# Simulation of Atmospheric Refraction

Thursday, November 22, 2018 - 21:50 | Author: wabis | Topics: Simulation, Interactive, Physics
The interactive Refraction Simulator on this page renders scenes as seen on atmospheric conditions you can provide. It is an accurate simulation of Atmospheric Refraction for both Globe Earth and Flat Earth. The simulation predicts what we can expect to observe in reality.

Link to here: walter.bislins.ch/RefractionSim

## The Refraction Simulation App

The Demos behind the black buttons are based on the excellent website about An Introduction to Mirages by Andrew T. Young. Please follow the links below the buttons shown as "more infos" to go to the specific page on his website where he explains the phenomena in detail (sometimes you have to press F5 to see the contents).

no scnene description
• Rendering
• Baro Observer
• Baro Target
• Graphs
• Targets
• Save/Restore

## How the Simulation works

The images of the simulation are composed of pixels. Each pixel has at least one assosiated light ray comming from the scene and carrying a color value from the surface of at least one object.

But the simulation works backwards. It calculates for each image pixel at least one light ray into the scene and checks where this ray hits the Targets. This principle as called a Ray Tracer.

Each light ray is divided into many Ray Segments. The start point and direction of the first Ray Segment is determined by the location of the image pixel with respect to the observer.

Then depending on the position and direction of the Ray Segment the Refraction Coefficient and from that the bending of the Ray Segment is calculated as described at How Custom Refraction is derived. From this bending the end position and direction of the Ray Segment can be calculated. This gives the start point of the next Ray Segment and so on.

Each Ray Segment is checked against all Targets of the scene. If the Ray Segment intersects a Target, the exact intersection point is calculated and which target coordinates are hit by the Ray Segment. The target coordinates are used to determine the color of the target at this coordinates. If the target is opaque, the calculation of the light ray ends here and the image pixel color is defined. If the Target is transparent, the color is stored for later and the calculation of the light ray continues until the next opaque Target or the Horizon is hit. Then all stored colors are combined and define the image pixel color of this light ray. It is also possible to assign multiple light rays to one pixel to get smoother edges in the image (antialiasing). This is accomplished with the Obersampling setting.

Because the simulation makes use of curved Ray Segments, only very few segments are needed, compared to simulations that use a straight Ray Segment approach, to achieve the same accuracy. This allows to simulate even Astronomical Refraction with very long light rays in a short time, see the Sunset Demo.

The simulation is written entirely in JavaScript and uses the HTML canvas element to display the rendered graphics. The Source Code of the simulation App alone, excluding the code of the modules for the graphic output, control panels ect., consists of about 5800 lines.

## How Custom Refraction is derived

The bending of each light ray is dependent on the Refraction Curve along the path of the light ray. The simulation splits each light ray into a number of segments and calculates the bending of each segment and joins the segments together to a single light ray.

The curvature of a Light Ray Segment depends on the Refraction Coefficient at the location of the Ray Segment. The Refraction Coefficient depends on the atmospheric conditions at the location of the Ray Segment and the vertical direction of the light ray segment.

The atmospheric conditions are derived from the inputs in the Barometric Settings panel. You can specify two different barometric parameter sets for two different locations, eg. for the location of the observer via Baro Measure Dist at the Baro Observer panel and the location of the farthest Target or the Horizon at the Baro Target panel. If you don't specify barometric data for the second panel, the parameters from the first panel are used along the whole light path. If you specify separate target parameters, the values between are interpolated and for distances beyond that the target parameters are kept fixed.

You can enter Pressure, Pressure Altitude and some pairs of (altitude, temperature). The simulation contains a default barometric model as defined by the Standard Atmosphere. The entered parameters modify this model accordingly:

1. The simulation calculates a smooth Temperature Curve through the entered parameter values and blends the curve slowly into the curve of the Standard Atmosphere where no values are specified. You are limited to enter values to an altitude of 9 km. Above that the Standard Atmosphere is used. This is not a big limitation, because Refraction is nearly zero at altitudes above anyway.
2. A Temperature Gradient Curve is derived from the calculated Temperature curve.
3. A Pressure Curve is derived from the user supplied Pressure and the Standard Atmosphere.
4. A Refraction Curve is derived from the Pressure, Temperature and Temperature Gradient curves.

You can display the Temperature Curve, Temperature Gradient Curve and Refraction Curve by selecting Display = Baro.

The simulation can now calculate the Refraction Coefficient for any point in the atmosphere between observer and Horizon. The Refraction Coefficient, together with the vertical angle of the Ray Segment at a certain position and altitude, determine the curvature of the Ray Segment.

To calculate the Refraction Coefficient from the barometric data and the vertical angle of the Ray Segment the following equation is used:

(1)
source
where'
 $k$ ' =' 'Refraction Coefficient of a light ray segment at a certain location between observer and target, depending on the altitude and the distance from the observer $P$ ' =' 'Air pressure at that location in mbar or hPa or 1/100 Pa, Standard = 1013.25 mbar $T$ ' =' 'Temperature at that location in Kelvin, Standard = 288.15 K = 15°C $\mathrm{d} T/\mathrm{d} h$ ' =' 'Temperature-Gradient at that location in K/m or °C/m, Standard = −0.0065°C/m $\theta$ ' =' 'Vertical angle of the Ray Segment at that location measured from the horizontal

Note: The constants in the equation above are slightly different in different parts of the world due to different mean atmospheric conditions. Their visual differences are minute but must be taken into account on precise survey measurements over long distances.

