horizon

Etymology

The word horizon derives from the Greek ὁρίζων κύκλος (horízōn kýklos) 'separating circle', where ὁρίζων is from the verb ὁρίζω (horízō) '(I) divide, (I) separate', which in turn derives from ὅρος (hóros) 'boundary, landmark'.

True horizon

The true horizon surrounds the observer and it is typically assumed to be a circle, drawn on the surface of a perfectly spherical model of the relevant celestial body, i.e., a small circle of the local osculating sphere. With respect to Earth, the center of the true horizon is below the observer and below sea level. Its radius or horizontal distance from the observer varies slightly from day to day due to atmospheric refraction, which is greatly affected by weather conditions. Also, the higher the observer's eyes are from sea level, the farther away the horizon is from the observer. For instance, in standard atmospheric conditions, for an observer with eye level above sea level by 1.8 metres (6 ft), the horizon is at a distance of about 4.8 kilometres (3 mi).

When observed from very high standpoints, such as a space station, the horizon is much farther away and it encompasses a much larger area of Earth's surface. In this case, the horizon would no longer be a perfect circle, not even a plane curve such as an ellipse, especially when the observer is above the equator, as the Earth's surface can be better modeled as an oblate ellipsoid than as a sphere.

Distance to the horizon

The distance to the true (geometric) horizon (not accounting for atmospheric refraction) from an observer at height

h

{\displaystyle h}

above the surface of a celestial body assumed to be perfectly spherical can be calculated using the formula:

d

=

2

R

h

+

h

2

{\displaystyle d={\sqrt {2Rh+h^{2}}}}

Where:

d

{\displaystyle d}

is the distance to the true (geometric) horizon;

R

{\displaystyle R}

is the radius of the assumed body, e.g., the Earth's arithmetic mean radius of 6,371,008.77138 meters (≈20,902,259.74862 ft);

h

{\displaystyle h}

is the height of the observer above the surface, e.g., the Earth's orthometric height (height above mean sea level).

Assuming no atmospheric refraction and a spherical Earth with radius R=6,371 kilometres (3,959 mi):

Related measures

Another relationship involves the great-circle distance s along the arc over the curved surface of the Earth to the horizon; this is more directly comparable to the geographical distance on maps.

It can be formulated in terms of γ in radians,

s

=

R

γ

;

{\displaystyle s=R\gamma \,;}

then

cos

⁡

γ

=

cos

⁡

s

R

=

R

R

+

h

.

{\displaystyle \cos \gamma =\cos {\frac {s}{R}}={\frac {R}{R+h}}\,.}

Solving for s gives

s

=

R

cos

−

1

Objects above the horizon

To compute the greatest distance DBL at which an observer B can see the top of an object L above the horizon, simply add the distances to the horizon from each of the two points:

DBL = DB + DL

For example, for an observer B with a height of hB=1.70 m standing on the ground, the horizon is DB=4.65 km away. For a tower with a height of hL=100 m, the horizon distance is DL=35.7 km. Thus an observer on a beach can see the top of the tower as long as it is not more than DBL=40.35 km away. Conversely, if an observer on a boat (hB=1.7 m) can just see the tops of trees on a nearby shore (hL=10 m), the trees are probably about DBL=16 km away.

Referring to the figure at the right, and using the approximation above, the top of the lighthouse will be visible to a lookout in a crow's nest at the top of a mast of the boat if

D

B

L

<

3.57

(

h

B

+

h

L

)

,

{\displaystyle D_{\mathrm {BL} }<3.57\,({\sqrt {h_{\mathrm {B} }}}+{\sqrt {h_{\mathrm {L} }}})\,,}

where DBL is in kilometres and hB and hL are in metres.

As another example, suppose an observer, whose eyes are two metres above the level ground, uses binoculars to look at a distant building which he knows to consist of thirty stories, each 3.5 metres high. He counts the stories he can see and finds there are only ten. So twenty stories or 70 metres of the building are hidden from him by the curvature of the Earth. From this, he can calculate his distance from the building:

Refracted horizon

Historically, the distance to the refracted horizon has long been vital to survival and successful navigation, especially at sea, because it determined an observer's maximum range of vision and thus of communication, with all the obvious consequences for safety and the transmission of information that this range implied. This importance lessened with the development of the radio and the telegraph, but 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. Pilots can also retain their spatial orientation by referring to the horizon.

