# Finding the Curvature of the Earth: Comments

Wednesday, October 28, 2020 - 20:29 | Author: wabis | Topics: FlatEarth, Calculator, Animation, Knowlegde

### 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.

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.

2Gerard Frensen 6/13/2017 | 23:32

Great site, very informative, very well done! Thanks for this great work

Gerard...

3Kelly White 7/19/2017 | 14:34

Excellent tool and information. I'm just having a comment conversation with someone who doesn't quite understand this, but your tool will really help. Thanks!

4Stephen hansen 7/19/2017 | 21:33

Very interesting Walter. You have a amazing mind.

5Rory 8/13/2017 | 09:40

This is awesome. :-)

6Herr Gnorts 8/14/2017 | 17:09

Super, das!

Aber kann ich irgendwo ablesen, wie *weit* der Horizont ist, links nach rechts?

7Risto 8/15/2017 | 18:34

Excellent work! Could it be possible to adjust the refraction parameter as well?

8Phil Laj 8/16/2017 | 01:33

It's a nice job and it's very impressive but I don't like how the yaw also gets lower and lower as the altitude rises. :'(

Going up in a vertical elevator/balloon in real life wouldn't look like that, re-think about that part because the rest is top notch! :D

Phil, choose option HorizView = Eye-Lvl to keep eye-level at the same position.

Herr Gnorts: siehe das neue Feld im Computed Values Panel DisplHorWith.

Risto: Refraction is now implemented, see Refraction Panel and some of the new Animations.

10Everett Anderson 9/4/2017 | 18:37

Refraction can sometimes have the reverse effect of making objects in the distance seem lower than they actually are. The phenomena is called "sinking", and it can sometimes actually cause distant objects to disappear behind the horizon when they actually don't. Flat Earthers have actually cited this as the explanation for ships disappearing beyond the horizon and the towers in Soundly's videos curving downward. How should rational people respond to this claim?

@Everett Anderson

### Refraction and Sinking

To produce Sinking instead of Raising, compared to Standard-Refraction, the atmosphere must have a steeper lapse rate than normal. Laps rate is the negative Temperature-Gradient dT/dh. However, there isn't much room to play with: the Standard Atmosphere already has a lapse rate of 6.5°C/km, but convection limits lapse rates in the free atmosphere to about 10°C/km. For Refraction to be 0 the lapse rate on Standard Atmosphere should be 34.3°C/km. To bend light upward it has to be even greater, which can only accour in thin layers. To get a temperature decrease of more than 34.4°/km you must have a hot surface with a layer of cold air above. Such conditions produce heavy distortions and mirages of differend kind rather than only Sinking, because the air is instable.

Because the density of undisturbed air increases with decreasing altitude light is generally bent only downwards. Only specific changes of temperature gradients near the surface can locally change the density gradient in such a way that distorted and mirrored layers and some Sinking may appear. On most images we see already streching and mirroring at the lowest layer even when the overall image is still lowered by the average Refraction. These distortions are caused by small layers of cold air above warm surfaces.

12indio007 9/5/2017 | 15:08

This sim has a fundamental flaw.
What you can see is limited to the aperture through which the light passes.
Where is that calculation?

I should also say refraction values is a wild guess at best. For a simple reason. There is an assumption of linearity. This is a misplaced and provably wrong assumption.

@indio007

### No Aperture in the Curvature App

Quote: What you can see is limited to the aperture through which the light passes. Where is that calculation?

First, Aperture does not limit or influence which part of a scene is depiced on the sensor. The loss of light when closing the aperture is compensated by longer exposure and higher ISO values. You see the exact same thing. With aperture you can influence the Depth of field.

So Aperture does not change the shapes or even the relative positions of objects in a scene. Look at the Animations and then look at the real images I linked above the App when an animation is choosen. Simulations of the Globe Earth and the images match, but the Flat Earth simulation does not match at all. And the simulated situations are taken from videos Flat Earthers provided, by the way.

Second: The App does not simulate a real camera, only the projection part of 3D objects to the focal plane of a camera, without aperture, exposure times and ISO settings. In such a projection there is no such thing as Aperture, as in a drawing there is no such thing as aperture. But the computed 2D image from 3D objects is accurate.

### Refraction derived from linearized Model

Quote: I should also say refraction values is a wild guess at best. For a simple reason. There is an assumption of linearity. This is a misplaced and provably wrong assumption.

It is not assumed that the density gradient of the atmosphere is linear in reality. But on sufficiently small scales, systems can always be approximated as linear (like the globe earth can be approximated as a flat plane on small scales). Linear approximations are the normal way most mathematical models of physical systems are derived. Its an application of calculus and results in differential equations that can then be applied on any (nonlinear) situation, within the limits the model is intended for.

So the math of how atmospheric layers bend light rays can be derived from a linearized model and this math can then also be applied to real nonlinear systems. Why is that so? Because a mathematical model is universal. It maps an input (density gradient) to an output (bending of light). You can derive the mapping function from linearized systems. But you are not restricted to use it only on linear density gradients. It also works on any gradient, because the math model is a representation of the real physical system and can be used to predict the outcome of any input.

