TheInfoList

The horizon is the apparent line that separates the surface of a
celestial body In astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses ...
from its
sky The sky is the panorama obtained from observing the universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter and energy. The Big Bang th ...

when viewed from the perspective of an observer on or near the surface of the relevant body. This line divides all viewing directions based on whether it intersects the relevant body's surface or not. The ''true horizon'' is actually a theoretical line, which can only be observed to any degree of accuracy when it lies along a relatively smooth surface such as that of Earth's oceans. At many locations, this line is obscured by
terrain Relief map of Sierra Nevada, Spain Terrain or relief (also topographical Topography is the study of the forms and features of land surfaces. The topography of an area could refer to the surface forms and features themselves, or a desc ...

, and on
Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

it can also be obscured by life forms such as
tree In botany, a tree is a perennial plant with an elongated Plant stem, stem, or trunk (botany), trunk, supporting branches and leaves in most species. In some usages, the definition of a tree may be narrower, including only wood plants with se ...

s and/or human constructs such as
building A building, or edifice, is a structure with a roof and walls standing more or less permanently in one place, such as a house or factory. Buildings come in a variety of sizes, shapes, and functions, and have been adapted throughout history for a ...

s. The resulting intersection of such obstructions with the sky is called the ''visible horizon''. On Earth, when looking at a sea from a shore, the part of the sea closest to the horizon is called the offing. Pronounced, "Hor-I-zon". 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
Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

. Its center is below the observer and below
sea level Mean sea level (MSL) (often shortened to sea level) is an average In colloquial, ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in th ...

. Its distance from the observer varies from day to day due to
atmospheric refraction Atmospheric refraction is the deviation of or other from a straight line as it passes through the due to the variation in as a function of . This refraction is due to the velocity of light through , decreasing (the increases) with increased ...
, which is greatly affected by
weather Weather is the state of the atmosphere An atmosphere (from the greek words ἀτμός ''(atmos)'', meaning 'vapour', and σφαῖρα ''(sphaira)'', meaning 'ball' or 'sphere') is a layer or a set of layers of gases surrounding a p ...

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 , the horizon is at a distance of about . When observed from very high standpoints, such as a
space station A space station, also known as an orbital station or an orbital space station, is a spacecraft File:Space Shuttle Columbia launching.jpg, 275px, The US Space Shuttle flew 135 times from 1981 to 2011, supporting Spacelab, ''Mir'', the Hubble S ...

, 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 In mathematics, a plane curve is a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought o ...
such as an ellipse, especially when the observer is above the equator, as the Earth's surface can be better modeled as an
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a Surface (mathemat ...
than as a sphere.

# Etymology

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

# Appearance and usage

Historically, the distance to the visible 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 Communication (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power o ...

, 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 Radio is the technology of signaling and telecommunication, communicating using radio waves. Radio waves are electromagnetic waves of frequency between 30 hertz (Hz) and 300 gigahertz (GHz). They are generated by an electronic device ...

and the
telegraph Telegraphy is the long-distance transmission of messages where the sender uses symbolic codes, known to the recipient, rather than a physical exchange of an object bearing the message. Thus flag semaphore Flag semaphore (from the Ancient ...
, but even today, when flying an
aircraft An aircraft is a vehicle that is able to flight, fly by gaining support from the Atmosphere of Earth, air. It counters the force of gravity by using either Buoyancy, static lift or by using the Lift (force), dynamic lift of an airfoil, or in ...

under
visual flight rules In aviation Aviation is the activities surrounding mechanical flight and the aircraft industry. ''Aircraft'' includes airplane, fixed-wing and helicopter, rotary-wing types, morphable wings, wing-less lifting bodies, as well as aerostat, li ...
, a technique called
attitude flying In aviation Aviation is the activities surrounding mechanical flight Flight or flying is the process by which an object (physics), object motion (physics), moves through a space without contacting any planetary surface, either within an at ...
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 Image:Change of axes.svg, Changing orientation of a rigid body is the same as rotation (mathematics), rotating the axes of a frame of reference, reference frame attached to it. In geometry, the orientation, angular position, attitude, or direction ...
by referring to the horizon. In many contexts, especially perspective drawing, the curvature of the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

is disregarded and the horizon is considered the theoretical line to which points on any
horizontal plane In astronomy, geography, and related sciences and contexts, a ''Direction (geometry, geography), direction'' or ''plane (geometry), plane'' passing by a given point is said to be vertical if it contains the local gravity direction at that point. ...
converge (when projected onto the picture plane) as their distance from the observer increases. For observers near
sea level Mean sea level (MSL) (often shortened to sea level) is an average In colloquial, ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in th ...

