![Sureal Horizon](https://upload.wikimedia.org/wikipedia/commons/5/5f/Sureal_Horizon.jpg)
The horizon is the apparent curve that separates the surface of a
celestial body 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 curve 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. 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 wo ...
, 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, i.e., a
small circle
A circle of a sphere is a circle that lies on a sphere. Such a circle can be formed as the intersection of a sphere and a plane, or of two spheres. Circles of a sphere are the spherical geometry analogs of generalised circles in Euclidean space ...
of the local
osculating sphere
Figure of the Earth is a term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. A sphere is a well-known historical approxima ...
. 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 standardise ...
. Its radius or horizontal distance from the observer varies slightly from day to day due to
atmospheric refraction
Atmospheric refraction is the deviation of light or other electromagnetic wave from a straight line as it passes through the atmosphere due to the variation in air density as a function of height. This refraction is due to the velocity of ligh ...
, 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 cloud cover, cloudy. On Earth, most weather phenomena occur in the lowest layer of the planet's atmos ...
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, 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 a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
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.
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 inqui ...
, 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 transmi ...
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 engine ...
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
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 ...
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 imagin ...
by referring to the horizon.
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
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. ...
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 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 ...
.
The expression can be simplified as:
:
where the constant equals ''k''.
In this equation, Earth's surface is assumed to be perfectly spherical, with ''R'' equal to about .
Examples
Assuming no
atmospheric refraction
Atmospheric refraction is the deviation of light or other electromagnetic wave from a straight line as it passes through the atmosphere due to the variation in air density as a function of height. This refraction is due to the velocity of ligh ...
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, from ground, and about above sea level, the horizon is at a distance of .
* For an observer atop
Mount Everest
Mount Everest (; Tibetic languages, Tibetan: ''Chomolungma'' ; ) is List of highest mountains on Earth, Earth's highest mountain above sea level, located in the Mahalangur Himal sub-range of the Himalayas. The China–Nepal border ru ...
( 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 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. In the English language, Mars is named for the Roman god of war. Mars is a terrestrial planet with a thin at ...
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 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) 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.
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
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 first defined in the British Weights and Measures Act 1824 and continued to be developed th ...
, 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 United States customary unit of distance; both are based on the older English unit of length equal to 5,280 Engli ...
s (as commonly used on land), and ''h'' in feet, the distance to the horizon is
:
.
If ''d'' is in
nautical miles, 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.
Related measures
Arc distance
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
Geographical distance or geodetic distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. ...
on a map.
It can be formulated in terms of ''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 tha ...
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 "highe ...
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 ''D''
BL 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:
:''D''
BL = ''D''
B + ''D''
L
For example, for an observer B with a height of ''h''
B1.70 m standing on the ground, the horizon is ''D''
B4.65 km away. For a tower with a height of ''h''
L100 m, the horizon distance is ''D''
L35.7 km. Thus an observer on a beach can see the top of the tower as long as it is not more than ''D''
BL40.35 km away. Conversely, if an observer on a boat (''h''
B1.7m) can just see the tops of trees on a nearby shore (''h''
L10m), the trees are probably about ''D''
BL16 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
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 b ...
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
storeys, 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
In sailing and warfare, hull-down means that the upper part of a vessel or vehicle is visible, but the main, lower body (hull) is not; the term hull-up means that all of the body is visible. The terms originated with sailing and naval warfare i ...
.
Effect of atmospheric refraction
Due to
atmospheric refraction
Atmospheric refraction is the deviation of light or other electromagnetic wave from a straight line as it passes through the atmosphere due to the variation in air density as a function of height. This refraction is due to the velocity of ligh ...
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.
![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 Standard may refer to:
Symbols
* Colours, standards and guidons, kinds of military signs
* Standard (emblem), a type of a large symbol or emblem used for identification
Norms, conventions or requirements
* Standard (metrology), an object th ...
. 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.
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, we ...
(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
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
; it is a
circular arc
Circular may refer to:
* The shape of a circle
* ''Circular'' (album), a 2006 album by Spanish singer Vega
* Circular letter (disambiguation)
** Flyer (pamphlet), a form of advertisement
* Circular reasoning, a type of logical fallacy
* Circular ...
. 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 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
In three-dimensional geometry, a parallel project ...
. 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 ...
to ''OS''. But
Brook Taylor (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' ...
:
: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, ...
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. Ad ...
where parallel lines meet. In her book ''Geometry of an Art'' (2007),
Kirsti Andersen
Kirsti Andersen (born December 9, 1941, Copenhagen), published under the name Kirsti Pedersen, is a Danish historian of mathematics. She is an Associate Professor of the History of Science at Aarhus University, where she had her Candidate exami ...
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 Australian mathematician on the faculties of the University of San Francisco and Monash University.
Biography
He was born in Melbourne, Australia and lived there until he went to the Massachusetts Institu ...
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 ''incide ...
, the modern abstract study of line intersection. Stillwell also ventured into
foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
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 who used synthetic geometry to provide a foundation for arithmetic.
Life and influence
Karl was born in the Free Imperial City of Rothenburg, which is n ...
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