That is true for all derived physical laws. E.g. Newtons law of gravity is not only applicable to simple linear systems but are universal, as long as gravity is not too strong and the speeds involved are much slower than the speed of light. Physicists know this limitations and know that Relativity must be applied on those conditions.

Please read my 2. paragraph of What is Refraction?. There I explain why and how average values, derived from many, many real measurements by surveyors, may be applied to get the average overall effect of Refraction. You can always approximate a real physical system by a simplified average version, superimposed by perturbations. The perturbations are often smaller than the simplified part and do only slightly perturbate the outcome. We see that in real images where Refraction takes place. The overall image of an object may be raised by the average part of Refraction but then some parts of it are streched, compressed or mirrored atop of it.

My App simulates only an undisturbed average Refraction, because you would have to provide me with the real atmospheric conditions (which change all the time) from the observer to the object for each light ray, so you can never get this data anyway. But the undisturbed approximation suffice to get the concept. A mountain is raised by the calculated amount of Refraction whether its image is disturbed or not.

If you want see simulations of Refraction with complicated density gradients, see Introduction to Superior-Mirage Simulations.

may I suggest a feature request? it would be nice to have the ability to have permalink to the simulation, with all parameters embedded in URL, maybe as a long JSON string in a URL parameter. or if that's not possible, the ability to copy or save all parameters in a JSON string.

This features are now implemented. See Save/Restore Panel below the App Window.

• You can save the App State in Json Format
• You can create an URL that restores the App State
• You can restore the App State from Json or Url Format
• You can auto-start a Demo by appending &demo=Name to the Url, where Name is the Text of a Demo Button, e.g. Curve

Check this out here:

16Grahame Rathbone 10/18/2017 | 23:52

Awesome work, Walter.

I was fumbling around with geogebra just to make simple views and had to give up a few times.

17Grahame Rathbone 10/19/2017 | 15:01

By the way, an idea: What about another that shows the relative sizes of the Earth and moon using distance and lens focal length, libration, eclipses, INCLUDING the specular aspect of it's surface - phases and the illumination of the moon from an angle almost right behind us...?

18George Hnatiuk 10/24/2017 | 21:44

Just came across your site. Upon first look it appears to be well done. I will look at it more critically in the days to come. I have also engaged in refraction theory and made many measurements / observations with a theodolite total station (Topcon GTS-3C) in varying atmospheric conditions and will check over your work.

Seems like you put together a worthwhile blog.
George Hnatiuk

19Grahame Rathbone 11/16/2017 | 13:13

BTW:

It would be VERY useful to be able to pass created demonstration diagrams in a url...
So the diagram parameters are ready and presented when someone visits...

20Michael Dunlavey 11/16/2017 | 17:39

This is a beautiful app. Nice job. (Is there a pause button I didn't see?)

Regarding horizon dip angle, it's not necessary to have an eye-level reference if opposite horizons can be seen at the same time. If a straightedge can be pointed at one horizon, it will point above the opposite horizon with twice the dip angle. This could be done with two small mirrors mounted on adjoining faces of a cube, with the whole thing mounted on a ruler. Then you just take a picture looking at the mirrors. I haven't built it. Of course there are plenty of other ways to determine "level", like a line between the tops of opposite windows in an aircraft cabin. (I'm sure the theodolite app is the simplest, but I like low-tech :)

Grahame, It would be VERY useful to be able to pass created demonstration diagrams in a url...

You can do this in the panel Save/Restore. Use the button Get App Url. Then copy the generated url into eg YouTube comments. Clicking this url will open the App and restore the corresponding state.

Note: you can also copy such an url into the text field in this panel and click Set App State. The button Get App State gives the currect state in an editable json format. You can change values and add text in DemoText and Description. Then click Set App State to show this state. Use Get App Url again to get the url with added text.

Do start an animation via an url to this page, simply add &demo=xxx where xxx is the name of the demo button.

22Nick 12/29/2017 | 03:45

Walter,

I really do want to thank you for this website. I have always been somewhat of a conspiracy theorist, so naturally, YouTube recommended me Flat Earth videos. About 6 months ago I watched quite a few, and while I did not think that such a claim could be possible, some of the videos were somewhat convincing; due to flat earthers taking things out of context and purposefully misrepresenting anomalies (such as refraction ) which can easily explain away their claims. I was turned on to your site from MetaBunk, and I am so glad I found it. It is constructed excellently, yet is simple. You clearly are very smart to be able to set up all of the simulations, yet you make them easy enough to use so that everyone can try them and actually learn things they likely never would have. I really do appreciate your site from saving me from the unintelligent Flat Earth rabbit hole, and I recommend it to Flat Earthers, as well as people in general..

Thanks again,

Nick

23ivan 12/30/2017 | 00:16

Very impressive but thats a lot of science that should never have been required. I don't understand the maths, but then again I don't need to. I know the earth is a globe, logic tells me it cant be anything other than a globe. For all the poo pooing by the FE brigade of any sensible explanation as to why the earth isn't flat, they wont argue why the internet isn't awash with photos taken through telescopes with captions such as heres my picture of the Cuban coastline taken from the coast of Ireland. or heres my photo of Australia taken from South Africa. Need I go on.

24globe 1/4/2018 | 09:49

well this calculator is false, refraction doesnt work on flat earth model here in this calculator, the models dont have sky, the calculator is therefor biased in favor of a globe.