the difference between this ''geometrical horizon'' (which assumes a perfectly flat, infinite ground plane) and the ''true horizon'' (which assumes a
spherical Earth Spherical Earth or Earth's curvature refers to the approximation of as a . The earliest documented mention of the concept dates from around the 5th century BC, when it appears in the writings of . In the 3rd century BC, established ...
surface) is imperceptible to the unaided eye (but for someone on a 1000-meter hill looking out to sea the true horizon will be about a degree below a horizontal line). 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 horizontal coordinate system is a celestial coordinate system In astronomy, a celestial coordinate system (or celestial reference system) is a system for specifying positions of satellites, planets, stars, galaxies, and other celestia ...

, the locus of points that have an
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference and a point or object. The exact definition and reference datum varies according to the context (e.g. ...
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.

# Distance to the horizon

Ignoring the
effect of atmospheric refraction Effect may refer to: * A result or change of something ** List of effects ** Cause and effect, an idiom describing causality Pharmacy and pharmacology * Drug effect, a change resulting from the administration of a drug ** Therapeutic effect, a benef ...
, distance to the true horizon from an observer close to the Earth's surface is about :$d \approx \sqrt \,,$ where ''h'' is height above
sea level Mean sea level (MSL) (often shortened to sea level) is an average In colloquial, ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in th ...

and ''R'' is the
Earth radius Earth radius is the distance from the center of Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. About 29% of Earth's surface is land consisting of continent A continent is o ...
. When ''d'' is measured in kilometres and ''h'' in metres, the distance is :$d \approx 3.57\sqrt \,,$ where the constant 3.57 has units of km/m½. When ''d'' is measured in miles (statute miles i.e. "land miles" of ) and ''h'' in feet, the distance is :$d \approx \sqrt \approx 1.22\sqrt \,.$ where the constant 1.22 has units of mi/ft½. In this equation
Earth's surface Earth is the third planet A planet is an astronomical body orbiting a star or Stellar evolution#Stellar remnants, stellar remnant that is massive enough to be Hydrostatic equilibrium, rounded by its own gravity, is not massive enough to c ...
is assumed to be perfectly spherical, with ''r'' equal to about .

## Examples

Assuming no
atmospheric refraction Atmospheric refraction is the deviation of or other from a straight line as it passes through the due to the variation in as a function of . This refraction is due to the velocity of light through , decreasing (the increases) with increased ...
and a spherical Earth with radius R=: * For an observer standing on the ground with ''h'' = , the horizon is at a distance of . * For an observer standing on the ground with ''h'' = , the horizon is at a distance of . * For an observer standing on a hill or tower above sea level, the horizon is at a distance of . * For an observer standing on a hill or tower above sea level, the horizon is at a distance of . * For an observer standing on the roof of the
Burj Khalifa The Burj Khalifa ( ar, برج خليفة, ; pronounced , literally "Khalifa Tower" in English), known as the Burj Dubai prior to its inauguration in 2010, is a skyscraper in Dubai, United Arab Emirates. With a total height of 829.8 m (2,722&nbs ...
, from ground, and about above sea level, the horizon is at a distance of . * For an observer atop
Mount Everest Mount Everest (Chinese Chinese can refer to: * Something related to China China, officially the People's Republic of China (PRC), is a country in East Asia. It is the List of countries and dependencies by population, world's ...

( in altitude), the horizon is at a distance of . * For an observer aboard a commercial passenger plane flying at a typical altitude of , the horizon is at a distance of . * For a U-2 pilot, whilst flying at its service ceiling , the horizon is at a distance of .

## Other planets

On terrestrial planets and other solid celestial bodies with negligible atmospheric effects, the distance to the horizon for a "standard observer" varies as the square root of the planet's radius. Thus, the horizon on
Mercury Mercury usually refers to: * Mercury (planet) Mercury is the smallest planet in the Solar System and the closest to the Sun. Its orbit around the Sun takes 87.97 Earth days, the shortest of all the Sun's planets. It is named after the Roman g ...

is 62% as far away from the observer as it is on Earth, on
Mars Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, being larger than only Mercury (planet), Mercury. In English, Mars carries the name of the Mars (mythology), Roman god of war and is often referred to ...

the figure is 73%, on the
Moon The Moon is Earth's only natural satellite. At about one-quarter the diameter of Earth (comparable to the width of Australia (continent), Australia), it is the largest natural satellite in the Solar System relative to the size of its plane ...

the figure is 52%, on Mimas the figure is 18%, and so on.

