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.
Link: walter.bislins.ch/CurveCalc
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.
Get App Url Set App State Clear
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.
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.
Obsever Height: Height of the observer above sea level.
Target Distance: Distance from observer to target along the surface.
Target Size: Height of the target from sea level to the top of the target.
Refraction: Refraction Coefficient k. See Panel Refraction for more parameters. If you click on Std then standard refraction is calculated according to the observer height and standard atmospheric conditions. For show k is calculated see RefractionCoefficient k.
Zoom, View∠: Zoom factor f = focal length in 35 mm equivalent units or viewing angle can be used to magnify the image. This two parameters are linked by the following equation:
(1) 
VP Dist: The distance to the vanishing point VP on the Flat Earth Perspective model can be influenced by this value. The distance to the vanishing point is the globe horizon distance + VP Dist.
In this panel some calculated object data is displayed. If multiple objects are selected, the data for the nearest object is displayed.
Visible, Hidden: how much of the object size is hidden behind the horizon and how much is visible.
Angular Size, Angular Visible, Angular Hidden: like above but in angular size. The angular size is arctan( size / distance ) in degrees.
Refraction Angle: How much of the object is lifted due to refraction expressed as an angular size. See RefractionAngle ρ how this angle is calculated.
Lift Absolute: absolute amount of apparent lift of the object with respect to eye level due to refraction.
Relative to Horizon: amount of apparent lift of the object with respect to the horizon due to refraction. The horizon appears lifted with respect to eye level by refraction too. If an object lies behind the horizon, its lift relative to the horizon is smaller than the absolute lift of the object with respect to eye level.
Target Top Angle, Target Top Angle FE: Angle α between target top and eye level for globe and flat earth (FE) respectively. The angle is positive if the target top is above eye level. Some theodolites measure a so called zenith angle ζ. The zenith angle is the angle between the vertical up and the target top. The correlation between this angles is α = 90° − ζ.
Angular Distance θ, Tilt 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:
(2) 
 
where^{'} 

Note that
Drop: is the amount the surface at the target has dropped from the tangent plane at the surface of the observer. This amount depends on the surface distance between observer and target. This distance is dependent on the Target Distance and the Side Pos of the target via Pythagoras.
Bulge Height: is the maximal amount the surface appears to bulge up from the direct line through the earth from the surface at the observer and the surface at the target. This distance is dependent on the Target Distance and the Side Pos of the target via Pythagoras. Note, because the surface bends down in every direction on the globe, the bulge is always lower than the plane tangential at the surface of observer.
Distance: is the distance along the surface from the base of the observer to the base of the target. It is dependent on the Target Distance and the Side Pos of the target via Pythagoras.
Dist on Surf: is the distance of the horizon line from the base of the observer along the surface.
Dip Angle: is the angle between the horizon line and the eye level line as measured at the observer.
Dist from Eye: is the line of sight distance of the horizon line from the observer.
Dist on EyeLvl: is the distance of the horizon measured on the eye level plane.
Drop from Surf: is the drop of the horizon line as measured down from the tangential plane with origin at the surface of the obsever.
Drop from EyeLvl: is the drop of the horizon line as measured down from the tangential plane with origin at the observer height. Drop from Eye−Lvl = Drop from Surf + Height.
Horizon Width: Is the amount of the horizon that is visible within the display. This amount depends on the distance of the horizon and the viewing angle or focal length.
Grid Spacing: is the distance between the grid lines of the globe model.
Radius Earth: is the radius of the earth used for all calculations. This value can be changed.
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:
The transformation T is such, that all Zcoordinates (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 ZCoordinates 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.
How much of an object is hidden behind the curvature of the earth, the so called hidden height h_{h}, depends on the distance of the object from the observer and from the height of the observers eye above the surface h_{O}. 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:
(3) 

(4) 
 
where^{'} 

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 · R_{earth} 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 d_{H} or s_{H} is less than the distance to the target d or s.
The exact distances to the horizon can be calculated with the following equations:
(5) 

(6) 
 
where^{'} 

If the observer height h_{O} is much smaller than the radius of the earth R, the Exact Equations for the Hidden Height can be simplified by the following approximation:
(7) 
 
for 
 
where^{'} 

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:
(8) 
 
where^{'} 

Conclusion: For all practical purposes while the observer is within the troposphere, so that the observer height h_{O} 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.