Globe: Flat Earther have no working math model of Refraction that I could incoorporate into the App, sorry, that is not my fault. If I apply the working Standard Physical Model of Refraction, the Flat Earth would look like a Hollow Earth. To make the Earth appear like wie see it, on the Flat Earth model light has to be bent upward by Refraction. This contradicts physical laws. Light gets alway bent toward the denser medium. The density of air decreases with increasing altitude, so light gets bent downward. The consequence is, that on the globe on strong Refraction it can look flat and mountains that are hidden behind the curvature can appear raised over the horizon.

If you want to simulate how the Flat Earth looks by using a false Refraction Model where light gets bent upwards, you can use the Globe Model and use values for k = 1 (no Refraction on FE, earth looks flat) to k = 0 (severe Refraction on FE, earth looks curved as a Globe with radius 6371 km).

I look forward to your calculator for Flat Earth Refraction.

26truth 1/4/2018 | 09:56

23indio007 12/29/2017 | 07:38
Your refraction section is complete nonsense. Yes the math is sound but it has no connection to physical reality. What empirical test where used to validate the model?????????????? NONE. While I am on it, you people think that the refractive index proscribes some universal bending of light. NO! that's not how it works. It is the CHANGE in the refractive index that bends the light. The real physics is this....

1. The refractive index is the ratio of the speed of light in the medium of interest to vacuum speed. The change in the speed of light from one medium to another is what causes the change in direction. Well, air is very close to vacuum. The difference is in the fourth decimal. The difference between different air refractive indices is in the 5th and 6th decimal. The change is very small therefore the bending is very small IN THE REAL WORLD.

2. Plus we generally are using a constant altitude for line of sight. There is little of no change in the refractive index because the density and pressure do not change at a constant altitude. You have to concoct a change in altitude by assuming a globe and pretending a beam of light traversing two mountain peaks goes high<>low<> high in altitude. You assume the globe and prove the globe which is no proof at all. If you assume flat there is no change in refractive index and therefore no bending of light.

3. You say standard refraction.. Ok how was that derived??? ahhh yes, they made a model by assuming a structure of the atmosphere and the making a model that should work based on the assumptions NOT EXPERIMENT. There is no empirical basis but you guys just run with it. Little do you know, that the refraction approximation can't be used at large zenith distances because the refractive index BLOWS UP at 89° from zenith. At 90° from zenith. (looking directly at the horizon) light should curve so much you should see the back of your head. That's not physical , so the model is NON-PHYSICAL Source: Page 108 and fig 15 Title: Understanding astronomical refraction
Authors: Young, A. T.
Journal: The Observatory, Vol. 126, p. 82-115 (2006)
Bibliographic Code: 2006Obs...126...82Y
(url)

4. Now that it's shown that the model actually models nothing and your model is a derivative of that model are you going to correct your error? You have no math and you have no empirical data. That means "standard refraction" applied to the horizon to 1° is inept. Un;less of course you think there is infinite refraction.....

5. there is no empirical data on refraction because it can't be measured consistently. The measurements don't match any model. Measurements are literally all over the place. Read this Empirical Modelling of Refraction Error in Trigonometric Heighting Using Meteorological Parameters (url)

6. Stop pretending that refractive index equations are certain and accurate. They are not. 2 seconds on google shower will show you 100's of papers wher people are still trying to come up with a model.

7. The fact is the refractive index is based on the dielectric constant. Traversing the boundary of 2 mediums with different dielectric constant is a complex equation in which the imaginary part is nonlinear but absolutely effects how a beam of light traverse the boundary. The refraction model is overly simplistic and doesn't model reality. it models a need to come up with something ... anything to validate the curve of the globe.

8. refraction in atmosphere was invented before the time of Pliny the Elder who wrote about the seemingly impossibility of the Moon and Sun being over the horizon during a lunar eclipse. Refraction of air was invented specifically to make the globe work!

9. Some things never change.

### Refraction: Is this App wrong?

@indio007, truth

Your refraction section is complete nonsense. Yes the math is sound but it has no connection to physical reality. What empirical test where used to validate the model?

Complete nonsense? Can you provide me a better model? Do Geodetic Surveyors measure the earth with false models since centuries? Do you deny that airplanes and ships find their destinations over thousands of miles exactly, by using spherical maps made by Geodetic Surveying? Can you present any Map that is demonstrably false?

I use the Standard Model for Refraction as used in Geodetic Surveying. The values used in Geodesy are empirically derived from many, many measurements in reality. The first common used value k = 0.13 was introduced by Gauss.

Sources I use are:

I'm aware, that Refraction is strongly dependent on the current atmospheric conditions along the line of sight to an object. As nobody can provide the exact variing refraction values along any line of sight, we can not simulate Refraction exactly. But we can make many measurements under different atmospheric conditions and derive from them empirical formulas for Terrestrial Refraction for certain atmospheric conditions. This is applied in Geodetic Survey. The empirical Formulas vary slightly between countries. I used the formula presented in Wikipedia.

