Editing Coordinate system (section)

==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}}
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 | image1 = Cartesian-coordinate-system.svg
 | caption1 = The [[Cartesian coordinate system]] in the plane
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 | image2 = Rectangular coordinates.svg
 | caption2 = The Cartesian coordinate system in three-dimensional space
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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.