Editing Coordinate system

{{Short description|Method for specifying point positions}}
{{redirect|Coordinate|coordinates on the Earth|Spatial reference system|other uses|Coordinate (disambiguation)}}
[[File:3D Spherical.svg|thumb|The [[spherical coordinate system]] is commonly used in ''physics''. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distance ''r'', polar angle ''θ'' ([[theta]]), and azimuthal angle ''φ'' ([[phi]]). The symbol ''ρ'' ([[rho]]) is often used instead of ''r''.]]
In [[geometry]], a '''coordinate system''' is a system that uses one or more [[number]]s, or '''coordinates''', to uniquely determine and standardize the [[Position (geometry)|position]] of the [[Point (geometry)|points]] or other geometric elements on a [[manifold]] such as [[Euclidean space]].<ref>Woods p. 1</ref><ref>{{MathWorld|title=Coordinate System|urlname=CoordinateSystem}}</ref> The order of the coordinates is significant, and they are sometimes identified by their position in an ordered [[tuple]] and sometimes by a letter, as in "the ''x''-coordinate". The coordinates are taken to be [[real number]]s in [[elementary mathematics]], but may be [[complex number]]s or elements of a more abstract system such as a [[commutative ring]]. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and ''vice versa''; this is the basis of [[analytic geometry]].<ref>{{MathWorld|title=Coordinates|urlname=Coordinates}}</ref>

==Common coordinate systems==

===Number line===
{{Main|Number line}}
{{Anchor|Number line}} <!-- [[Negative distance]] redirects here. I considered the main but this is better. Courtesy note per [[WP:SPECIFICLINK]]. -->
The simplest example of a coordinate system is the identification of points on a [[line (geometry)|line]] with real numbers using the ''[[number line]]''. In this system, an arbitrary point ''O'' (the ''origin'') is chosen on a given line. The coordinate of a point ''P'' is defined as the signed distance from ''O'' to ''P'', where the signed distance is the distance taken as positive or negative depending on which side of the line ''P'' lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point.<ref>{{cite book |last1=Stewart |first1=James B. |last2= Redlin |first2= Lothar |last3=Watson |first3=Saleem |author-link=James Stewart (mathematician) |title=College Algebra |publisher=[[Brooks Cole]] |year=2008 |edition= 5th |pages=13–19 |isbn=978-0-495-56521-5}}</ref>
[[File:Number-line-right-arrow.svg|center|The number line]]

===Cartesian coordinate system===
{{Main|Cartesian coordinate system}}
{{multiple image
 | total_width = 450
 | image1 = Cartesian-coordinate-system.svg
 | caption1 = The [[Cartesian coordinate system]] in the plane
 | alt1 =
 | image2 = Rectangular coordinates.svg
 | caption2 = The Cartesian coordinate system in three-dimensional space
 | alt2 =
}}
The prototypical example of a coordinate system is the [[Cartesian coordinate system]]. In the [[plane (geometry)|plane]], two [[perpendicular]] lines are chosen and the coordinates of a point are taken to be the signed distances to the lines.<ref>{{cite book
 | last1 = Anton | first1 = Howard
 | last2 = Bivens | first2 = Irl C.
 | last3 = Davis | first3 = Stephen
 | year = 2021
 | title = Calculus: Multivariable
 | url = https://books.google.com/books?id=001EEAAAQBAJ&pg=PA657
 | page = 657
 | publisher = [[John Wiley & Sons]]
 | isbn = 978-1-119-77798-4
}}</ref> In three dimensions, three mutually [[Orthogonality|orthogonal]] planes are chosen and the three coordinates of a point are the signed distances to each of the planes.<ref>{{cite book |vauthors=Moon P, Spencer DE |year=1988 |chapter=Rectangular Coordinates (x, y, z) |title=Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions |edition=corrected 2nd, 3rd print |publisher=Springer-Verlag |location=New York |pages=9–11 (Table 1.01) |isbn=978-0-387-18430-2}}</ref> This can be generalized to create ''n'' coordinates for any point in ''n''-dimensional Euclidean space.

Depending on the direction and order of the [[#Coordinate axis|coordinate axes]], the three-dimensional system may be a [[right-hand rule|right-handed]] or a left-handed system.