My App does not claim to simulate Refraction with all distortions in various layers of air. It only gives a rough picture how a certain Refraction affects a scene. With my App you can estimate how strong Refraction has to be to bring a hidden mountain into view as seen on some pictures. You can simulate the overall effect of Refraction on certain scenes. Not more, not less.

While I am on it, you people think that the refractive index proscribes some universal bending of light. NO! that's not how it works. It is the CHANGE in the refractive index that bends the light. The real physics is this....

Did you read my section about What is Refraction? Cite: The density of the atmosphere generally decreases exponentially with increasing altitude. Any density change causes a refraction. If the density change is not abrupt but continuous as in the atmosphere, the light is not refracted but bent, but we call it Refraction anyway.

The refractive index is the ratio of the speed of light in the medium of interest to vacuum speed. The change in the speed of light from one medium to another is what causes the change in direction. Well, air is very close to vacuum. The difference is in the fourth decimal. The difference between different air refractive indices is in the 5th and 6th decimal. The change is very small therefore the bending is very small IN THE REAL WORLD.

Yes indeed. Standard Refraction bends light in the atmosphere on an arc with a radius of about 7 times the radius of the earth. A very, very small bending indeed. But: the effect of this bending increases with increasing distance to the object and the angular size of objects descrease with increasing distance. So even a very small bending can raise a very distant mountain above the horizon. See Demo Canigou.

Plus we generally are using a constant altitude for line of sight. There is little of no change in the refractive index because the density and pressure do not change at a constant altitude.

On the globe a straigt line of sight always passes through layers of different altitudes due to the curvature of the earth. That causes different densities along the path, different refractivity and so bending of light.

You have to concoct a change in altitude by assuming a globe and pretending a beam of light traversing two mountain peaks goes high<>low<> high in altitude. You assume the globe and prove the globe which is no proof at all. If you assume flat there is no change in refractive index and therefore no bending of light.

I do not assume a globe. I present two models to make predictions: Globe Earth and Flat Earth. Refraction is only an additional feature, that can explain certain scenes like distant mountains and cities raised into view. You can enter known values for objects and observer and the App displays predictions for both models. Then you can compair the predictions with reality. Reality prooves one or the other model. My App does not assume a globe. My App proves nothing. My App makes predictions.

You say standard refraction.. Ok how was that derived??? ahhh yes, they made a model by assuming a structure of the atmosphere and the making a model that should work based on the assumptions NOT EXPERIMENT.

That is a false claim. Where did you get this false infos from? The values for Standard Refraction are derived using controlled setups of simultaneous reciprocal vertical angle measurements. Empirical formulas for different atmospheric conditions are then derived from this real measurements.

I don't know where you guys get this false infos, that scientists make their theories out of nothing. Every theory is developed to explain observed facts, not the other way around. First you observe and measure a fact, then you make a hypothesis that could explain the facts, then you build a mathematical model to quantify the hypothesis to make further predictions to check the hypothesis against reality. If the hypothesis always matches reality, does not contradict other theories and explains all current data, then it finally becomes a Scientific theory. A scientific theory falls or must be adjusted as soon it contradicts new findings. A scientific theory is not proof. It's the simplest and best description/model we have for certain observations.

the refractive index BLOWS UP at 89° from zenith. At 90° from zenith. (looking directly at the horizon) light should curve so much you should see the back of your head.

That is possible indeed, except we cannot see a distance all around the globe, because of obstacles, diffusion of light and variing Refraction. If Refraction k = 1, light is bent along the surface of the earth for hundreds of miles. This can be observed in reality quite often under certain conditions. But you confused Terrestrial Refraction with Astronomical Refraction.

Now that it's shown that the model actually models nothing and your model is a derivative of that model are you going to correct your error?

"My" Globe Model is not based on Refraction! My Globe Model is based on a sphere with radius 6371 km. There is nothing to correct. Refraction is only an additional feature. You can set Refraction to zero anytime. It does not change the Globe Model.

refraction in atmosphere was invented before the time of Pliny the Elder who wrote about the seemingly impossibility of the Moon and Sun being over the horizon during a lunar eclipse. Refraction of air was invented specifically to make the globe work!

Refraction is not an invention but an observation, see Monitoring of the refraction coefficient in the lower atmosphere using a controlled setup of simultaneous reciprocal vertical angle measurements.

Stop pretending that refractive index equations are certain and accurate.

Nowhere do I pretend this. You cannot falsify the implemented Globe Model by refering to (well known) Refraction, which plays a minor role here anyway (I implemented this feature only in a later version). The Globe Model itself has nothing to do with Refraction. It is simply a geometrical Model of a Sphere. As simple as the Flat Earth Model is a geometrical Model of a Plane.

Forget Refraction and compair the Globe and Flat Earth predictions with the pictures presented. You won't find any match with the Flat Earth Model, with or without Refraction.

And why do you deny the observed dip of the horizon as presented on this page at all? This can not be explained by Flat Earth Model at all.

28Rob Smith 1/16/2018 | 14:17

I love this App, it shows so much but is still simple to use. I have seen a number of people suggest that Soundlys videos of the causeway indicate that the earth is actually a tube of 10km diameter, that's why the horizon is flat but the causeway curves. it would be interesting to model this just to show what nonsense that is.