## Derivation

If the Earth is assumed to be a featureless sphere (rather than an
oblate spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimension thumb , 236px , The first four spatial dimensions, repres ...

) with no atmospheric refraction, then the distance to the horizon can easily be calculated. The
secant-tangent theorem The tangent-secant theorem describes the relation of line segments created by a secant and a tangent line with the associated circle. This result is found as Proposition 36 in Book 3 of Euclid's Euclid's Elements, ''Elements''. Given a secant ' ...
states that :$\mathrm^2 = \mathrm \times \mathrm \,.$ Make the following substitutions: * ''d'' = OC = distance to the horizon * ''D'' = AB = diameter of the Earth * ''h'' = OB = height of the observer above sea level * ''D+h'' = OA = diameter of the Earth plus height of the observer above sea level, with ''d, D,'' and ''h'' all measured in the same units. The formula now becomes :$d^2 = h\left(D+h\right)\,\!$ or :$d = \sqrt =\sqrt\,,$ where ''R'' is the radius of the Earth. The same equation can also be derived using the
Pythagorean theorem In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

. At the horizon, the line of sight is a tangent to the Earth and is also perpendicular to Earth's radius. This sets up a right triangle, with the sum of the radius and the height as the hypotenuse. With * ''d'' = distance to the horizon * ''h'' = height of the observer above sea level * ''R'' = radius of the Earth referring to the second figure at the right leads to the following: :$\left(R+h\right)^2 = R^2 + d^2 \,\!$ :$R^2 + 2Rh + h^2 = R^2 + d^2 \,\!$ :$d = \sqrt \,.$ The exact formula above can be expanded as: :$d = \sqrt \,,$ where ''R'' is the radius of the Earth (''R'' and ''h'' must be in the same units). For example, if a satellite is at a height of 2000 km, the distance to the horizon is ; neglecting the second term in parentheses would give a distance of , a 7% error.

## Approximation

If the observer is close to the surface of the earth, then it is valid to disregard ''h'' in the term , and the formula becomes- :$d = \sqrt \,.$ Using kilometres for ''d'' and ''R'', and metres for ''h'', and taking the radius of the Earth as 6371 km, the distance to the horizon is :$d \approx \sqrt \approx 3.570\sqrt \,$. Using
imperial units The imperial system of units, imperial system or imperial units (also known as British Imperial or Exchequer Standards of 1826) is the system of units A system of measurement is a collection of units of measurement A unit of measureme ...
, with ''d'' and ''R'' in
statute mile The mile, sometimes the international mile or statute mile to distinguish it from other miles, is a British imperial unit and US customary unit United States customary units (U.S. customary units) are a system of measurements commonly u ...
s (as commonly used on land), and ''h'' in feet, the distance to the horizon is :$d \approx \sqrt \approx \sqrt \approx 1.22 \sqrt$. If ''d'' is in
nautical mile A nautical mile is a unit of length A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are the metric system, metric units, used in every country gl ...
s, and ''h'' in feet, the constant factor is about 1.06, which is close enough to 1 that it is often ignored, giving: :$d \approx \sqrt h$ These formulas may be used when ''h'' is much smaller than the radius of the Earth (6371 km or 3959 mi), including all views from any mountaintops, airplanes, or high-altitude balloons. With the constants as given, both the metric and imperial formulas are precise to within 1% (see the next section for how to obtain greater precision). If ''h'' is significant with respect to ''R'', as with most
satellites In the context of spaceflight, a satellite is an object that has been intentionally placed into orbit. These objects are called artificial satellites to distinguish them from natural satellites such as Earth's Moon. On 4 October 1957, the So ...
, then the approximation is no longer valid, and the exact formula is required.

# Other measures

## Arc distance

Another relationship involves the
great-circle distance The great-circle distance, orthodromic distance, or spherical distance is the shortest distance between two points on the surface of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical ob ...
''s'' along the arc over the curved surface of the Earth to the horizon; with ''γ'' in
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

s, :$s = R \gamma \,;$ then :$\cos \gamma = \cos\frac=\frac\,.$ Solving for ''s'' gives :$s=R\cos^\frac \,.$ The distance ''s'' can also be expressed in terms of the line-of-sight distance ''d''; from the second figure at the right, :$\tan \gamma = \frac \,;$ substituting for ''γ'' and rearranging gives :$s=R\tan^\frac \,.$ 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'').