===Polar coordinate system===
{{Main|Polar coordinate system}}
Another common coordinate system for the plane is the ''polar coordinate system''.<ref>{{cite book| last=Finney|first=Ross| author2=George Thomas|author3=Franklin Demana|author4=Bert Waits|title=Calculus: Graphical, Numerical, Algebraic|edition=Single Variable Version|date=June 1994|publisher=Addison-Wesley Publishing Co.|isbn=0-201-55478-X|url-access=registration| url=https://archive.org/details/calculusgraphica00ross}}</ref> A point is chosen as the ''pole'' and a ray from this point is taken as the ''polar axis''. For a given angle ''θ'', there is a single line through the pole whose angle with the polar axis is ''θ'' (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is ''r'' for given number ''r''. For a given pair of coordinates (''r'',&nbsp;''θ'') there is a single point, but any point is represented by many pairs of coordinates. For example, (''r'',&nbsp;''θ''), (''r'',&nbsp;''θ''+2''π'') and (−''r'',&nbsp;''θ''+''π'') are all polar coordinates for the same point. The pole is represented by (0, ''θ'') for any value of ''θ''.
{{Clear}}

===Cylindrical and spherical coordinate systems===
{{Main|Cylindrical coordinate system|Spherical coordinate system}}
[[File:Cylindrical Coordinates.svg|thumb|Cylindrical coordinate system]]
There are two common methods for extending the polar coordinate system to three dimensions. In the '''cylindrical coordinate system''', a ''z''-coordinate with the same meaning as in Cartesian coordinates is added to the ''r'' and ''θ'' polar coordinates giving a triple (''r'',&nbsp;''θ'',&nbsp;''z'').<ref>{{cite book |last1=Margenau |first1=Henry |author-link1=Henry Margenau |last2=Murphy |first2=George M. |year=1956 |title=The Mathematics of Physics and Chemistry |url=https://archive.org/details/mathematicsofphy0002marg |url-access=registration |publisher=D. van Nostrand |location=New York City |page=[https://archive.org/details/mathematicsofphy0002marg/page/178 178] |lccn=55010911|oclc=3017486}}</ref> Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (''r'',&nbsp;''z'') to polar coordinates (''ρ'',&nbsp;''φ'') giving a triple (''ρ'',&nbsp;''θ'',&nbsp;''φ'').<ref>{{cite book |author-link1= Philip M. Morse |last1=Morse |first1=PM |author-link2=Herman Feshbach |last2=Feshbach |first2=H |year= 1953 |title= Methods of Theoretical Physics, Part I |publisher= McGraw-Hill |location= New York |page= 658 |lccn= 52011515}}</ref>

===Homogeneous coordinate system===
{{Main|Homogeneous coordinates}}
A point in the plane may be represented in ''homogeneous coordinates'' by a triple (''x'',&nbsp;''y'',&nbsp;''z'') where ''x''/''z'' and ''y''/''z'' are the Cartesian coordinates of the point.<ref>{{cite book |title=An Introduction to Algebraical Geometry|first=Alfred Clement|last=Jones
|publisher=Clarendon|year=1912}}</ref> This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the [[projective plane]] without the use of [[infinity]]. In general, a homogeneous coordinate system is one where only the ratios of the coordinates are significant and not the actual values.

===Other commonly used systems===
Some other common coordinate systems are the following:
* [[Curvilinear coordinates]] are a generalization of coordinate systems generally; the system is based on the intersection of curves.
** [[Orthogonal coordinates]]: [[coordinate surface]]s meet at right angles
** [[Skew coordinates]]: [[coordinate surface]]s are not orthogonal
* The [[Log-polar coordinates|log-polar coordinate system]] represents a point in the plane by the logarithm of the distance from the origin and an angle measured from a reference line intersecting the origin.
* [[Plücker coordinates]] are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as [[homogeneous coordinates]].
* [[Generalized coordinates]] are used in the [[Lagrangian mechanics|Lagrangian]] treatment of mechanics.
* [[Canonical coordinates]] are used in the [[Hamiltonian mechanics|Hamiltonian]] treatment of mechanics.
* [[Barycentric coordinate system]] as used for [[ternary plot]]s and more generally in the analysis of [[triangle]]s.
* [[Trilinear coordinates]] are used in the context of triangles.

There are ways of describing curves without coordinates, using [[intrinsic equation]]s that use invariant quantities such as [[curvature]] and [[arc length]]. These include:
* The [[Whewell equation]] relates arc length and the [[tangential angle]].
* The [[Cesàro equation]] relates arc length and curvature.

==Coordinates of geometric objects==
Coordinates systems are often used to specify the position of a point, but they may also be used to specify the position of more complex figures such as lines, planes, [[circle]]s or [[sphere]]s. For example, [[Plücker coordinates]] are used to determine the position of a line in space.<ref>{{cite book
| last= Hodge
| first= W.V.D.
| author-link= W. V. D. Hodge
| author2=[[D. Pedoe]]
| title= Methods of Algebraic Geometry, Volume&nbsp;I (Book&nbsp;II)
| publisher= [[Cambridge University Press]]
| year= 1994
| isbn= 978-0-521-46900-5
| orig-year= 1947}}</ref> When there is a need, the type of figure being described is used to distinguish the type of coordinate system, for example the term ''[[line coordinates]]'' is used for any coordinate system that specifies the position of a line.