### Perspective on a Globe

Rob Smith: I have seen a number of people suggest that Soundlys videos of the causeway indicate that the earth is actually a tube of 10km diameter, that's why the horizon is flat but the causeway curves. it would be interesting to model this just to show what nonsense that is.

See my new Blog Article Perspective on a Globe. In this article you can play with an App showing a grid of a Globe Model to see how it is distorted by perspective especially when viewing from low altitudes.

30Doug 3/22/2018 | 15:58

Thanks for all the work on this it obviously took a lot of time. Hopefully some middle schools or high schools are getting to use this.

I took a picture of the San Mateo bridge in the San Francisco bay from Seal Point. The bridge is the longest in California and has a 5 mile (8km) flat causeway section. I was about 8 miles away (13 km) and it looked very much like a Soundly picture.

31Frank 4/7/2018 | 14:13

Just dropping by to say brilliant work, it wont do a thing to convince idiots that their magic god disk "theory" is bullshit. But it's still brilliant.

32Paul 6/2/2018 | 14:17

It is obvious that NASA does have the same or similar computer model of a globe where they project the cgi composits of the earth on it and project it under the (fake) ISS station. The live images under the earth under the (fake) ISS is computer generated and not real. Why dont you compare your ball to the fake images from the moon since we "know" how far the moon is away from us? Since the believe of a ball earth is based on NASAs fake moon landings, why dont we go back to the foundation of it all? You think is some sociopath does lie to you once he will not lie to you twice?

### All about NASA Apollo Missions

Paul. There are thousands of pre-Photoshop images of the earth from space and the moon and hundreds of analog videos. There are tens of thousands of pages of documentations of the Apollo project, the missions and the science experiments. There are hundreds of original audio clips of the missions. The rockets were built and tracked by astronomers all over the world to the moon and back. We have hundreds of kg moon rocks that are analyzed by thousands of people world wide. To fake the rocks it would take hundreds of years. We have science experiments on the moon that have sent results to earth even when the Apollo programm was finished already. You can read the reports online. We have retro reflectors on the moon to reflect lasers from earth to measure the distance to the moon almost daily with sub cm accuracy.

It is cheaper to go to the moon and land on it than to fake all this.

We have currently space probes in low moon orbits that send images of the landing sites where you can see the remains of the missions and even the paths of the astronauts. They send images of the earth too. We have non NASA landers on the moon right now that send images from the surface that look exactly the same as the footage from the Apollo missions (eg. no stars, no thrust crater, same color of the moon).

So, why would NASA produce all this and put it online for moon hoax believers that are too lazy to read them anyway?

Hoaxers call all images fake, because they look different as they expect. Of course they look different. Space and moon are different. If you had a basic understanding of physics you would expect exactly what we can see on all the footage.

Apollo Lunar Surface Journal from the National Space Science Data Center NSSDC

All mission reports, images, videos, audios, transcripts, science reports, sample cataloges, technical debriefings, biomedical results, scientific results and many, many more:

Preliminary Science Reports Apollo 11-17

Master Catalog

The NSSDC Master Catalog is available for the queries pertaining to information about data that are archived at the NSSDC as well as for supplemental information:

National Space Science Data Center

"The National Space Science Data Center serves as the permanent archive for NASA space science mission data. "Space science" pertains to astronomy and astrophysics, solar and space plasma physics, and planetary and lunar science. As the permanent archive, NSSDC teams with NASA's discipline-specific space science "active archives" which provide access to data to researchers and, in some cases, to the general public. NSSDC also serves as NASA's permanent archive for space physics mission data. It provides access to several geophysical models and to data from some non-NASA mission data."

34Khaled Salah 6/28/2018 | 04:12

i have an issue with all the available curvature calculators that i ran into, non of them seems to account for the relative tilting of extremely tall objects over huge distances.

Example : 2 observation points with same height across quarter the circumference of the earth with the line of sight tangent to the earth's surface.

i realized that when i was testing my spreadsheet curvature calculator. i trust my calculator but i still need some assurance or confirmation and if i'm right about my conclusion then why dont they account for it (for me accounting for the tilt was not an objective but was just derived naturally as part of my equations)

my level of knowledge with geometry is not up to your level, i'm just clever in using the little that i know, so i hope you can bare that in mind if you are going to explain stuff for me

35Michael Hannes Zuntag 7/18/2018 | 19:33

I have an issue with your opening statement... "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 plane rather than a globe."

It is not that the earth simply looks flat.

It is because we are able to see things at a distance that should be obscured by the curvature of the earth, especially large lakes and the ocean.

We would not believe the earth is flat simply because it looks flat. In fact, we have been conditioned to falsely see the curvature at the point of the horizon, and it does seem that objects go over the horizon until zoomed in on.

Thanks for your work on this calculator and app... I will be using it with local observations I have made at 100km over the ocean.

### Take Observer Altitude and Refraction into account

Michael: It is not that the earth simply looks flat. It is because we are able to see things at a distance that should be obscured by the curvature of the earth.

How do you figure out what should be obscured by the curvature of the earth?

It heavily depends on the distance, observer height and refraction. If you take all this into account, you can compute exactly how much of an object in the distance is hidden, eg by using my Advanced Earth Curvature Calculator. And you would be surprised, how much more should be expected to be seen in the distance.

