![Sureal Horizon](https://upload.wikimedia.org/wikipedia/commons/5/5f/Sureal_Horizon.jpg)
The horizon is the apparent line that separates the surface of a
celestial body
An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical object, physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''bod ...
from its
sky
The sky is an unobstructed view upward from the surface of the Earth. It includes the atmosphere and outer space. It may also be considered a place between the ground and outer space, thus distinct from outer space.
In the field of astronomy, ...
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 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
The ocean (also the sea or the world ocean) is the body of salt water that covers approximately 70.8% of the surface of Earth and contains 97% of Earth's water. An ocean can also refer to any of the large bodies of water into which the worl ...
. At many locations, this line is obscured by
terrain
Terrain or relief (also topographical relief) involves the vertical and horizontal dimensions of land surface. The term bathymetry is used to describe underwater relief, while hypsometry studies terrain relative to sea level. The Latin word ...
, and on Earth it can also be obscured by life forms such as trees and/or human constructs such as buildings. 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 relevant celestial body. With respect to Earth, the center of the true horizon is below the observer and below
sea level
Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardised g ...
. Its distance from the observer varies from day to day due to
atmospheric refraction, which is greatly affected by
weather
Weather is the state of the atmosphere, describing for example the degree to which it is hot or cold, wet or dry, calm or stormy, clear or cloudy. On Earth, most weather phenomena occur in the lowest layer of the planet's atmosphere, the ...
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 is a spacecraft capable of supporting a human crew in orbit for an extended period of time, and is therefore a type of space habitat. It lacks major propulsion or landing systems. An orbital station or an orbital space station i ...
, 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
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...
than as a sphere.
Etymology
The word ''horizon'' derives from the Greek () 'separating circle', where is from the verb
ὁρίζω () 'to divide, to separate', which in turn derives from () 'boundary, landmark'.
Appearance and usage
![Sailing_Boat_Horizon](https://upload.wikimedia.org/wikipedia/commons/d/d9/Sailing_Boat_Horizon.JPG)
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 la, communicare, meaning "to share" or "to be in relation with") is usually defined as the transmission of information. The term may also refer to the message communicated through such transmissions or the field of inquir ...
, 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 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 called a transmit ...
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 is a method of telegraphy, whereas p ...
, but even today, when flying an
aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air. It counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines ...
under
visual flight rules
In aviation, visual flight rules (VFR) are a set of regulations under which a pilot operates an aircraft in weather conditions generally clear enough to allow the pilot to see where the aircraft is going. Specifically, the weather must be better ...
, 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
In geometry, the orientation, angular position, attitude, bearing, or direction of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it occupies.
More specifically, it refers to the imagina ...
by referring to the horizon.
In many contexts, especially
perspective drawing
Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of 3D projection, graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate r ...
, the curvature of the Earth 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'' or ''plane'' passing by a given point is said to be vertical if it contains the local gravity direction at that point.
Conversely, a direction or plane is said to be hor ...
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
Spherical Earth or Earth's curvature refers to the approximation of figure of the Earth as a sphere.
The earliest documented mention of the concept dates from around the 5th century BC, when it appears in the writings of Greek philosophers. I ...
surface) is imperceptible to the unaided eye. However, for someone on a hill looking out across the 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 that uses the observer's local horizon as the fundamental plane to define two angles: altitude and azimuth.
Therefore, the horizontal coordinate system is sometimes called as th ...
, 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 datum and a point or object. The exact definition and reference datum varies according to the context ...
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, distance to the true horizon from an observer close to the Earth's surface is about
[
]
:
where ''h'' is height above sea level and ''R'' is the
Earth radius
Earth radius (denoted as ''R''🜨 or R_E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly (equatorial radius, deno ...
.
When ''d'' is measured in kilometres and ''h'' in metres, the distance is
:
where the constant 3.57 has units of km/m
½.
When ''d'' is measured in miles (statute miles i.e. "land miles" of
) and ''h'' in feet, the distance is
:
where the constant 1.22 has units of mi/ft
½.
In this equation Earth's surface is assumed to be perfectly spherical, with ''r'' equal to about .
Examples
Assuming no
atmospheric refraction 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, , Khalifa Tower), known as the Burj Dubai prior to its inauguration in 2010, is a skyscraper in Dubai, United Arab Emirates. It is known for being the world’s tallest building. With a total height ...
, from ground, and about above sea level, the horizon is at a distance of .
* For an observer atop
Mount Everest
Mount Everest (; Tibetan: ''Chomolungma'' ; ) is Earth's highest mountain above sea level, located in the Mahalangur Himal sub-range of the Himalayas. The China–Nepal border runs across its summit point. Its elevation (snow heig ...
( 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 commonly refers to:
* Mercury (planet), the nearest planet to the Sun
* Mercury (element), a metallic chemical element with the symbol Hg
* Mercury (mythology), a Roman god
Mercury or The Mercury may also refer to:
Companies
* Merc ...
