BildImage 3

Height = 120 km, Focal Length: f = 18 mm, Camera GoPro4

The horizon has exactly the same curvature on all images at every position after applying the lens correction.

The Horizon is not at Eye Level

Flat-Earther claim that the horizon is always at eye level, which it would be if the earth were flat. The definition of eye-level is that a line from the eye of the observer to a distant point at the same height forms exactly a 90° angle to the vertical at the observer. The distant horizon of a flat earth would apparently reach to the eye-level and thus form a 90° angle.

A dip-angle from eye-level to the real horizon can not be estimated with the naked eye, since a corresponding reference is missing on the horizon. Just looking straight at the horizon and claiming that it is at eye level, so forms exactly a 90° angle to the vertical, is a false claim. This is true approximately for low altitudes only. In an aircraft at an altitude of 11 km, the horizon drops 3,36° (see DipAngle in the simulation). This is a clear drop, but not recognisable with the naked eye because there is no reference.

The following photo was taken with the Theodolite App with an iPhone. The aircraft flew at an altitude of 33 709 ft, as noted in the picture at the top/center. The iPhone was aligned so that the crosshair shows eye-level on the horizon. This is the case when the ELEVATION ANGLE shows 0.

Informationen zum BildComparison of the effective horizon line with eye-level in an airplane. .
Height = 10,275 km; 35mm focal lengt f = 33,9 mm; Aspect Ratio: 16:9; LineSpacing 8,035 km; Original Image
Informationen zum BildComparison of the effective horizon line with eye-level in an airplane. .
Height = 10,275 km; 35mm focal lengt f = 33,9 mm; Aspect Ratio: 16:9; LineSpacing 8,035 km; Original Image
  • Swap Images
  • With Grid
  • Without Grid

The calculation results in a dip angle of 3,252°. The horizon is 20,53 km below Eye-Level and is at a distance of 361,6 km. The overlayed grid lines have a spacing of 8,035 km. These values are all calculated by the simulation.

I did not take the picture myself but found it at BlogSpot. There is a copy of it on my website. I own the app Theodolite on my iPhone and I know how it works. I calculated the focal length of the iPhone by measuring the angle of view, which I could do with the app. The calculated angle of view of about 65° for the diagonal coincides with data on the Internet. It corresponds to a 35mm focal length of 33,9 mm.

The values: Height = 10,275 km, angle of view 65° and display aspect ratio 16:9 I entered in the simulation. Then I cut out the simulation image along the black frame, scaled it to the same size as the photo and overlayed them with the blending mode multiply. As you can see, the calculated image fits exactly to the photo and shows exactly where the horizon of the Earth is with respect to Eye-Level. Note that a very slight curvature is barely visible on the grid but because of the haze at the horizon not as visible on the photo.

Overhead Displays show Eye-Level

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ZoomInformationen zum BildRockwell Collins’ Head-up Guidance System (HGS™) incorporates critical flight information into the pilot's field of vision and provides flight path support at all stages of the flight. Source: Rockwell Collins; With kind permission

Airplanes can be equipped with overhead displays. These displays are pushed between the pilot and the front window. When the pilot looks out of the window through this glass screen, he can see all critical flight informations like artificial horizon, speed, altitude, vertical speed, heading, even the runway, and also the terrain like on a night vision device. It is remarkable that the displayed graphics moves in sync with the head movement of the pilot. It looks like the graphics are projected onto the terrain.

If the aircraft is now cruising at high altitudes, in the image at 39 000 ft, the real horizon lies about 3,5° below the eye level due to the earths curvature. The display projects a horizontal line at eye-level into the scene. In the picture you can clearly see the distance between the eye-level line and the real horizon.

The stylized airplane in the display shows the effective flight direction. In the picture, the symbol lies on the horizontal eye-level line, which means that the aircraft neither climbs nor descends. It is located to the left of the center, which means that the aircraft does not fly straight ahead but is pushed sideways to the left from the wind (see arrow on the top left). The aircraft must correct for this deviation by pointing the nose into the wind according the arrow, so that it does not miss the destination. The autopilot automatically performs this correction.

Kommentare

1wabiswalter@bislins.ch (Walter Bislin, Autor dieser Seite) 08.03.2017 | 17:50

Influence of atmospheric Refraction

The line of sight to the horizon is rarely a straight line as assumed by the simple formulas, but is curved downwards due to the temperature and pressure changes of the atmosphere near the ground (refraction). This means that you can see much further than the calculations with the straight line suggests.

In extreme cases, e.g. if warm air is above cold water the refraction can lead the light hundreds of kilometers along the water surface! The result is that the earth is seemingly flat.

Source Wikipedia: https://en.wikipedia.org/wiki/Horizon#Effect_of_atmospheric_refraction

This fact has been known for centuries among land surveyors and seafarers.

Note: You can trust your eyes only at short distances. Over large distances, the light path through the atmosphere is disturbed in an unpredictable way. It's nothing like it seems!

Occlusion due to earth curvature with refraction

In the excellent video FLAT EARTH - EXPERIMENT - TELESCOPE from 01.08.2016 the author Alex Chertnik shows how to measure and document measurements with the telescope over water the right way. He measures over three similar distances on different days and at different times of the day, how much of 4 about 300 m high chimneys is hidden by the curvature of the earth.

In contrast to all flat-earth videos he considers the refraction in his calculations. His measurements correspond exactly to the calculations for a globe earth with a radius of 6371 km, taking into account the standard refraction.

The video shows clearly how the image wobbles and flickers due to the fluctuations of the refraction, and that the Horzont is not a clear horizontal line, but shows wavy distortions. These waves come only to a small extent from the water itself, but arise through the refraction. The occlusion fluctuates by many meters due to these refraction waves.

Note: the refraction directly above water can be much higher than the standard 7%!

The video proves very clearly that the earth must be a ball.

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