The curvature of the Ray Segment is then calculated by the following equation:

(2)
where'
 $c_\mathrm{R}$ ' =' 'curvature of the light Ray Segment $R_\mathrm{R}$ ' =' 'radius of the light ray segment $R_\mathrm{E}$ ' =' 'radius of the earth $k$ ' =' 'Refraction Coefficient as calculated with (1)

Note: The same equations are used for Globe and Flat Earth, because the physics of Refraction is not dependent on the shape of the earth. The curvature or flatness of the air gradient is taken into account by the simulator. The equations do NOT assume a globe earth! The radius of the earth RE in the equation is due to the way the Refraction Coefficient k is defined. k could be defined as the curvature of the light ray without scaling it with the readius of the earth. You can modify (1) accordingly so that you can calculate the curvature of the light ray directly without any reference to the radius of the earth:

(3)
where'
 $c_\mathrm{R}$ ' =' 'curvature of the light Ray Segment $R_\mathrm{R}$ ' =' 'radius of the light ray segment $P$ ' =' 'atmospheric pressure in mbar $T$ ' =' 'temperature in Kelvin $\mathrm{d} T / \mathrm{d} h$ ' =' 'temperature gradient in °C/m $\theta$ ' =' 'vertical angle of the Ray Segment measured from the horizontal

## URL Parameters

The App can be called with some URL parameters to set a specific stored App State:

On the Save/Restore Panel click Get App URL to get an URL from the current state of the App. Copy this URL into any Web-Form (e.g. YouTube comment). Clicking the link will call this page with the saved state.

• To start a certain Demo with the URL, append
&demo=DemoName to the page URL, where DemoName is one of the following names:
Intro, Std2km, Std5km, Std10km, Looming, Towering, Stooping, InferiorMirage, SuperiorMirage, ChicagoStrongLooming, ChicagoSuperiorMirage, Sunset, FECurved, GodRays
• To open a specific Panel, append
&tab=TabName, where TabNam is one of the following names:
Rendering, BaroObserver, BaroTarget, Graphs, Targets, SaveRestore

The parameters may be combined in one URL, eg. &demo=Sunset&tab=Targets

## Specialist Terms

### Target

A Scene is composed of a list of Targets. A Target can be an Image or a Pattern.

Each Target can be positioned and rotated individually.

A Target can be sized as needed. This is important if you use images that show a certain scene like a skyline.

A Target can have Boundaries/Limits or it can be infinetely extended in any direction. If a Target is an image without boundary limits, the image is repeated.

Targets can be made partially transparent. Simulated Light Rays are sent through the scene until they hit an opaque Target or exceed the Horizon limit.

To create a new Target, open the Targets panel and click on New. A copy of an internal Target Template is created. You can also create a Copy of the current selected Target by clicking Clone and then change some parameters. Or you can create one of the Preset Targets.

Note: Targets are treated differently in the Model Flat Earth and Globe Earth. On the Flat Earth model the targets are always flat planes. On the Globe Model the same targets are wraped around the globe, so targets that are not entirely vertical are curved. This allows to create targets like water images that are flat on the Flat Earth display but curved on the Globe Earth display.

Note: For some scenes the order of the Targets may be important. You can move the Targets in the sequence with the buttons Move Up and Move Down.

### Horizon

Because each simulated Light Ray is dividet into many Ray Segments, a limit must be defined so that the calculation is stopped if no opaque Target is hit. This limit is called the Horizon.

A color gradient can be specified for the Horizon, so that a light ray hitting the horizon has a defined color assigned to it. The specified Horizon Distance limits how far a light ray has to be calculated.

### Ray Segment

In the simulation each light ray is dividet into many Ray Segments. For each Ray Segment the Ray Curvature is calculated separately from the Refraction Coefficient at the position of the Ray Segment, see How Custom Refraction is derived.

### Ray Curvature

The Ray Curvature is the bending of a simulated Ray Segment. The Ray Curvature is derived from the Refraction Coefficient and the vertical angle of the Ray Segment at its position.

### Refraction Coefficient

The Refraction Coefficient is a measure of the Ray Curvature of a light ray with respect to the radius of the earth. The Refraction Coefficient is defined as:

(4)
where'
 $k$ ' =' 'Refraction Coefficient $R_\mathrm{E}$ ' =' '6371 km = radius of the earth $R_\mathrm{R}$ ' =' 'radius of a light ray segment

A value of k = 0 means the Ray Segment is straight. A value of k = 1 means the Ray Segment has the same curvature radius as the earth. Ray segments with negative Coefficients are bent upwards.

The Refraction Coefficient can be calculated from atmospheric properties such as Pressure, Temperature and Temperature Gradient at any position of a light ray segment. It also depends on the vertical angle of the ray segment, see equation (1).

The Refraction Coefficient in reality is slightly different at each location of a light ray from the observer to the Target. In practical applications like survey a mean Refraction Coefficient may be used as an approximation. The simulation calculates the individual Refraction Coefficient for each Ray Segment along the whole way from the observer to a target.

### Refraction Curve

The Refraction Curve describes the Refraction Coefficient as a function of altitude, so each altitude is a certain Refraction Coefficient assigned. The Refraction Curve is derived from the Barometric Settings, see How Custom Refraction is derived.

The simulation calculates two Refraction Curves, one for the observer location and one for a Target location the user can specify. For all locations between observer and target the Refraction Coefficient is derived by interpolating the Refraction Coefficient from this two curves according to the altitude and location of a Ray Segment. If no Barometric Settings are defined for a target location, the observer Refraction Curve is used everywhere.

### Pressure Curve

The Pressure changes with altitude. Such a change is described by a Pressure Curve, which gives the Pressure as a function of altitude.

The simulation uses the Standard Atmosphere as the basis for the Pressure Curve but modifies it according to the user specified Pressure entered in the Barometric Settings.

### Temperature Curve

The temperature changes with altitude. Such a change is described by a Temperature Curve, which gives the Temperature as a function of altitude.

The simulation uses some discrete user specified temperature values to calculate a smooth Temperature Curve through these points. The calculated curve blends into the curve as defined by the Standard Atmosphere above the highest specified Temperature point.

### Temperature Gradient

The Temperature Gradient describes the change in Temperature at each altitude. Mathematically it is the Derivative of the Temperature Curve with respect to altitude.

The simulation calculates a Temperature Gradient Curve from the Temperature Curve.

### Temperature Gradient Curve

The Temperature Gradient Curve is a mathematically curve that connects all Temperature Gradient values for a specific altitude and is derived from the Temperature Curve. The Temperature Gradient Curve is used in the calculation of the Refraction Curve.

### Standard Atmosphere

The Standard Atmosphere defines curves for Temperature, Pressure, air Density and Temperature Gradient for each altitude up to about 85 km. These curves are derived from mean values and used as a reference in many applications, eg. in aviation.