Effect of atmospheric refraction

Due to atmospheric refraction the distance to the visible horizon is further than the distance based on a simple geometric calculation. If the ground (or water) surface is colder than the air above it, a cold, dense layer of air forms close to the surface, causing light to be refracted downward as it travels, and therefore, to some extent, to go around the curvature of the Earth. The reverse happens if the ground is hotter than the air above it, as often happens in deserts, producing mirages. As an approximate compensation for refraction, surveyors measuring distances longer than 100 meters subtract 14% from the calculated curvature error and ensure lines of sight are at least 1.5 metres from the ground, to reduce random errors created by refraction.

If the Earth were an airless world like the Moon, the above calculations would be accurate. However, Earth has an atmosphere of air, whose density and refractive index vary considerably depending on the temperature and pressure. This makes the air refract light to varying extents, affecting the appearance of the horizon. Usually, the density of the air just above the surface of the Earth is greater than its density at greater altitudes. This makes its refractive index greater near the surface than at higher altitudes, which causes light that is travelling roughly horizontally to be refracted downward. This makes the actual distance to the horizon greater than the distance calculated with geometrical formulas. With standard atmospheric conditions, the difference is about 8%. This changes the factor of 3.57, in the metric formulas used above, to about 3.86. For instance, if an observer is standing on seashore, with eyes 1.70 m above sea level, according to the simple geometrical formulas given above the horizon should be 4.7 km away. Actually, atmospheric refraction allows the observer to see 300 metres farther, moving the true horizon 5 km away from the observer.

This correction can be, and often is, applied as a fairly good approximation when atmospheric conditions are close to standard. When conditions are unusual, this approximation fails. Refraction is strongly affected by temperature gradients, which can vary considerably from day to day, especially over water. In extreme cases, usually in springtime, when warm air overlies cold water, refraction can allow light to follow the Earth's surface for hundreds of kilometres. Opposite conditions occur, for example, in deserts, where the surface is very hot, so hot, low-density air is below cooler air. This causes light to be refracted upward, causing mirage effects that make the concept of the horizon somewhat meaningless. Calculated values for the effects of refraction under unusual conditions are therefore only approximate. Nevertheless, attempts have been made to calculate them more accurately than the simple approximation described above.

Astronomical horizon

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 that have an altitude of zero degrees. While similar in ways to the geometrical horizon, in this context a horizon may be considered to be a plane in space, rather than a line on a picture plane.

Perspective

In many contexts, especially perspective drawing, the curvature of the Earth is 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. For observers near sea level, 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 imperceptible to the unaided eye. However, for someone on a 1,000 m (3,300 ft) hill looking out across the sea, the true horizon will be about a degree below a horizontal line.

Vanishing points

The horizon is a key feature of the picture plane in the science of graphical perspective. Assuming the picture plane stands vertical to ground, and P is the perpendicular projection of the eye point O on the picture plane, the horizon is defined as the horizontal line through P. The point P is the vanishing point of lines perpendicular to the picture. If S is another point on the horizon, then it is the vanishing point for all lines parallel to OS. But Brook Taylor indicated that the horizon plane determined by O and the horizon was like any other plane:

The term of Horizontal Line, for instance, is apt to confine the Notions of a Learner to the Plane of the Horizon, and to make him imagine, that that Plane enjoys some particular Privileges, which make the Figures in it more easy and more convenient to be described, by the means of that Horizontal Line, than the Figures in any other plane;…But in this Book I make no difference between the Plane of the Horizon, and any other Plane whatsoever...

The peculiar geometry of perspective where parallel lines converge in the distance, stimulated the development of projective geometry which posits a point at infinity where parallel lines meet. In her book The Geometry of an Art (2007), Kirsti Andersen described the evolution of perspective drawing and science up to 1800, noting that vanishing points need not be on the horizon. In a chapter titled "Horizon", John Stillwell recounted how projective geometry has led to incidence geometry, the modern abstract study of line intersection. Stillwell also ventured into foundations of mathematics in a section titled "What are the Laws of Algebra ?" The "algebra of points", originally given by Karl von Staudt deriving the axioms of a field was deconstructed in the twentieth century, yielding a wide variety of mathematical possibilities. Stillwell states

This discovery from 100 years ago seems capable of turning mathematics upside down, though it has not yet been fully absorbed by the mathematical community. Not only does it defy the trend of turning geometry into algebra, it suggests that both geometry and algebra have a simpler foundation than previously thought.