The problem with many flat earther is, that they ignore refraction completely, get the observer altitude wrong, because they think it does not matter a lot, and even use wrong formulas to calculate the hidden part (8" per mile squared).

So you must know the distance, the observer height and refraction to predict how much is visible and hidden.

Many flat earther think, refraction is an excuse and can not raise whole mountains or cities into view. But they don't know, that the apparent raise due to refraction is more, the farther away an object is, because refraction acts on a longer distance, and objects appear smaller with increasing distance (angular size decreases). This two effects work together. For example, if you have a small refraction angle of 0.1° and the angular size of a mountain in the distance is only 0.2° (that is a 873 m high mountain in 250 km distance), it get raised 436 m, ie half of its height! See Mountain Canigou Demo.

By measuring the earth in geodesy over long distances they always apply refraction to the measurements. They either measure the temperature gradient, humidity and pressure of the air and calculate refraction from that, or yet better, they use special instruments or special configurations, that are able to measure refraction directly along the line of sight. Nowadays the use GPS, which eliminates the effect of terrestrial refraction. Astronomical refraction is still taken into account, because this influences the signal between GPS satellite and receiver. But this is done by the stations automatically. We get the positions of such stations to cm accuracy -- and find without any doubt, the earth is a globe with some small irregularities.

Refraction is often underestimated. At heigher altitudes above 50 m it gets quite standard between 0.13 and 0.17 in most atmospheric conditions, see Refraction Coefficient as a Function of Altitude. But in winter near the ground eg. at 1.5 m you get mean refraction of 0.8 quite often, as studies have shown. Go lower and you achieve refraction values even greater than 1. Which means, the light follows the surface of the earth for hundreds of miles. The earth looks perfectly flat in this cases and you can see a laser from the ground.

So you have to factor in all relevant parameters to calculate the hidden/visible part of objects in the distance! You have always to apply at least standard refraction for altitudes above 50 m. Below it can get much higher or even negative (mirages).

Please use my Advanced Earth Curvature Calculator to compute what is expected to be seen taking refraction (and angular size) into account.

37Oscar Gunther 7/20/2018 | 19:36

You ROCK STAR! I am in awe. Well done!

38Pavel 8/3/2018 | 14:36

Very useful app. Thx.

But there is another thing that should be included. Geoid.
It is another thing why are objects visible at a long distance.

### Geoid Variations

Pavel: But there is another thing that should be included. Geoid. It is another thing why are objects visible at a long distance.

Geoid variations are neglegtable compaired to the effect of Standard Refraction, especially at dinstances over 50 km. The greatest Geoid variations I have found is less than 5 m / 100 km, corresponding to an angular size of less than 0.003°. Standard Refraction raises an object at 100 km about 127 m, which corresponds to a refraction angle of 0.072°. So the Geoid variation over 100 km is at most 4% of standard refraction. Over longer distances the Geoid variation compared to refraction gets even much smaller (see table) because refraction lift increases with the distance squared, while Geoid deviations are at most increasing linear with distance:

Note: In the table standard Refraction k = 0.17 and observer height 2 m is assumed. Refraction lift is the apparent raising of the target with respect to the horizon line, lift with respect to eye level is even more.

 Distance Max Geoid variation Std Refraction lift Geoid/Refraction 10 km 20 km 50 km 100 km 200 km 500 km 0.50 m 1.00 m 2.50 m 5.00 m 10.0 m 25.0 m 0.63 m 3.94 m 29.9 m 127 m 522 m 3313 m 79% 25% 8.4% 3.9% 1.9% 0.8%

Over shorter distances than 10 km refraction lift can be ignored, because objects are not hidden behind the horizon. If you know the Geoid variation you can correct the hidden value accordingly. But if the Geoid variation between observer and target is not a hump or valley, but rather a constant slope, then the result of my App is the same as without Geoid variation. If the Geoid is a hump then you can lower the refraction value a bit to account for the fact, that more is hidden. If the Geoid is a valley you can increase the refraction value to account for it.

As we generally don't know the exact refraction coefficient anyway, we can simply ignore Geoid variations.

40thetruthis 8/30/2018 | 14:03

I'm absolutely not convinced of all your explanation sir, knowing that any kind of videos, images and pixels are not scientific proofs, and as I know gravity is still not proved in 2018, or, just show me the gravity equation that magically curve the water of all oceans.

Also I would like to know why in 2018 we, everyday people are still not experiencing space, as promised 10, 20 and 30 years ago? something wrong maybe? let's continue to ask questions as long as we don't experiencing the reality they show us in TV.

thetruthis: knowing that any kind of videos, images and pixels are not scientific proofs

There are no scientific proofs. Science can not and does not prove anything. Math can prove. Science delivers evidence and explanations and makes mathematical models that describe observations and lets us predict observations. You need to have predictability to be able to falsify a hypothesis. You have to know what outcome of an experiment you expect precisely to be able to compare it width observational data.

I have created a computer model (is also a kind of math model) that predicts the outcome of observations. Images of objects are observations. You can measure the shape of observed objects from images quantitatively. If you know that an image of an object has no or very little distortion, you can compare it 1:1 with the prediction of the computer model, which I have done here. If the image matches with the globe model it is evidence that the model is correct. In this case, the image eg. of the causeway matches exactly the model, so this is evidence that the model of a globe earth with radius 6371 km is right. The image does not match the prediction of the flat earth model at all, so the flat earth model is falsified - the wrong model. I can even measure the radius or the refraction from an image and the model.