The simulation contains the Standard Atmosphere model and uses it for regions the user does not provide atmospheric data data.

## Comments

1Nicolas 11/28/2018 | 09:17

Great work!

Nevertheless, dispite maybe it would be antiscientific, but it would be very curious and even funny to see the picture with flat earth and negative refraction with its radius equal to the radius of spherical earth

2wabis (Walter Bislin, Author of this Page) 11/28/2018 | 20:03

Nocolas, already implemented!

Press the button FE Curved above the displays. To see the correspondig Rays bending upwards select Display = Rays below the right display.

To get the same observations on the Flat Earth due to refraction there must be an impossible Temperature Gradient of −15°C/100m every day!

3Nicolas 11/30/2018 | 04:42

It's super great! The same picture on the Flat Earth really impossible due to the properties of atmosphere and it's gasses. But there is vacuum and maybe there is aether. And it could be more dense closer to Earth surface. And lightspeed could be faster in this denser aether. A littlebit. And produce very stable negative refraction. Hypothetically.

4Nicolas 12/7/2018 | 13:12

I really don't believe that you've surrendered to the problem of measuring of terrestrial refraction. Moreover, you was very close to the truth. Some correction of method and formulas,maybe... I guess, one middle point only needed.

5wabis (Walter Bislin, Author of this Page) 12/7/2018 | 23:02

Nicolas, if you know a method to align at least 3 measuring rods exactly parallel at a distance of about some 100 m or more each, let me know. The only way I can think of is to measure the angle to a fix star at each location at the same latitude at the same time. But this only will work if you believe that stars are very far away, much more than what flat earther believe.

We have instruments that measure refraction directly using 2 lasers with different wave lengths which are reflected from a target. But this method is nothing a layman can use. This instruments are cutting edge technology and have to be extremely precise, because the difference in the light paths are extremely small. This instruments are very expensive.

But anyway, we have multiple ways to measure refraction and correct the measurements accordingly. And we always get a drop with distance as expected by the WGS84 globe model.

Nowadayas we use directly GPS to acquire 3D coordinates of any point on earth or in space. Jesse drove over the Causeway bridge with his car equipped with GPS recorders. He then plotted the data. And guess what, the bridge curves with a radius corresponding to the accepted radius of the earth. This experiment will be repeated with much more details this winter.

6Nicolas 12/8/2018 | 09:53

Oh, I've just mistaken too with your method, with what I wanted to change with it. You are right.
But I have another method. It measures change of declination of vertical line (or not exactly vertical)due to the distance of measuring. It is just modified method of measuring the distances between top and bottom points of bridge vertical supports. I propose to use mountain slope. Distance between two points depends from refraction, but difference doesn't so much. (Really, this method is not so simple. For example, if to measure distances to low and upper point from lower point, the measurement will show declining due to refraction, not for declining itself)
Really the fixed star is the most precise method, of course

7Nicolas 12/8/2018 | 23:50

And there is more simple method to detect the declining of axes: using vertical declination measured between two points in both directions, and geometric properties of parallel lines crossed by third line, but refraction will affect the result. Meanwhile the first method doesn't depend of refraction.

8Nicolas 12/9/2018 | 09:44

If to talk about Flat Earth and our beliefs, my model of Flat Earth ( really I'm not a flatearther, but I've built the model of Flat Earth, or rather flat part of huge jupiterlike planet, on my view, the only possible, if to be flat). According to my model, all terrestrial bodies ( the Sun and Moon too) that we see, or don't exist at all, or situated in absolutely other places (but in the same configuration ). And there is Dome above Earth, that doesn't rotate, that wasn't built, but had grown itself long time ago, that is a living organism, and it has a holographic function and other functions. And there is no any conspiracy, spacecrafts work, and show us real photos, but Dome works too. And we may rely on stars that Dome shows, but it is only our belief, that it is shown in that way, that we need.

9NickMet 2/14/2019 | 18:11

Hi, I guess I found an error in refr calc.
https://ibb.co/Tb4kTQ6

Water texture does not go to true horizon level, check several times.

10wabis (Walter Bislin, Author of this Page) 2/14/2019 | 23:24

Nick, thank you for your feedback.

It's not an error but something I can improve. You have to insert a value at Horizon Dist other than auto on the Rendering panel, because auto sets the horizon too near if the target is not behind the curvature of the globe. The rendering of the ground stops at the horizon sphere, which is in your case too near.

The horizon in the simulator is a sphere with radius Horizon Dist. If a light ray hits this sphere the ray tracer stops there, even if the real globe or FE horizon is much farther away. This limit is there to speed up the calculation, because it limits the number of light segments that have to be calculated. So depending on the scene you are trying to create, you have to manually enter a suitable horizon distance depending on the observer height. Try 50 to 100 km. This should do it.

I will improve the calculation of the auto horizon to set it some factor behind the globe horizon.

11Ivo 3/9/2019 | 20:06

The atmospheric refraction works the opposite way - it lowers distant objects, and it also widens them at the very end, which is why the Sun and the Moon look elliptical at rise and set. The formula for the Earth's "curvature" is the formula for the atmospheric refraction! It explains the bending of rays in your Dome model, too!

12wabis (Walter Bislin, Author of this Page) 3/9/2019 | 21:31

### Atmospheric Refraction is not the Curvature of the Earth

Ivo: The atmospheric refraction works the opposite way - it lowers distant objects, and it also widens them at the very end, which is why the Sun and the Moon look elliptical at rise and set.

What is your evidence? Show me your measurements or references to experiments that support your claim. Because surveyors know since centuries its not as you claim. It's measured uncountable times. Here is the proof that not light is responsible for the apparent curve but the earth itself is curved. The data points you see are recorded by GPS, which is not dependent on refraction and shows the real physical locations of the points in space, independent of what the shape of the earth is. I can click on the points to calculate the curvature radius that the data points describe. It matches the expected radius of the earth for this locations.

The laws of refraction state that light gets bent to the denser medium, which in the atmosphere is commonly at the ground. So light gets bent down except in small layers over warm ground at temperature inversions, which causes mirages. If light gets bent down, objects appear higher, the earth appeas flatter than it is.

