Editing Horizon (section)

==Related measures==

===Arc distance===
Another relationship involves the [[great-circle distance]] ''s'' along the [[arc (geometry)|arc]] over the [[figure of the Earth|curved surface of the Earth]] to the horizon; this is more directly comparable to the [[geographical distance]] on a map.

It can be formulated in terms of ''γ'' in [[radian]]s,

:<math>s = R \gamma \,;</math>

then

:<math>\cos \gamma = \cos\frac{s}{R}=\frac{R}{R+h}\,.</math>

Solving for ''s'' gives

:<math>s=R\cos^{-1}\frac{R}{R+h} \,.</math>

The distance ''s'' can also be expressed in terms of the line-of-sight distance ''d''; from the second figure at the right,

:<math>\tan \gamma = \frac {d} {R} \,;</math>

substituting for ''γ'' and rearranging gives

:<math>s=R\tan^{-1}\frac{d}{R} \,.</math>

The distances ''d'' and ''s'' are nearly the same when the height of the object is negligible compared to the radius (that is, ''h''&nbsp;≪&nbsp;''R'').

===Zenith angle===
[[File:HomogSphElevObsZmax.png|thumb|250px|right|Maximum zenith angle for elevated observer in homogeneous spherical atmosphere]]
When the observer is elevated, the horizon [[zenith angle]] can be greater than 90°. The maximum visible zenith angle occurs when the ray is tangent to Earth's surface; from triangle OCG in the figure at right,

:<math>\cos \gamma =\frac{R}{R+h}</math>

where <math>h</math> is the observer's height above the surface and <math>\gamma</math> is the angular dip of the horizon. It is related to the horizon zenith angle <math>z</math> by:

:<math>z = \gamma +90{}^\circ </math>

For a non-negative height <math>h</math>, the angle <math>z</math> is always ≥ 90°.