### Gravity is an established Sientific Theory

thetruthis: and as I know gravity is still not proved in 2018

Sunlight is still not proved too. But both sunlight and gravity can be observed by our senses and measured. You know if you are standing up or upside down because you feel gravity. Both are real. Do you see how ridiculous your comment is?

And again science does not prove anything.

But we have a Scientific theory of gravity, which is one of the best tested theories we have. A scientific theory is an explanation of an aspect of the natural world that can be repeatedly tested, in accordance with the scientific method, using a predefined protocol of observation and experiment. Established scientific theories have withstood rigorous scrutiny and embody scientific knowledge.

### How Gravity bends Water around the Earth

thetruthis: or, just show me the gravity equation that magically curve the water of all oceans.

See my Earth Gravity Calculator for gravity equations for the earth.

How is water bent around the globe by gravity?

This has to do with the fact that the surface of any liquid in equilibrium always lies everywhere on the same equipotential surface, no matter what shape the equipotential surface has. An equipotential surface is a surface where the gravitational attraction has exactly the same magnitude everywhere on this surface. Each equipotential surface on earth is a sphere (or more accurate an ellipsoid) around the center of the planet, because on such a sphere the distance to the center of the earth is the same everywhere and thus the gravitational potential (attraction) on this sphere is the same everywhere. The equipotential spheres build layers of spheres with decreasing potential (attraction) with increasing distance from the center of the earth. So every equipotential surface is a sphere around the earth.

Now, any local deviation in height of a liquid from the equipotential sphere the surrounding liquid is lying on, causes the deviation to be attracted to the earth differently then the surrounding. This causes tangential forces at that location which produce surface streams of the liquid until the whole surface of the water is on an sphere with the same potential everywhere so that no tangential forces arise anymore. Water on an equipotential surface is said to be at the same level everywhere. It is level, but not flat.

So it is a natural process described and predicted by the laws of gravity that water curves around a planet.

You can fill water in a bucket and rotate the bucket around its vertical axes. This produces centrifugal forces in addition to the gravitational force. These forces together form parabolic shaped equipotential surfaces. So the water surface in the bucket conforms to a parabolic shape of revolution. Water is not always flat!

### Space Flights for Everyone?

thetruthis: Also I would like to know why in 2018 we, everyday people are still not experiencing space, as promised 10, 20 and 30 years ago? something wrong maybe?

It's because everyday people do not earn enough money to pay such flights? Such flight are still much too expensive for any commercial flights. Who would spend millions, risk their life and probably get sick only to see earth from space for some time? But if you have the money you can go to space right now. Or become an astronaut. There were more than 500 people in space already, even a billionaire who is not an astronaut. As I said, if you have the money...

But now we are changing the topic.

42Ivo Busko 8/30/2018 | 19:31

In my opinion this is the best "flat earth calculator" site in existence. You not only made a superb job in building a very detailed, accurate, and easy to use simulator, but also augmented it with discussions and explanations on the geometry and physics behind not-so-well-known topics such as refraction and perspective. These details are often misused (on purpose) by flat earthers in clunky attempts to justify their beliefs. It's good to have access to your collection of facts whenever one has to debunk some outlandish claim made by them.

I myself tried my hand at your source code, as a pastime exercise on java script. Attempting to add clouds to the simulated landscape (building on the objects infrastructure, but with a variable height). I have noting presentable so far, but maybe (when I retire) I will be able to come up with some usable piece of code.

Best, and keep up the excellent job!

any chance for fish-eye / barrel distortion simulation for a future feature?

@Priyadi, this is not possible with vector graphics like this App is based on. But that is a good idea for a future refraction simulator.

45Nicolas 11/9/2018 | 11:22

We, living creatures on the Earth, can detect with our appliances and observation of sky, that our earth is spherical. But what if the sky is not what we are thinking about, and if there is negative refraction, though we are thinking it is positive? Due to ethernal nature.

@Nicolas, Refraction in generall and atmospheric refraction in particular are well known physical effects. Refraction is taken into account in astronomical obserevations and survey since ever.

Negative refraction is only possible in thin layers directly over the ground when its temperature is hotter than the air above. It produces a layer of inferior mirage, mirroring. Above that layer the refraction is always positive, because the air cools with altitude. This makes the earth even looking more flat than it is. So refraction is NOT the cause of the curvature measured, on the contrary. Surveyors and astronomers therefore never make measurements in low levels over the ground and always take refraction into account.

Did you know that Refraction is even measured and taken into account in measurements of large structures as wings of airplanes in montage halls, eg. by Boeing. Here again the scientific theory of refraction is confirmed to be correct. Temperature gradient in hall can be big, cold floor, hot a roof. For example: Temperature gradient in the hall: 0.5°C/m, wing size: 30 m, Refraction deflection: 0.4 mm.