I have never observed things at the horizon getting wider. Did you ever measure this on a video of a moon set or sunset with solar filter? Refraction gets stronger near the surface, so the bottom parts of sun and moon get lifted more then the top parts, which makes them appear vertically squeezed. It is modeled in my App here. Left simulation without refraction, right with custom refraction that causes mirages at the horizon. Note that the whole sun is raised from below the horizon by refraction. It is common knowledge that the sun has already set when it appears just above the horitzon, which proves that the simulation is accurate.

Ivo: The formula for the Earth's "curvature" is the formula for the atmospheric refraction! It explains the bending of rays in your Dome model, too!

##### Equation for Earths Curvature Drop
(5)
where'
 $y$ ' =' 'drop from the tangent line at the observer at the horizontal distance x $x$ ' =' 'horizontal distance from the observer along the tangent line $R$ ' =' '6371 km = radius of the earth

Note: the equation 8 inches per miles squared is an approxomation to this exact equation for the curvature drop.

##### Equation for Refracted Light Drop
(6)
with
where'
 $y$ ' =' 'drop from the tangent line at the observer at the horizontal distance x $x$ ' =' 'horizontal distance from the observer along the tangent line $c_\mathrm{R}$ ' =' 'curvature of the light ray due to refraction $R_\mathrm{R}$ ' =' 'radius of the light ray $P$ ' =' 'atmospheric pressure in mbar $T$ ' =' 'temperature in Kelvin $\mathrm{d} T / \mathrm{d} h$ ' =' 'temperature gradient in °C/m

Look this equations the same? No! Similar yes, but the earth's curvature is commonly not the same as the curvature of light due to refraction. And the light ray is bent down in the same direction as the curvature of the earth, in the direction of denser air, not upward. This means objects appear raised on normal refraction and the globe looks flatter than it is.

You can have negative refraction in small layers on temperature inversions. To get negative refraction as demandet to let the flat earth appear like the globe, you need a temperature gradient of at least -15° Celsius per 100 m, all the way up to about 10 km every day. You have to agree that this is physically impossible.

I have simulated a refraction for the flat earth that makes it look like the globe for the lower 300 m which requires a temperature drop of −45°C from the surface to 300 m. Click here for this simulation.

Choose Refraction = Zero below the left image to see how this looks like without refraction on the flat earth. Choose Refraction = Standard and Model = Globe to compare with the globe model under standard refraction.

13Ivo 3/10/2019 | 04:25

The evidence is all the evidence of apparent disappearing bottoms of objects in the distance, which globalists love to point out, but forget to observe how looking left or right nothing is curving, whatsoever. So, if objects are disappearing from the bottom in the distance, but nothing is curving left or right of them, then the explanation is obvious - atmospheric refraction, which lowers them to the observer. This is their apparent look to an observer with a naked eye. If an observer has a tool, like a high zoom camera, he may be able to observe more of the object. The atmospheric refraction depends on the conditions of the atmosphere, too, but those are small effects on short distances. For long distances by average you can use this formula to calculate it: x * X / 8000 in miles!
You can't use globalist GPS to prove the globe - that's redundant! So, objects get lower and lower with distance, is this earth's curvature or is it atmospheric refraction? It's all relative on how you want to look at it!
You have a picture of elliptical Sun at sunset - it's wider! Yes, I have pictures of sunsets with a solar filter, and they are all elliptical. Same with the Moon! Think of it as smaller Sun at sunset, but wider. In some pictures, the Sun is even wider than at noon time! The close it gets to the horizon, the wider it gets, and then it disappears. In some cases, there are mirages that stretch it down and make it look like a half sun, but that is an illusion!

14wabis (Walter Bislin, Author of this Page) 3/10/2019 | 20:58

### About Left to Right Curvature

Ivo: The evidence is all the evidence of apparent disappearing bottoms of objects in the distance, which globalists love to point out, but forget to observe how looking left or right nothing is curving, whatsoever.

This is not true. First if you are near the surface, you can't see left to right curvature because you are in the center of your horizon. The horizon is the same distance all around you and raises to the same level all around you. So how do you expect to see the curvature if the horizon is only some miles away from you? But if you gain altitude, you start to see the circular horizon around you more and more from above, so there will be a left to right curvature increase with altitude.

Watch Finding the curvature of the Earth for an explanation with animations and example images that show how much curvature you can expect to see from left to right and under what conditions.

### Light is bending down, not up

Ivo: So, if objects are disappearing from the bottom in the distance, but nothing is curving left or right of them, then the explanation is obvious - atmospheric refraction, which lowers them to the observer.

The physics of light propagation through a medium with a density gradient as in the atmosphere implies that light gets bend down, which makes distant objects appear raised, not lowered. That's why we can see farther than the geometric horizon of the earth.

### Silly Formula for Refraction

Ivo: The atmospheric refraction depends on the conditions of the atmosphere, too, but those are small effects on short distances. For long distances by average you can use this formula to calculate it: x * x / 8000 in miles!

What does this formula calculate exactly? The apparent drop of objects due to refraction? You are telling correctly that refraction depends on the atmospheric conditions. But I don't see any term for pressure, temperature, density, temperature gradient or humidity in your formula. An equations that depends on atmospheric parameters has this parameters in it. Your equation is not even a valid equation (what units has x, what is it equating, where is the left hand side of the equation?)

### GPS is based on x,y,z Space Coordinates

Ivo: You can't use globalist GPS to prove the globe - that's redundant!

You don't know how GPS works and you seem to assume it is bases on a spherical model. That is not quite right.

GPS consists of more than 24 satellites whose positions and velocities in space are known extrem precisely. A GPS receiver measures its distance from multiple satellites in view and calculates from this distances its position in space. It is irrelevant whether you are on a flat earth, a globe or in outer space. This works the same way for any location.

So the primary measurements of GPS is a position in space as x,y,z coordinates with the origin at the center of the orbits of the satellites (which is the center of the earth, what ever shape it has). If you measure million points distributed over the surface of the earth, you get millions x,y,z coordinates in space. This coordinate system is called ECEF (earth centered earth fixed).

