# WaBis

## Walter Bislin's Blog (En)

Friday, August 31, 2018 - 17:54 | Author: wabis | Topics: FlatEarth, Calculator, Knowlegde, Geometry

Use this calculator to calculate and visualize what we can see of a target of a certain size and distance from any observer altitude, taking refraction into account. You can compare the results between flat earth and globe earth. Many more values are calculated and you can customize and store the settings.

• Basics
• View
• Target 1
• Target 2
• Refraction⇒
• Std
• 0
• Units
• Save/Restore
• Reset All

Please read the paragraph on Refraction to get familiar with this panel.

Use this Formular to convert between different lengh units. You can Copy/Paste the results into input fields in the other Forms.

Use Get App Url to get an URL containing the current App State. Click Set App State oder copy the URL into any browser address field to go to this page and display the current App State.

Curvature Calculator showing Angles
Curvature Calculator showing No Refraction
Curvature Calculator showing Refraction
• Angles
• No Refraction
• Refraction

Note: Values marked with a * are not dependent on Refraction in reality. The marked values show the apparent values if refraction is not zero. So to display the real values, set Refraction = 0. This is true for all Horizon Data as well.

Angular Distance θ: is the angle between the observer, the center of the earth and the nearest target. This angle is used in some Drop calculators, as the Drop x is:

(1)
where'
 $x$ ' =' 'drop from the surface level $R$ ' =' 'radius of the earth $\theta$ ' =' 'angle between observer, the center of the earth and the target

Note that $\theta$ is the same angle as the Tilt of the target object.

## Flat Earth Perspective

If the Flat Earth model is selected you can choose 2 different "Perspective models" for the Flat Earth, although there exists only one single law of perspective in reality:

• Normal applies the real law of Perspective on the Flat Earth model to map the 3D model onto the screen. This is the same law of perspective that is applied on the Globe model. You can't change the Perspective model on the Globe model.
• Flerspective is a special transformation that is unique to this App. It applies a transformation T to the 3D model before the normal law of perspective is applied additionally to map the 3D model onto the screen. It is the perspective model that Flat Earthers proclaim.

The transformation T is such, that all Z-coordinates (vertical direction) of the 3D model are changed to linearly converge to eye level at the vanishing point (VP) distance in all directions. This corresponds to the side view Flat Earthers draw, where all objects and lines converge to a single VP. All lines that pass the vanishing distance are clipped. Objects behind the VP are not drawn.

Note: if the VP is placed infinitely far away, like it is in the real law of Perspective, then the Z-Coordinates would keep their values and the result is real Perspective.

The distance of the VP in the App is calculated such that it is the same as the Globe horizon distance, dependent on the observers altitude, plus the value of the VP Dist slider. This way the VP distance increases automatically with increasing altitude of the observer.

Note, that T only scales the vertical size of objects. The horizontal size remains fixed and only shrinks on the screen with increasing distance of an object due to the normal law of perspective. If the horizontal sizes were also scaled by T, then the gaps bewteen objects would get bigger when they move on parallel tracks into the distance, which can not be observed in reality. So T is not an affine transformation and has nothing to do with the law of perspective used in drawings and computer graphics. It is implemented for educational purposes only.

There are situations, like boats or bridges over water, where Flerspective seems to match observations: objects near eye level in the distance disappear, similar as on the Globe model, where they get hidden buttom first by the horizon. But zooming in on the horizon shows that Flerspective does not match observations. Most observations look very different than what Flerspective produces.

Flat Earthers may object that I did not implement the right kind of Perspective. But no Flat Earther is able to give me a precise mathematical description of Flerspective that could be implemented in code. Even in my version of Flerspective I had to fudge the numbers to match observations as good as possible. This is not necessary on the normal law of Perspective together with the Globe model.

