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 to here: walter.bislins.ch/CurveCalc
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 (see Angle of view):
(1) 
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.
Sagitta (Bulge): 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.
Frame Width: Is the width of the frame as measured with a scale at the center of the horizon. This amount depends on the distance of the horizon and the viewing angle or focal length. Compare with LeftRight Width.
Grid Spacing: is the distance between the grid lines of the globe model.
LeftRight Width specifies the horizontal distance between the 2 points where the curved horizon meets the border of the frame. The line between the 2 points passes through the earth. The distance of the line between this 2 points is smaller than the distance to the center of the horizon Dist from Eye.
LeftRight Drop (Angle) are the apparent drop height and angle respectively from the horizon tangent to the line between the 2 points where the curved horizon meets the border of the frame. The drop height is calculated from the drop angle using the distance to the center of the horizon Dist from Eye.
For a derivation of how LeftRight values are calculated see Calculating leftright Horizon Drop.
Radius Earth: is the radius of the earth used for all calculations. This value can be changed.
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.
The angle α to the target top is measured from the eye level line of the observer to the top of the nearest target. If the target top is below eye level, the angle is negative, else positive.
(9) 
 
width  
where^{'} 

Note: this equation is robust and gives positive and negative angles α correctly without the need of handling multiple cases.
Derivation of Angle to Target Top Equation
How much the horizon drops at each end of the image depends on the focal length (zoom) of the camera. The curve of the horizon is barely noticable if the viewing angle is narrow (high zoom value). Due to perspective distortions, if we are not thousands of kilometers away from earth, the apparent curve of the horizon is only approximately a circular arc. This gets obvious on very wide angle lenses beyond a viewing angle of 70°. But at the center of each image the apparent radius of the horizon is exactly the refracted radius of the earth (about 7/6 R for standard refraction). The reason is because we are looking tangential to a great circle passing through the horizon point from left to right.
Why does the curvature appear different from different altitudes? Because as we increase the observer altitude the horizon distance gets bigger. Due to perspective, everything at the horizon appears smaller. This includes the radius of the earth at the horizon, which is the apparent radius mentioned here. So although the curvature of the horizon is always the refracted R, it appears different due to perspective and the distance to the horizon.