If you plot this x,y,z coordinates in a 3D software the point cloud exhibits the physical shape of the surface of the earth. If the earth is flat, then all points would lie on a clearly recognizable plane in space. If the earth is a globe, then all points would lie on a clearly recognizable sphere.

Guess what: The points in space exhibit a shperical shape. All this points confirm the measurements global surveyors have made in the last few centuries.

Here is an example of a flight around the world showing 18,857 with GPS recorded points in x,y,z coordinates. They lie all in some altitude above a sphere.

The reason why most people think GPS is globe based is because they get latitude, longitude and elevation from their devices, not x,y,z coordinates. But this is only to simplify things for us humans. The raw x,y,z coordinates are useless for us on the surface of the planet. So they are converted into spherical coordinates latitide, longitude and elevation. This is a transformation of the coordinate system and does not move the points in space onto a globe or flat earth. The points stay where the are, but their location in space is described in another coordinate system, that is more usefull for us to navigate around. This globe model is called en.wikipedia.org WGS84. The corresponding Geoid, that defines mean sea level for each point on earth depending on the local gravitaional field is called Earth Gravity Model EGM96. Yes, gravity is an important part of GPS to calculate accurate elevations. Without the taking the Geoid into account the elevations would be hundreds of meters wrong.

Copyright: Danilo Pivato, Source: NASA

So using GPS is the easiest way to figure out the real shape of anything in space by measuring many points of the surface of the thing. You can do this for mountains, lakes, rivers, roads, countries or the whole world. We have measured the shape of the earth in this way to cm accuracy in the last few decades and it confirms the measurements of the globe by centuries of geodetic surveyors.

Today all navigation is based on the WGS84 globe model of GPS, using internally the x,y,z space coordinates to calculate paths and distances on the globe. If you would project those points onto a flat earth, all distances would be wrong, because projections move the points and the distance between moved points are commonly not the same anymore. That's why nobody ever can come up with a flat earth map, that gets all distances right.

### Width of Sun at Sunset is constant

Ivo: You have a picture of elliptical Sun at sunset - it's wider! Yes, I have pictures of sunsets with a solar filter, and they are all elliptical. Same with the Moon! Think of it as smaller Sun at sunset, but wider. In some pictures, the Sun is even wider than at noon time! The close it gets to the horizon, the wider it gets, and then it disappears.

The image on the right shows that the sun does not get wider but squeezed vertically. It's width is constant. Note that as the sun approaches the horizon, it gets also pushed up. At the end the real sun has already set while it's image is just touching the horizon, somewhat depending on refraction of course. On standard refraction the apparent angular lift at the horizon is about 0.5°, which is about the angular size of the sun.

15NickMet 5/12/2019 | 20:13

Hi, can you add to Advanced Earth Curvature Calculator the outline (contour) option of all continents for flat earth (azimutal projection) and globe? It would be nice to see continents on FE from 100 km height.

16NickMet 5/20/2019 | 12:25

It is me again. Can you add target image from URL to your Advanced Earth Curvature Calculator like in the refraction simulator?

17Roberto Scur 12/30/2019 | 00:53

wabis - ”Roberto Scur, no you don ‘t know me for sure!”

rs - Yes, I do know you, I know your arguments posted on your web site, I know how you think, how you create your sophistry, I studied your logic and the variables of your formulas, otherwise I would not understand how your fallacies are made. I'm a computer science graduate, I've been working with CAD, 2D and 3D software for 28 years - I know you very well, WB.

wabis - ”What scientific experiments and measurements can you show that prove the earth is flat?”

rs - The failure of cosmology to present the curvature of the earth is the most important proof that the earth is not a space spinning ball, in my opinion. The experiments are done all over the world, thanks to the powerful zoom cameras available today and the restless questioners of official science. I took some of them and did my mathematical and physical analysis, stripped down fundamental principles in these subjects that demonstrate beyond reasonable doubt that the earth is not a globe.

wabis - ”All experiments and measurements I have done show without a doubt the earth is a globe.”

rs - Nope, what you did was followed the rules and, maybe, invented other ones to fit on your software codes and mainly purposes. By the way, who pays you to do this job?

wabis - ”Not to speak of millions of images of the whole globe from space since more than half a century.”

rs - Yes, of course, this is the worst try to convince a professional like me, specialized on graphics software with 3D approach. All those “millions of images” are computer-generated imagery.

wabis - ”All you can do is denying all measurements that geodetic surveyors have done since centuries, taking refraction into account.”

rs - They do not take refraction into account at all, sir., give me a break.

wabis - ”Today we can use differential GPS to measure the shape of the earth to cm accuracy not affected by refraction.”

rs - That’s what I just told you, no refraction considered, buddy!

wabis - ”… And guess what, all measured benchmarks (millions) are remeasured again and again with increasing accuracy and the different methods used match each other. You can download the historical record of each benchmark. I have the data of 1500 benchmarks from Kansas, which is stated to be flatter than a pancake. When I import the 3D ECEF cartesian coordinates of the benchmarks into a 3D software it shows curvature. I have measured the radius of the arcs any 3 points in a row span, it is always around 6400 km. Get over it, the earth is measured repeatedly and is a globe.”

rs - I would be really surprise if 3D ECEF does not show you an spherical or elliptical result, this data are trash, this data has already been deformed when converted to a geodetic reference datum or geodetic reference system. Did you checked out data in nature as topography collect them?

wabis - ”Concerning density: Look at the gas law. Density depends on pressure and temperature. It’s easier to measure pressure and temperature and calculate density from that or directly use the equation for the refraction coefficient, which is derived from the index of refraction gradient, which depends on wavelength, pressure and temperature gradient (=density gradient), and a little bit on humidity and CO2 concentration.”

rs - No mr. Bislins, density depends on how many molecules there are in the air, MAINLY WATER VAPOR MOLECULES DECREASING THE DENSITY OF THE ATMOSPHERIC LAYERS - the closer to sea level you are, more water there is in the air and less dense is the air. The Avogadro Law explains that the number of molecules remains the same when molecules are exchanged. When water vapor replaces N2 or O2 molecules, the air will be LESS Dense, so the atmospheric layers organize from bottom to top, from least dense to denser, because there is more H2O in the form of steam the lower it is. regardless of pressure and temperature only. ALL measurements found in cosmology IGNORE water vapor in their tables. This is the key to illusion.