[1]
Law of Perspective Fallacies
https://www.theflatearthsociety.org/forum/index.php?topic=15589.0
[2]
The Flat Earth And The Law Of Perspective
http://coconutrevival.com/?p=1983
[3]
Everything is perspective on flat earth
https://www.thedailyplane.com/everything-is-perspective-on-flat-earth/
[4]
Sun setting illusion on the flat earth
https://flatgeocentricearth.wordpress.com/2015/12/02/sun-setting-illusion-on-the-flat-earth/

## Exact Equations for the Hidden Height

How much of an object is hidden behind the curvature of the earth, the so called hidden height Hh, depends on the distance of the object from the observer and from the height of the observers eye above the surface hO. The distance can be expressed as the line of sight d to the object, tangent to the horizon, or as the arc length s along the surface of the earth between observer and target.

Note: To calculate the hidden height you must not use the famous equation 8 inches per miles squared! This equation is an approximation to calculate the drop of the earth surface from a tangent line on the surface at the observer. It calculates not the hidden part of an object.

Depending on whether you know the line of sight distance d or the distance along the surface s the following equations calculate the hidden height exactly:

(2)
(3)
where'
 $h_\mathrm{h}$ ' =' 'hidden height $d$ ' =' 'line of sight distance between observer and target, tangential to the surface of the globe $s$ ' =' 'distance between observer and target along the surface of the globe $h_\mathrm{O}$ ' =' 'eye height of the observer measured from the surface of the globe $R$ ' =' '6371 km = radius of the earth without refraction; 7433 km = radius to use for standard refraction 7/6 Rearth; 7681 km = radius for standard refraction k = 0.17 (sea level)

The same equations can be used to calculate the hidden height with and without refraction. You simply have to choose the corresponding value for R. Because under standard refraction the earth looks less curved, you can use a bigger radius for the earth than it is in reality. For standard refraction 7/6 · Rearth use R = 7433 km. For standard refraction k = 0.17 use R = 7681 km.

There are multiple slightly different values for standard refraction in use. Near the ground the bigger value is more accurate. For higher altitudes the smaller value is more accurate. If you press the button Std the App calculates refraction depending on the observer altitude. Near sea level refraction is about k = 0.17.

The hidden height equations are only valid if the object lies behind the horizon. That is if the distance to the horizon dH or sH is less than the distance to the target d or s.

The exact distances to the horizon can be calculated with the following equations:

(4)
(5)
where'
 $d_\mathrm{H}$ ' =' 'line of sight distance to the horizon $s_\mathrm{H}$ ' =' 'distance to the horizon along the surface of the globe $h_\mathrm{O}$ ' =' 'eye height of the observer measured from the surface of the globe $R$ ' =' '6371 km = radius of the earth without refraction; 7433 km = radius to use for standard refraction 7/6 Rearth; 7681 km = radius for standard refraction k = 0.17 (sea level)

## Approximation Equations for the Hidden Height

If the observer height hO is much smaller than the radius of the earth R, the Exact Equations for the Hidden Height can be simplified by the following approximation:

(6)
for

$h_\mathrm{O} \ll R$

where'
 $h_\mathrm{h}$ ' =' 'hidden height $h_\mathrm{O}$ ' =' 'eye height of the observer measured from the surface of the globe $d$ ' =' 'distance between observer and target, line of sight or along the surface $R$ ' =' '6371 km = radius of the earth without refraction; 7433 km = radius to use for standard refraction 7/6 Rearth; 7681 km = radius for standard refraction k = 0.17 (sea level)

Note: for observer height much less than the radius of the earth, the line of sight distance d and the surface distance s are identical for all practical purposes. So the equation above holds for both cases.

The distance to the horizon can also be approximated by the following equation:

(7)
where'
 $d_\mathrm{H}$ ' =' 'distance to the horizon $h_\mathrm{O}$ ' =' 'eye height of the observer measured from the surface of the globe $R$ ' =' '6371 km = radius of the earth without refraction; 7433 km = radius to use for standard refraction 7/6 Rearth; 7681 km = radius for standard refraction k = 0.17 (sea level)

Conclusion: For all practical purposes while the observer is withing the troposphere, so that the observer height hO is much less than the radius of the earth R, you can use the approximation equations. In this case for the distance between observer and target you can use the line of sight or the distance along the surface. They are practically identical.