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, only being larger than Mercury (planet), Mercury. In the English language, Mars is named for the Mars (mythology), Roman god of war. Mars is a terr ...
the figure is 73%, on the
Moon
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
the figure is 52%, on
Mimas
Mimas may refer to:
*Mimas (Giant), son of Gaia in Greek mythology, one of the Gigantes
* Mimas (''Aeneid''), a son of Amycus and Theono, born the same night as Paris, who escorted Aeneas to Italy
*Karaburun, a town and district in Turkey, formerl ...
the figure is 18%, and so on.
Derivation
![Horizons](https://upload.wikimedia.org/wikipedia/commons/7/7c/Horizons.svg)
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 surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circ ...
) with no atmospheric refraction, then the distance to the horizon can easily be calculated.
The
tangent-secant 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 ''Elements''.
Given a secant ''g'' intersectin ...
states that
:
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
:
or
:
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 between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
.
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:
:
:
:
The exact formula above can be expanded as:
:
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
![horizon_distance_graphs](https://upload.wikimedia.org/wikipedia/commons/4/44/Horizon_distance_graphs.svg)
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-
:
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
:
.
Using
imperial units, with ''d'' and ''R'' in
statute miles (as commonly used on land), and ''h'' in feet, the distance to the horizon is
:
.
If ''d'' is in
nautical mile
A nautical mile is a unit of length used in air, marine, and space navigation, and for the definition of territorial waters. Historically, it was defined as the meridian arc length corresponding to one minute ( of a degree) of latitude. Today ...
s, and ''h'' in feet, the constant factor is about 1.06, which is close enough to 1 that it is often ignored, giving:
:
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, 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 distance along a great circle.
It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a ...
''s'' along the
arc
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
over the
curved surface of the Earth to the horizon; with ''γ'' in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s,
:
then
:
Solving for ''s'' gives
:
The distance ''s'' can also be expressed in terms of the line-of-sight distance ''d''; from the second figure at the right,
:
substituting for ''γ'' and rearranging gives
:
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
![HomogSphElevObsZmax](https://upload.wikimedia.org/wikipedia/commons/b/b7/HomogSphElevObsZmax.png)
When the observer is elevated, the horizon
zenith angle
The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction (plumb line) opposite to the gravity direction at that location (nadir). The zenith is the "highest" ...
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,
:
where
is the observer's height above the surface and
is the angular dip of the horizon. It is related to the horizon zenith angle
by:
:
For a non-negative height
, the angle
is always ≥ 90°.
Objects above the horizon
![HorizonDistance](https://upload.wikimedia.org/wikipedia/commons/5/5a/HorizonDistance.png)
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 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 ships, or land by ...
at the top of a mast of the boat if
:
where ''D''
BL is in kilometres and ''h''
B and ''h''
L are in metres.
![Curvatura 2](https://upload.wikimedia.org/wikipedia/commons/c/c7/Curvatura_2.gif)
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) or story (American English) is any level part of a building with a floor that could be used by people (for living, work, storage, recreation, etc.). Plurals for the word are ''storeys'' (UK) and ''stories'' (US).
T ...
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:
:
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:
:
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:
:
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.
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
mirage
A mirage is a naturally-occurring optical phenomenon in which light rays bend via refraction to produce a displaced image of distant objects or the sky. The word comes to English via the French ''(se) mirer'', from the Latin ''mirari'', meanin ...
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.
![Pale Horizon](https://upload.wikimedia.org/wikipedia/commons/1/18/Pale_Horizon.jpg)
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
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
and
refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
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
A mirage is a naturally-occurring optical phenomenon in which light rays bend via refraction to produce a displaced image of distant objects or the sky. The word comes to English via the French ''(se) mirer'', from the Latin ''mirari'', meanin ...
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 is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...
(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
:
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
:
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
:
where
is the angle between the ray and a line through the center of the Earth. The angles ''ψ'' and
are related by
:
;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
:
Taking the radius of the Earth as 6371 km, with ''d'' in km and ''h'' in m,
:
with ''d'' in mi and ''h'' in ft,
:
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
![A colorful view of airglow layers at Earth's horizon](https://upload.wikimedia.org/wikipedia/commons/3/37/A_colorful_view_of_airglow_layers_at_Earth%27s_horizon.jpg)
From a point above Earth's surface, the horizon appears slightly
convex; it is a
circular arc. The following formula expresses the basic geometrical relationship between this visual curvature
, the altitude
, and Earth's radius
:
:
The curvature is the reciprocal of the curvature
angular radius
The angular diameter, angular size, apparent diameter, or apparent size is an angular distance describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the visual angle, and in optics, it is ...
in radians. 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 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 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
![TwoPointPerspective](https://upload.wikimedia.org/wikipedia/commons/f/f5/TwoPointPerspective.png)
The horizon is a key feature of the
picture plane
In painting, photography, graphical perspective and descriptive geometry, a picture plane is an image plane located between the "eye point" (or '' oculus'') and the object being viewed and is usually coextensive to the material surface of the w ...
in the science of
graphical perspective
Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
. 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 is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of IBM ...
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 w ...
(1719) indicated that the horizon plane determined by ''O'' and the horizon was like any other
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
:
: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
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adj ...
where parallel lines meet. In her book ''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
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''incidenc ...
, 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
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 grass ...
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.
See also
References
Further reading
*
{{Authority control
Horizontal coordinate system
Astronomical coordinate systems