It does not matter what Cosmology creates, Just now you are including this mathematical novelty in your refractive calculus, aren't you, Mr Bislins?

I will give you some advice used here in Brazil: "hurry eats raw"!Just now you are including this mathematical novelty in your refractive calculus, aren't you, Mr Bislins?

I will give you some advice used here in Brazil: "hurry eats raw"!

You anticipated citing your Ciddor equation without including it in your curvature calculator. There, as far as the user is concerned, he can change the temperature and pressure as much as he wants and it will not affect how much the object will be visible or invisible, it all comes down to feeding the sophistry of an R '(apparent radius) to stretch your Earth Ball to the fullest, so it can encompass any appearance of objects on the observed horizon.

The Ciddor Equation is the work of reality-defrauding cosmology, so I studied it now, and as such could not miss the constant ill-fated to fix the holes of the ball; In this equation alone there is a table with ten pre-established "K", very convenient to deceive those interested.

I did not insult any topographer. Miss this addiction to sophistry, this professionals don't use curvature for anything in practice, but they can use real refraction.

You are my target, your work do not offers any benefit because you distort the reality of the world we live in, you propagate Earth ball baits by camouflaging your software and texts with dogmatic formulas and lies, this kind of dirty job have nothing to see with computers, gps and anything really useful. I am happy to unmask people like you!

I know you, mr. Bislins, I really know you. Eyes on you.

18wabis (Walter Bislin, Author of this Page) 12/30/2019 | 18:11

Roberto Scur: I studied your logic and the variables of your formulas, otherwise I would not understand how your fallacies are made. I'm a computer science graduate, I've been working with CAD, 2D and 3D software for 28 years.

And I was working as a senjor software development engineer, developing CAD, GIS, 2D and 3D applications, beside others like programming languages, since 1983. I have studied physics, electronics and computer science and have a masters degree in engineering. How is that relevant to the shape of the earth again?

And you have not understood anything I have presented.

### Laser Experiments to prove the Flat Earth are flawed

Roberto Scur: The failure of cosmology to present the curvature of the earth is the most important proof that the earth is not a space spinning ball, in my opinion. The experiments are done all over the world, thanks to the powerful zoom cameras available today and the restless questioners of official science. I took some of them and did my mathematical and physical analysis, stripped down fundamental principles in these subjects that demonstrate beyond reasonable doubt that the earth is not a globe.

...and thousands of geodetic surveyors measured the curvature of the earth all over the world since centuries for a living. Their results are essential in many fields. Wrong measurements would be obvious and not accepted by anyone.

Flat earther first have to learn how experiments are done. They not only have no model, can't make any predictions for what they should expect to measure, nor do they measure anything. This is not even pseudo science. This is confirmation bias like shown here:

All the flat earth laser and mirror experiments are done in the surface layer near the ground where refraction is the strongest as you can see as inferior mirages and compressions that are only visible directly above the surface. Surface refraction bends light along the curvature of the earth for hundreds of miles as soon as the surface is colder than the air above. I have observed such an extreme phenomenon over a frozen lake on Strong Refraction at Bedford Targets, where the earth appeared even convex.

This extreme bending of light is a well known fact by seafarers and geodetic surveyors. Go higher where refraction is minimal (standard refraction), out of the surface layer, and you will see that all observations match the predictions of the Curvature Calculators taking refraction into account. Geodetic surveyors never measure Geodetic control networks in the surface layer. They have built high survey towers to get out of the ground layer for a reason. See also Refraction Coefficient as a Function of Altitude.

### Horizon does not raise to Eye Level

If Flat Earthers where honest, they would only take observations from some meters altitude above the surface on minimal refraction where the air is very steady and clear and use an autolevel or theodolite. They would see that the horizon is not at eye level, because of the curvature of the earth. You can observe this very easily in an airplane on a Head-up Guidance System. There you can measure the horizon drop and calculate the radius of the earth from it:

(7)
where'
 $R$ ' =' 'measured radius of the earth $h$ ' =' 'altitude of the airplane $\varphi$ ' =' 'drop angle of the horizon from eye level (e.g. on the Head-up Guidance System)

Here are some videos from an airline pilot showing the drop of the horizon due to earths curvature:

### I'm not interested in Money

Roberto Scur: ”All experiments and measurements I have done show without a doubt the earth is a globe.” Nope, what you did was followed the rules and, maybe, invented other ones to fit on your software codes and mainly purposes. By the way, who pays you to do this job?

I didn't invent anything to fit a purpose. My software is free to download and to inspect. I'm not payed for what I do here. I'm not in for the money. Money is irrelevant in my life. So much for how well you know me.

### Images of the Earth from Space are not CGI

Roberto Scur: ”Not to speak of millions of images of the whole globe from space since more than half a century.” Yes, of course, this is the worst try to convince a professional like me, specialized on graphics software with 3D approach. All those “millions of images” are computer-generated imagery.

I know that today images can be produced artificially on computers in astonishing details. But can they show the world wide weather exactly as observed from the ground anywhere, anytime in all details? The possibility to create CGI does not prove that all images of the earth from space are faked. Using the same logic, all images you take with your cameras are fake too, because it could be CGI as well.

It is not yet possible to fake images generated every 10 minutes in 11,000 x 11,000 pixels resolution from one satellite alone (Himawari 8), showing the weather of the whole half globe in 16 visible and infrared channels in realtime. You would have to collect data from over 1,000 high altitude balloons (at least 100,000 ft) at fixed positions. There is not a single evidence that such an armada of balloons exist. They would be visible with a P1000 easily.

But you can prove the weather satellites are up there in space:

And there are thousands of images from space taken by cameras on film from times where no CGI existed, hundreds from the Apollo missions alone. And as of 2018, there are 71 different government space agencies in existence beside NASA.