It may occur that systems of coordinates for two different sets of geometric figures are equivalent in terms of their analysis. An example of this is the systems of homogeneous coordinates for points and lines in the projective plane. The two systems in a case like this are said to be ''dualistic''. Dualistic systems have the property that results from one system can be carried over to the other since these results are only different interpretations of the same analytical result; this is known as the ''principle of [[duality (mathematics)|duality]]''.<ref>Woods p. 2</ref>

==Transformations==
{{Broader|Geometric transformation}}
{{Main list|List of common coordinate transformations}}
{{See also|Active and passive transformation}}

There are often many different possible coordinate systems for describing geometrical figures. The relationship between different systems is described by ''coordinate transformations'', which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates (''x'',&nbsp;''y'') and polar coordinates (''r'',&nbsp;''θ'') have the same origin, and the polar axis is the positive ''x'' axis, then the coordinate transformation from polar to Cartesian coordinates is given by ''x''&nbsp;=&nbsp;''r''&nbsp;cos''θ'' and ''y''&nbsp;=&nbsp;''r''&nbsp;sin''θ''.

With every [[bijection]] from the space to itself two coordinate transformations can be associated:
* Such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation)
* Such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation)

For example, in [[dimension|1D]], if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to −3, so that the coordinate of each point becomes 3 more.

==Coordinate lines/curves==
{{Anchor|Coordinate axis|Coordinate line|Coordinate curve}}
{{Redirect-distinguish|Coordinate line|Line coordinates}}

Given a coordinate system, if one of the coordinates of a point varies while the other coordinates are held constant, then the resulting curve is called a '''coordinate curve'''. If a coordinate curve is a [[straight line]], it is called a '''coordinate line'''. A coordinate system for which some coordinate curves are not lines is called a ''[[curvilinear coordinate system]]''.<ref>{{cite book |title=Mathematical Methods for Engineers and Scientists |volume=2|first=K. T.|last=Tang|publisher=Springer|year=2006|isbn=3-540-30268-9|page=13}}</ref>
''[[Orthogonal coordinates]]'' are a special but extremely common case of curvilinear coordinates.

A coordinate line with all other constant coordinates equal to zero is called a '''coordinate axis''', an [[oriented line]] used for assigning coordinates. 
In a [[Cartesian coordinate system]], all coordinates curves are lines, and, therefore, there are as many coordinate axes as coordinates. Moreover, the coordinate axes are pairwise [[orthogonal]]. 

A polar coordinate system is a curvilinear system where coordinate curves are lines or [[circle]]s. However, one of the coordinate curves is reduced to a single point, the origin, which is often viewed as a circle of radius zero. Similarly, spherical and cylindrical coordinate systems have coordinate curves that are lines, circles or circles of radius zero.

Many curves can occur as coordinate curves. For example, the coordinate curves of [[parabolic coordinates]] are [[parabola]]s.

==Coordinate planes/surfaces==
{{Anchor|Coordinate plane|Coordinate surface}}
{{Redirect-distinguish|Coordinate plane|Plane coordinates}}
[[File:Parabolic coordinates 3D.png|thumb|left|Coordinate surfaces of the three-dimensional paraboloidal coordinates.]]

In three-dimensional space, if one coordinate is held constant and the other two are allowed to vary, then the resulting surface is called a '''coordinate surface'''. For example, the coordinate surfaces obtained by holding ''ρ'' constant in the [[spherical coordinate system]] are the spheres with center at the origin. In three-dimensional space the intersection of two coordinate surfaces is a coordinate curve. In the Cartesian coordinate system we may speak of '''coordinate planes'''.
Similarly, '''coordinate hypersurfaces''' are the {{nowrap|(''n'' − 1)}}-dimensional spaces resulting from fixing a single coordinate of an ''n''-dimensional coordinate system.<ref> {{cite book |title=A Computational Differential Geometry Approach to Grid Generation |first=Vladimir D.|last=Liseikin|publisher=Springer|year=2007|isbn=978-3-540-34235-9|page=38}}</ref>

==Coordinate maps==
{{Main|Coordinate map}}
{{Further|Manifold}}
The concept of a ''coordinate map'', or ''coordinate chart'' is central to the theory of manifolds. A coordinate map is essentially a coordinate system for a subset of a given space with the property that each point has exactly one set of coordinates. More precisely, a coordinate map is a [[homeomorphism]] from an open subset of a space ''X'' to an open subset of '''R'''<sup>''n''</sup>.<ref>Munkres, James R. (2000) ''Topology''. Prentice Hall. {{ISBN|0-13-181629-2}}.</ref> It is often not possible to provide one consistent coordinate system for an entire space. In this case, a collection of coordinate maps are put together to form an [[atlas (topology)|atlas]] covering the space. A space equipped with such an atlas is called a ''manifold'' and additional structure can be defined on a manifold if the structure is consistent where the coordinate maps overlap. For example, a [[differentiable manifold]] is a manifold where the change of coordinates from one coordinate map to another is always a differentiable function.

