Horizon
From Exampleproblems
- For other uses of horizon, see Horizon (disambiguation)
The horizon is the line that separates earth from sky. More precisely, it is the line that divides all of the directions you can possibly look into, into two categories: those which intersect the Earth, and those which do not. At many locations, the true horizon is obscured by trees, buildings, mountains, etc. The resulting intersection of earth and sky is instead known as the visible horizon. However, if you are on a ship at sea, the true horizon is strikingly apparent. Historically, the distance to the visible horizon has been extremely important as it represented the maximum range of communication and vision before the development of the radio and the telegraph. Even today, when flying an aircraft under Visual Flight Rules, a technique called attitude flying is used to control the aircraft, where the pilot uses the relationship between the aircraft's nose and the horizon to control the aircraft. He also retains his spatial orientation by referring to the horizon.
In many contexts, in particular perspective drawing, the curvature of the earth is typically disregarded and the horizon is considered the theoretical line to which points on any horizontal plane converge (when projected onto the picture plane) as their distance from the viewer increases. Note that, for viewers near the ground, the difference between this geometrical horizon (which assumes a perfectly flat, infinite ground plane) and the true horizon (which assumes a spherical Earth surface) is typically imperceptibly small. That is, if the Earth were truly flat, there would still be a visible horizon line, and, to ground based viewers, its position and appearance would not be significantly different than what we see on our curved Earth.
The distance of the horizon on earth, in a plain (standing on the ground or on a tower, or from a airplane) or on a hill or mountain surrounded by plains, is approximately 
kilometers, where h is the height in meters of the eyes.
Examples:
- standing on the ground with h = 1.70 m, the horizon is at a distance of 4.7 km
- standing on a hill or tower of 100 m height, the horizon is at a distance of 36 km
These figures indicate theoretical visibility (what can be seen depends also on how clear the air is, of course) of objects at ground level. To compute to what distance the tip of a tower, the mast of a ship or a hill is above the horizon, add the horizon distance for that height. For example, standing on the ground with h = 1.70 m, one can see, weather permitting, the tip of a tower of 100 m height at a distance of 41 km.
The Imperial version of the formula for the distance to the horizon is 
miles, where h is the height of the eye in feet.
The metric formula is reasonable (and the Imperial one is actually quite precise) when h is much smaller than the radius of the Earth (6371 km). The exact formula for distance from the viewpoint to the horizon, applicable even for satellites, is
where R is the radius of the Earth (note: both R and h in this equation are in kilometers). A different formula is given by
This formula gives the arc length distance s along the curved surface of the Earth to to bottom of object, whereas the above formula is for the straight line of sight distance to the top of the object of view. Both formulas agree when the height of the object is negligible compared to the radius.
See also
External links
- Derivation of the distance to the horizon
- Derivation of the distance to the horizon (with some questions answered)
- An artistic treatment of the horizon
Acknowledgements
The first version of this article originates from Jason Harris' Astroinfo which comes along with KStars, a Desktop Planetarium for Linux/KDE. See http://edu.kde.org/kstars/index.phtml