Here are the NASA archives of fotos taken on film. I stopped counting after 400 images of the earth I found in this archives alone. You can download most of the scanned images in a resolution of 3900x3900 pixels:

And here are the first television images from the first satellite designed to observe clouds, TIROS-1, or the Television InfraRed Observation Satellite, launched on April 1, 1960. The satellite carried two tiny television cameras:

TIROS-1 orbited from pole to pole, snapping photos every 30 seconds. The images were stored on tape recorders on the satellite until TIROS-1 came into range of data stations in Fort Monmouth, New Jersey, or Kaena Point, Hawaii. No CGI existed back then.

### Geodetic Surveyors take Refraction and Curvature into account

Roberto Scur: ”All you can do is denying all measurements that geodetic surveyors have done since centuries, taking refraction into account.” They do not take refraction into account at all, sir., give me a break.

Oh yes, surveyors do take refraction into account on zenith angle measurements during traverse and network observations, or they use methods that cancel the refraction error (simultaneous reciprocal zenith angle measurements). Modern theodolites have refraction corrections built in and you can provide actual refraction values calculated from temperature gradient measurements to account for other than standard refraction:

Here is a collection of citations of old surveying books concerning curvature and refraction (source Mick West (Metabunk)):

An articel in Wikipedia on techniques to eliminate Levelling Refraction. "A hi-tech approach to measure refraction is dispersometry using two different wavelengths of light."

Or read any book on geodetic surveying. They all state that the earth is a globe and explain how refraction and curvature is taken into account.

Here are some articles about how refraction is measured and how it influences geodetic survey:

Or put your fingers in your ears and cry "the earth is flat".

### GPS and Refraction

Roberto Scur: ”Today we can use differential GPS to measure the shape of the earth to cm accuracy not affected by refraction.” That’s what I just told you, no refraction considered, buddy!

GPS is not affected by refraction the same way as surveying with a theodolite.

GPS delivers raw x,y,z coordinates with the origin at the center of the earth, so called Earth Centered Earth Fixed (ECEF) cartesian coordinates. Professional grade GNSS receivers can export the raw x,y,z data together with the measured distances to all satellites in view and other parameters in the RINEX data exchange format. No measured distance to any satellite is less than 20,200 km. So they are definitively not on earth.

But even GPS has to take atmospheric and particularly ionospheric refraction into account. But this is built in. The end user has not to deal with it. But it is responsible for the relatively pure accuracy of only about 5 m for consumer grade GPS receivers. The method of Differential GPS can correct for this refraction error and achieve cm accuracy.

The latitude, longitude and elevation you get from your navigation device is calculated from the x,y,z coordinates for your convenience only. All calculations e.g. in navigation devices are done in the measured x,y,z coordinates, not in lat,long,elev.

### How GPS works

Roberto Scur: I would be really surprise if 3D ECEF does not show you an spherical or elliptical result, this data are trash, this data has already been deformed when converted to a geodetic reference datum or geodetic reference system. Did you checked out data in nature as topography collect them?

Then explain how airplanes can navigate using GPS to meter accuracy or how surveyor can measure to cm accuracy using Differential GPS, if the data is trash. You have no clue about GPS.

A GPS receiver measures the distances to all satellites of the multiple Global Navigation Satellite Systems (GNSS) in view and calculates from that your 3D position in space with respect to the center of the earth by solving an over-determined system of linear equations. The results are x,y,z ECEF cartesian coordinates. This are coordinates that have nothing to do with the shape of the earth. They are locations in space. The ISS for instance uses such coordinates for docking maneuvers.

I have tens of thousands of such data points measured from many locations around the world, even from 2 circumanavigations. They all lie on a sphere of radius R, not on a plane. Here is an example of 1527 benchmark points of the state Kansas, which is stated to be flatter than a pancake. They obvisoudly lie on a sphere (click and drag to change the viewing angle).

Note the coordinates in the table at the end are x,y,z not lat,long,elev. Maps are constructed from such data and maps are proven to be accurate. In the Description column are the names of the benchmarks listed.

Only after the x,y,z coordinates are calculated by the GPS receiver, they are transform into latitude, longitude and ellipsoid height in end user devices. The ellipsoid height is further corrected to mean sea level elevation using the data of the Geoid, which is incoorporated in all modern GPS devices. A source of inaccuracies is mainly due to atmospheric disturbances, which geodetic GNSS receivers can correct for by the method of Differential GPS. Airplanes equippied with approved GPS autolanding systems use differential GPS too, which is limited to airports that support this method.

For informations about GPS/GNSS see:

GPS: An introduction to Satellite Navigation
Online course from Standford University - 13 October 2014

NAVSTAR GPS USER EQUIPMENT INTRODUCTION
detailed GPS specifications

### Gas Law and Pressure, Temperature and Density

Roberto Scur: No mr. Bislins, density depends on how many molecules there are in the air, MAINLY WATER VAPOR MOLECULES DECREASING THE DENSITY OF THE ATMOSPHERIC LAYERS

Nonsense, Mr. Scur! Pressure, temperature and density are related to each other by the Ideal gas law and are unique for each composition of gas. If you know the composition of the gas (its specific gas constant RS) and its pressure and temperature you can calculate its density.

(8)
where'
 $\rho$ ' =' 'density $P$ ' =' 'pressure $T$ ' =' 'temperature $R_\mathrm{S}$ ' =' 'specific gas constant, 287.058 J/kg/K for dry air

If we assume a mean temperature gradient with a lapse rate of 6.5°C/km (see International Standard Atmosphere), we can derive the following equation for air density up to an altitude of 11 km using basic laws of physics (note the little g there):

(9)
with
where'
 $\rho(h)$ ' =' 'air density at altitude h in kg/m3 $h$ ' =' 'altiude above sea level (elevation) in meter $\rho_\mathrm{o}$ ' =' '1.225 kg/m2 = air density at sea level $T_\mathrm{o}$ ' =' '288.15 K = absolute temperature at sea level $\alpha$ ' =' '−0.0065 K/m = linear temperature gradient from sea level up to 11 km altitude $g$ ' =' '9.80665 m/s2 = mean gravitational acceleration at sea level $R_\mathrm{S}$ ' =' '287.058 J/kg/K = spezific gas constant of dry air

This is incoorporated in the equation for the refraction coefficient k as shown at Deriving Equations for Atmospheric Refraction. It results in a standard refraction coefficient of k = 0.17 at sea level.