==Orientation-based coordinates==
In [[geometry]] and [[kinematics]], coordinate systems are used to describe the (linear) position of points and the [[orientation (geometry)|angular position]] of axes, planes, and [[rigid body|rigid bodies]].<ref>{{cite book |title=Analytical Mechanics of Space Systems |author1-link=Hanspeter Schaub |author2-link=John Junkins |author1=Hanspeter Schaub |author2=John L. Junkins |chapter=Rigid body kinematics |page=71 |chapter-url=https://books.google.com/books?id=qXvESNWrfpUC&pg=PA71 |isbn=1-56347-563-4 |year=2003 |publisher=American Institute of Aeronautics and Astronautics}}</ref> In the latter case, the orientation of a second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system). For instance, the orientation of a rigid body can be represented by an orientation [[Matrix (mathematics)|matrix]], which includes, in its three columns, the [[Cartesian coordinates]] of three points. These points are used to define the orientation of the axes of the local system; they are the tips of three [[unit vector]]s aligned with those axes.

==Geographic systems==
{{main | Spatial reference system}}

The Earth as a whole is one of the most common geometric spaces requiring the precise measurement of location, and thus coordinate systems. Starting with the Greeks of the [[Hellenistic period]], a variety of coordinate systems have been developed based on the types above, including:
* [[Geographic coordinate system]], the [[Spherical coordinate system|spherical coordinates]] of [[latitude]] and [[longitude]]
* [[Grid reference system|Projected coordinate systems]], including thousands of [[cartesian coordinate system]]s, each based on a [[map projection]] to create a planar surface of the world or a region.
* [[Geocentric coordinate system]], a three-dimensional [[cartesian coordinate system]] that models the earth as an object, and are most commonly used for modeling the orbits of [[satellite]]s, including the [[Global Positioning System]] and other [[satellite navigation]] systems.

==See also==
{{cols|colwidth=21em}}
* [[Absolute angular momentum]]
* [[Alphanumeric grid]]
* [[Axes conventions]] in engineering
* [[Celestial coordinate system]]
* [[Coordinate frame]]
* [[Coordinate-free]]
* [[Fractional coordinates]]
* [[Frame of reference]]
* [[Galilean transformation]]
* [[Grid reference]]
* [[Nomogram]], graphical representations of different coordinate systems
* [[Reference system]]
* [[Rotation of axes]]
* [[Translation of axes]]
{{colend}}

===Relativistic coordinate systems===
{{cols}}
* [[Eddington–Finkelstein coordinates]]
* [[Gaussian polar coordinates]]
* [[Gullstrand–Painlevé coordinates]]
* [[Isotropic coordinates]]
* [[Kruskal–Szekeres coordinates]]
* [[Schwarzschild coordinates]]
{{colend}}

==References==
===Citations===
{{Reflist}}

===Sources===
{{Refbegin}}

* {{SpringerEOM
 |title= Coordinates
 |id= C/c026470
 |last= Voitsekhovskii
 |first=M.I.
 |last2=Ivanov
 |first2=A.B.
}}

* {{cite book
 |title= Higher Geometry
 |first=Frederick S.
 |last=Woods
 |publisher=Ginn and Co.
 |year=1922
 |pages= 1ff
 |url= https://books.google.com/books?id=3ZULAAAAYAAJ&pg=PA1
}}

* {{cite book
 |title= Geometry of Differential Forms
 |last1=Shigeyuki Morita
 |last2=Teruko Nagase
 |last3=Katsumi Nomizu
 |author-link3=Katsumi Nomizu
 |page=12
 |url= https://books.google.com/books?id=5N33Of2RzjsC&pg=PA12
 |isbn=0-8218-1045-6
 |year=2001
 |publisher=AMS Bookstore
}}
{{Refend}}

==External links==
{{Wiktionary|coordinate system|coordinate}}
{{Commons category|Coordinate systems}}
* [http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/AV0405/MARTIN/Hex.pdf Hexagonal Coordinate Systems]

{{Tensors}}
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[[Category:Coordinate systems| ]]
[[Category:Analytic geometry]]