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French

Pronunciation

Noun

The horizon (Ancient
Greek ὁ ὁρίζων, /ho horídzôn/, from ὁρίζειν, "to limit") is the
apparent line that separates earth from sky.

More precisely, it is the line that divides all
of the directions one can possibly look into two categories: those
which intersect the Earth's surface, and those which do not. At
many locations, the true horizon is obscured by nearby trees,
buildings, mountains and so forth. The resulting intersection of
earth and sky is instead described as the visible horizon.

Appearance and usage

For observers aboard 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 visual relationship between the aircraft's nose and the horizon
to control the aircraft. A pilot can also retain their spatial
orientation by referring to the horizon.

In many contexts, especially 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 observer increases. Note that, for observers 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, because of the relative size of the observer.
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 from what we
see on our curved Earth.

In astronomy the horizon is the horizontal plane
through (the eyes of) the observer. It is the fundamental
plane of the
horizontal coordinate system, the locus of points which have an
altitude
of zero degrees. While similar in ways to the geometrical horizon
described above, in this context a horizon may be considered to be
a plane in space, rather than a line on a picture plane.

Distance to the horizon

The straight line of sight distance
d in kilometers to the true horizon on earth is approximately

d = \sqrt,

where h is the height above ground or sea level
(in meters) of the eye of the observer. Examples:

For an observer standing on the ground with h = 1.70 m (average
eye-level height), the horizon appears at a distance of 4.7 km.

For an observer standing on a hill or tower of 100 m in height,
the horizon appears at a distance of 36 km.

To compute the height of a tower, the mast of a
ship or a hilltop visible 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 4.7+36 ≈ 41 km.

In the Imperial
version of the formula, 13 is replaced by 1.5, h is in feet and d
is in miles. Examples:

For observers on the ground with eye-level at h = 5 ft 7 in
(5.583 ft), the horizon appears at a distance of 2.89 miles.

For observers standing on a hill or tower 100 ft in height, the
horizon appears at a distance of 12.25 miles.

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

d = \sqrt,

where R is the radius of the Earth (note: both R
and h in this equation must be given in the same units (e.g.
kilometers), but any consistent units will work).

Another relationship involves the arc length
distance s along the curved surface of the Earth to the bottom of
object:

\cos\frac=\frac.

Solving for s gives the formula

s=R\cos^\frac.

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

As a final note, the actual visual horizon is
slightly farther away than the calculated visual horizon, due to
the slight refraction of light rays due to the atmospheric density
gradient. This effect can be taken into account by using a "virtual
radius" that is typically about 20% larger than the true radius of
the Earth.

Curvature of the horizon

From a point above the surface the
horizon appears slightly bent. There is a basic geometrical
relationship between this visual curvature \kappa, the altitude and
the Earth's radius. It is

\kappa=\sqrt\ .

The curvature is the reciprocal of the
curvature angular radius in radians. A curvature of 1
appears as a circle of an angular radius of 45° corresponding to an
altitude of approximately 2640 km above the Earth's surface. At an
altitude of 10 km (33,000 ft, the typical cruising altitude of an
airliner) the mathematical curvature of the horizon is about 0.056,
the same curvature of the rim of circle with a radius of 10 metres
that is viewed from 56 centimetres. However, the apparent curvature
is less than that due to refraction of light in the atmosphere and
because the horizon is often masked by high cloud layers that
reduce the altitude above the visual surface.