There are many, many stduies about refraction, both theoretical and with practical experiments, to research how refraction influences surveying. Here are a few examples:

47Nicolas 11/11/2018 | 10:04

Negative refraction, or hypothetical negative refraction, is very thin and brittle thread, on which very opaque and weak hypothetical Flat Earth uncertainly hangs. And this thread could easily be cut... or approoved, if it is true. But, alas, in these materials, given in your links, I haven't still found such a scissors. In experiments and measurements, in which goes about terrestrial refraction, the known radius of Earth is the base point. But if we suspect some possibility of Flat Earth, whatever weak, we can't base our conclusions on known radius of Earth. Really, it is possible, but it was not done. Or, really, excuse me, I have not still seen it.

@Nicolas. Refraction is a well studied phenomenon. We can measure Refraction and correct the measurements accordingly. There are many methods to do so. Nowadays we have methods like GPS which are not subject to refraction. All benchmarks (reference points) are measured again and again over the centuries using different devices and different methods. They agree all within the error margins of the devices used. GPS is based on the WGS84 ellipsoidal model. You can download the full GPS specification so you can build your own navigation system. There is no doubt since centuries, that the earth is a globe. The fact that all maps and navigation systems, based on the globe, are correct proves the globe.

In experiments and measurements, in which goes about terrestrial refraction, the known radius of Earth is the base point.

That's because we know for sure that the earth is a globe since we began to measure the globe. But if you want you can still measure refraction without assuming a globe and correct your measurements accordingly. If you do so you will always measure that all targets in the distance drop due to the curvature of the earth, not by a random amound as expected due to refraction, but exactly as predicted by the globe model with radius 6371 km. If you know for sure the size of the globe, you can make your life as a surveyor easier by using spherical calculations. You don't have to prove the globe at each single measurement, if it was done already long ago and confirmed many times.

<comment> Here I have developed A Method to Measure Terrestrial Refraction without assuming a globe. </comment>

How have you got to equation 3 (Refraction-Coefficient k) ?

### About the Refraction Equation for k

Amir, read the line above equation 3, there is the link to the source in Wikipedia. But I found it in some documents about survey too. See Terrestrial Refraction in Wikipedia.

We can derive a mathematical model of the atmosphere from basic physical laws, experiments and observations. Such a model is very complex but can be approximated by simpler equations for practical purposes. But because the atmosphere is a complex chaotic system which can not be predicted beyond a certain accuracy anyway, we can just derive an equation from a simplyfied model of the atmosphere directly.

We know that light gets bent in a complex curve through the atmosphere, depending on local atmospheric conditions along the light rays. But we can approximate the complex curve by a circular arc with enough accuracy for practical purposes on normal atmospheric conditions (eg. no mirrages). This is how the equation 3 is derevied. The constants in this equations are gathered from measurements. They may be slightly different in different locations, times and seasons. But for the purpose of geodesy the equation is accurate enough if you know in which conditions it may be applied.

Comparing equation 3 with more sophisticated models showed, that you don't get more accuracy with complexer models in practice. You can't measure the atmospheric conditions over the whole path to calculate refraction precisely. But we can measure the actual refraction with instruments and check agains the calculations. As we know the conditions under which equation 3 is accurate enough, we simply have to ensure, that it is only applied on those conditions. If you need more accuracy, you have to measure the current refraction with dedicated instruments.

I have implemented an App that can simulate complex refractions like mirages from atmospheric conditions you have to provide: Simulation of Atmospheric Refraction

51Olivier RT 5/25/2019 | 10:04

Hello, Great work!

I especially love the curvature calculator : Advanced Earth Curvature Calculator.

I have to say that I also made a very simple version of it, before I discovered yours: https://repl.it/@olivierrt/HorizonCurveCalculator?lite=1&outputonly=1

After using it, I have a question about the focal computation you implemented:

Why focal to view angle ratio correspond to 43.2mm wide sensor? No sensor of this size exists, and this value correspond to the diagonal of very common 24x36mm sensor. Why didn't you use 36mm instead 43.2mm?

### 35 mm equivalent focal length

The focal length can be used to give the magnification of a lens. But focal length specifications depend on the size of the sensor or film frame. Because many cameras use different sensor sizes, it would be difficult to compare different lenses for different cameras. The most common film size had a width of 35 mm and an aspect ratio of 1.5. This gives a diagonal of 43.2 mm. It is not true that no sensor of this size exists. Cameras using sensors of this size are called full frame sensor cameras and are very common in the professional sector. Each big camera manufacturer sells full frame sensor cameras.

To be able to compare the magnification of different lens/sensor combinations it is common practice to give a 35 mm equivalent focal length for each lens. By convention they have chosen to use the diagonal of a 35 mm film as the reference size (43.2 mm) but call it 35 mm equivalent anyway.

I could have used any sensor size in my calculations. But I chosed the same 35 mm equivalent like lenses use. This way you can enter the focal length published for an image, like it is stated in the EXIF data of an image, and my App displays the scene exactly like the camera/lens combination used. In this manner I can render images than can be superimposed onto a camera images and they match. On the other hand, I can find out the focal length used with my App by matching a camera image with the rendered image.

53Roger L Morton 6/26/2019 | 14:52

Presumably the data describing the scenes are saved in some sort of file format. What format, what tool do you author those in?
I am impressed that all this is done in javascript. Real impressive.