## Zenith angle

When the observer is elevated, the horizon
zenith angle The zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere In astronomy and navigation, the celestial sphere is an abstraction, abstract sphere that has an arbitrarily large radius and is concen ...
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, :$\cos \gamma =\frac$ where $h$ is the observer's height above the surface and $\gamma$ is the angular dip of the horizon. It is related to the horizon zenith angle $z$ by: :$z = \gamma +90^\circ$ For a non-negative height $h$, the angle $z$ is always ≥ 90°.

## Objects above the horizon

To compute the greatest distance at which an observer can see the top of an object above the horizon, compute the distance to the horizon for a hypothetical observer on top of that object, and add it to the real observer's distance to the horizon. For example, for an observer with a height of 1.70 m standing on the ground, the horizon is 4.65 km away. For a tower with a height of 100 m, the horizon distance is 35.7 km. Thus an observer on a beach can see the top of the tower as long as it is not more than 40.35 km away. Conversely, if an observer on a boat () can just see the tops of trees on a nearby shore (), the trees are probably about 16 km away. Referring to the figure at the right, the top of the lighthouse will be visible to a lookout in a
crow's nest A crow's nest is a structure in the upper part of the Mast (sailing), main mast of a ship or a structure that is used as a lookout point. On ships, this position ensured the widest field of view for lookouts to spot approaching hazards, other sh ...
at the top of a mast of the boat if :$D_\mathrm < 3.57\,\left(\sqrt + \sqrt\right) \,,$ where ''D''BL is in kilometres and ''h''B and ''h''L 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
storey A storey (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar and usage ...
s, each 3.5 metres high. He counts the storeys he can see, and finds there are only ten. So twenty storeys 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: :$D \approx 3.57\left(\sqrt+\sqrt\right)$ which comes to about 35 kilometres. It is similarly possible to calculate how much of a distant object is visible above the horizon. Suppose an observer's eye is 10 metres above sea level, and he is watching a ship that is 20 km away. His horizon is: :$3.57 \sqrt$ kilometres from him, which comes to about 11.3 kilometres away. The ship is a further 8.7 km away. The height of a point on the ship that is just visible to the observer is given by: :$h\approx\left\left(\frac\right\right)^2$ which comes to almost exactly six metres. The observer can therefore see that part of the ship that is more than six metres above the level of the water. The part of the ship that is below this height is hidden from him by the curvature of the Earth. In this situation, the ship is said to be
hull-down In sailing and warfare, hull-down means that the upper part of a vessel or vehicle is visible, but the main, lower body (Hull (watercraft), hull) is not; the term hull-up means that all of the body is visible. The terms originated with sailing an ...
.

# Effect of atmospheric refraction

Due to atmospheric
refraction In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...

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
mirage A mirage is a naturally-occurring in which light rays bend via to produce a displaced image of distant objects or the sky. The word comes to via the ''(se) mirer'', from the ''mirari'', meaning "to look at, to wonder at". Mirages can be c ...

s. 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 , whose
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

and
refractive index In optics Optics is the branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or ...

vary considerably depending on the temperature and pressure. This makes the air 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 In optics Optics is the branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or ...

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 Standard may refer to: Flags * Colours, standards and guidons * Standard (flag), a type of flag used for personal identification Norm, convention or requirement * Standard (metrology), an object that bears a defined relationship to a unit of ...
. 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 A mirage is a naturally-occurring in which light rays bend via to produce a displaced image of distant objects or the sky. The word comes to via the ''(se) mirer'', from the ''mirari'', meaning "to look at, to wonder at". Mirages can be c ...

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. Outside the visual wavelength range, refraction will be different. For
radar Radar (radio detection and ranging) is a detection system that uses radio waves to determine the distance (''ranging''), angle, or velocity of objects. It can be used to detect aircraft, Marine radar, ships, spacecraft, guided missiles, motor ...