### Density decreases with Altitude, Water Vapor and CO2 Influence

Roberto Scur: the closer to sea level you are, more water there is in the air and less dense is the air. When water vapor replaces N2 or O2 molecules, the air will be LESS Dense, so the atmospheric layers organize from bottom to top, from least dense to denser, because there is more H2O in the form of steam the lower it is. regardless of pressure and temperature only. ALL measurements found in cosmology IGNORE water vapor in their tables. This is the key to illusion.

It's true that density of moist air at the same temperature is less than density of dry air. But if you make the calculation you can see that water vapor can not make the air generally less dense on the ground than above, except perhaps on a small layer, if the air at the surface is saturated and gets dry as quickly as possible. The air can not have less humidity than 0, so either humidity effects are limited to small surface layers or are neglegtable over long height ranges compared to other atmospheric parameters.

Generally the density gradient depends mostly on the temperature gradient, then on pressure and temperature and only very little on humidity gradients. A constant humidity has no influence on refraction, a decreasing humidity with altitude makes refraction a little less, an increaing humidity a little stronger, but only marginal. The surface layer is the only place where refraction can vary considerably due to strong temperature gradients and to some extend due to humidity gradients. Above the surface layer density always decreases according to (9). Therefor refraction does not change much and bends light always down, so the earth appears less curved (looming). See also Refraction Coefficient as a Function of Altitude.

The density of moist air can be calculated as follows (source):

(10)
where'
 $\rho_\mathrm{ma}$ ' =' 'density of moist air $\rho_\mathrm{da}$ ' =' 'density of dry air $K_\mathrm{w}(x)$ ' =' 'correction factor depending on humidity ratio $x$ ' =' 'humidity ratio $R_\mathrm{a}$ ' =' '287.1 J/kg/K = specific gas constant of dry air $R_\mathrm{w}$ ' =' '461.5 J/kg/K = specific gas constant of water vapor

The maximal humidity ratio for air at 15°C is x = 0.01062 (source). The corresponding correction factor Kw due to maximum saturation humidity at 15°C is then:

(11)

So saturated moist air density is 0.9936 times the density of dry air. Moist air is maximal 0.64% less dense than dry air at 15°C. The general influence on the density gradient is neglegtable.

For all practical purposes in refraction calculations the water vapor influence is neglegtable. And as the air gets less dense with altitude, not the other way around, it causes refraction to bend light down which causes looming. The earth appears less curved than it is, about 7/6 R.

As the equations for the index of refraction show (see Calculator for Refractivity based on Ciddor Equation), the influence of water vapor and CO2 is marginal compared to pressure, temperature and the wavelength of light. So humidity and CO2 are ignored or a mean value is choosen, which does change refraction calculations only very marginal in the range of some percent.

### My Derivation of the Equation for Atmospheric Refraction

Roberto Scur: It does not matter what Cosmology creates, Just now you are including this mathematical novelty in your refractive calculus, aren't you, Mr Bislins?

No Mr. Scur, this is not novelty invented by me. This is accepted scientific fact. My equations are exactly the same as published by science. I derived the equtions myself to find out whether I can do it and came to exactly the same results as accepted by common science and applied in practice.

### Ciddor Equation is incorporated in my Curvature Calculator

Roberto Scur: You anticipated citing your Ciddor equation without including it in your curvature calculator.

The Ciddor equation is included in the refraction calculation. The equation for the refraction coefficient has the Ciddor equation incorporated, see Deriving Equations for Atmospheric Refraction.

### Refraction is independent of the Radius of the Earth

Roberto Scur: There, as far as the user is concerned, he can change the temperature and pressure as much as he wants and it will not affect how much the object will be visible or invisible, it all comes down to feeding the sophistry of an R '(apparent radius) to stretch your Earth Ball to the fullest, so it can encompass any appearance of objects on the observed horizon.

Temperature and pressure have only little influence on refraction compared to the temperature gradient. You have to increase your altitude to about 9000 m to half the refraction caused by pressure and temperature changes due to altitude. But relative small changes in the temperature gradient (the change of temperature with altitude) can cause big changes of refraction, which is common near the surface.

You can calculate refraction without any reference to R. R is only included in the refraction coefficient k, because k is defined to be the ratio of the curvatature of light to the curvature of the earth. You can calculate the curvature of light without referencing R. That's what I have done in my Refraction Simulator. The curvature of light is independent of R. Only the k-value uses R as a reference. But this does not change the physics. It's only a convenient definition and simplifies some refraction calculations.

### How Ciddor Equation and Constants are derived

Roberto Scur: The Ciddor Equation is the work of reality-defrauding cosmology, so I studied it now, and as such could not miss the constant ill-fated to fix the holes of the ball; In this equation alone there is a table with ten pre-established "K", very convenient to deceive those interested.

Really? Do you really think that's how scientists work?

The K values of the Ciddor equation are derived from very accurate measurements of reality. If they would not produce the results we can observe, they would be useless and never published and used in practice.

You can model complex connections by using polynomial as an approximation to any desired order. Then you make experiments to measure the coefficients (K-values) and the accuracy of the equations in the desired range of parameters like pressure, temperature ect. This is how empirical equations like the Ciddor equation and it's K-coefficients are derived.

### Geodetic versus Plane Surveying

Roberto Scur: I did not insult any topographer..., this professionals don't use curvature for anything in practice, but they can use real refraction.

I was not talking about topographers. I was talking about geodetic surveyors. You have to distinguish between geodetic and plane surveying:

Plane surveyors work on small patches of the earth which can be approximated by a flat plane for all practical purposes. Geodetic surveyors however measure the big picture and they definitively have to take the curvature of the earth (and refraction) into account. If they wouldn't, the measured triangles would not fit together, because on a globe the sum of the angles of any triangle is more than 180°, and that is what is measured on big triangulations.

The Geodetic control networks are then broken down (projected) to plane maps for plane surveyors. This does not mean that the earth is flat. It is flat enough locally on a small scale for all practical purposes like planing bridges and buildings.

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