(e.g. for wavelengths 300 to 3 mm i.e. frequencies between 1 and 100 GHz) the radius of the Earth may be multiplied by 4/3 to obtain an effective radius giving a factor of 4.12 in the metric formula i.e. the radar horizon will be 15% beyond the geometrical horizon or 7% beyond the visual. The 4/3 factor is not exact, as in the visual case the refraction depends on atmospheric conditions. ;Integration method—Sweer If the density profile of the atmosphere is known, the distance ''d'' to the horizon is given by :$d=\left\left( \psi +\delta \right\right) \,,$ where ''R''E is the radius of the Earth, ''ψ'' is the dip of the horizon and ''δ'' is the refraction of the horizon. The dip is determined fairly simply from :$\cos \psi = \frac \,,$ where ''h'' is the observer's height above the Earth, ''μ'' is the index of refraction of air at the observer's height, and ''μ''0 is the index of refraction of air at Earth's surface. The refraction must be found by integration of :$\delta =-\int_^ \,,$ where $\phi\,\!$ is the angle between the ray and a line through the center of the Earth. The angles ''ψ'' and $\phi\,\!$ are related by :$\phi =90^\circ -\psi \,.$ ;Simple method—Young A much simpler approach, which produces essentially the same results as the first-order approximation described above, uses the geometrical model but uses a radius . The distance to the horizon is then :$d=\sqrt \,.$ Taking the radius of the Earth as 6371 km, with ''d'' in km and ''h'' in m, :$d \approx 3.86 \sqrt \,;$ with ''d'' in mi and ''h'' in ft, :$d \approx 1.32 \sqrt \,.$ Results from Young's method are quite close to those from Sweer's method, and are sufficiently accurate for many purposes.

# Curvature of the horizon

From a point above Earth's surface, the horizon appears slightly
convex Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to: Science and technology * Convex lens A lens is a transmissive optics, optical device which focuses or disperses a light beam by me ...
; it is a
circular arc Circular may refer to: * The shape of a circle * Circular (album), ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fal ...
. The following
formula In , a formula is a concise way of expressing information symbolically, as in a mathematical formula or a . The informal use of the term ''formula'' in science refers to the . The plural of ''formula'' can be either ''formulas'' (from the mos ...

expresses the basic geometrical relationship between this visual curvature $\kappa$, the altitude $h$, and Earth's radius $R$: :$\kappa=\sqrt\$ The curvature is the reciprocal of the curvature in
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

s. A curvature of 1.0 appears as a circle of an angular radius of 57.3° corresponding to an altitude of approximately above Earth's surface. At an altitude of , the
cruising Cruising may refer to: * Cruising, on a cruise ship *Cruising (driving), driving around for social purposes, especially by teenagers *Cruising (maritime), leisurely travel by boat, yacht, or cruise ship *Cruising for sex, the process of searching i ...
altitude of a typical airliner, the mathematical curvature of the horizon is about 0.056, the same curvature of the rim of circle with a radius of 10 m that is viewed from 56 cm directly above the center of the circle. However, the apparent curvature is less than that due to
refraction In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...

of light by the atmosphere and the obscuration of the horizon by high cloud layers that reduce the altitude above the visual surface.

# Vanishing points

The horizon is a key feature of the
picture plane In painting Painting is the practice of applying paint, pigment, color or other medium to a solid surface (called the "matrix" or "support"). The medium is commonly applied to the base with a brush, but other implements, such as knives, spo ...
in the science of
graphical perspective Graphics () are visual perception, visual images or designs on some surface, such as a wall, canvas, screen, paper, or stone, to inform, illustration, illustrate, or entertain. In contemporary usage, it includes a pictorial representation of dat ...
. 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 Parallel may refer to: Computing * Parallel algorithm In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their a ...

to ''OS''. But
Brook Taylor Brook Taylor (18 August 1685 – 29 December 1731) was an English mathematician best known for creating Taylor's theorem and the Taylor series, which are important for their use in mathematical analysis. Life and work Brook Taylor ...
(1719) indicated that the horizon plane determined by ''O'' and the horizon was like any other
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
: :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 In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, proj ...
which posits a
point at infinity 150px, The real line with the point at infinity; it is called the real projective line. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic ...
where parallel lines meet. In her book ''Geometry of an Art'' (2007),
Kirsti Andersen Kirsti Andersen (born December 9, 1941, Copenhagen Copenhagen ( da, København ) is the capital and most populous city of Denmark. As of 1 January 2020, the city had a population of 794,128 with 632,340 in Copenhagen Municipality, 104,305 in ...
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 John Colin Stillwell (born 1942) is an Australia Australia, officially the Commonwealth of Australia, is a Sovereign state, sovereign country comprising the mainland of the Australia (continent), Australian continent, the island of Tasman ...

recounted how projective geometry has led to
incidence geometry Incidence may refer to: Economics * Benefit incidence, the availability of a benefit * Expenditure incidence, the effect of government expenditure upon the distribution of private incomes * Fiscal incidence, the economic impact of government tax ...
, the modern abstract study of line intersection. Stillwell also ventured into
foundations of mathematics Foundations of mathematics is the study of the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental r ...
in a section titled "What are the Laws of Algebra ?" The "algebra of points", originally given by
Karl von Staudt Karl Georg Christian von Staudt (24 January 1798 – 1 June 1867) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
deriving the axioms of a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
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.