Editing Cartesian coordinate system (section)

==Description==
{{Anchor|Cartesian coordinates in one dimension}}
===One dimension===
{{Main|Number line}}

An [[affine line]] with a chosen Cartesian coordinate system is called a ''number line''. Every point on the line has a real-number coordinate, and every real number represents some point on the line.

There are two [[degree of freedom|degrees of freedom]] in the choice of Cartesian coordinate system for a line, which can be specified by choosing two distinct points along the line and assigning them to two distinct [[real number]]s (most commonly zero and one). Other points can then be uniquely assigned to numbers by [[linear interpolation]]. Equivalently, one point can be assigned to a specific real number, for instance an ''origin'' point corresponding to zero, and an [[Curve orientation|oriented]] length along the line can be chosen as a unit, with the orientation indicating the correspondence between directions along the line and positive or negative numbers.{{efn|Consider the two [[ray (geometry)|rays]] or half-lines resulting from splitting the line at the origin. One of the half-lines can be assigned to positive numbers, and the other half-line to negative numbers.}} Each point corresponds to its signed distance from the origin (a number with an absolute value equal to the distance and a {{math|+}} or {{math|−}} sign chosen based on direction).

A [[geometric transformation]] of the line can be represented by a [[function of a real variable]], for example [[translation (geometry)|translation]] of the line corresponds to addition, and [[scaling (geometry)|scaling]] the line corresponds to multiplication. Any two Cartesian coordinate systems on the line can be related to each-other by a [[linear function]] (function of the form {{nobr|<math>x \mapsto ax + b</math>)}} taking a specific point's coordinate in one system to its coordinate in the other system. Choosing a coordinate system for each of two different lines establishes an [[Affine transformation|affine map]] from one line to the other taking each point on one line to the point on the other line with the same coordinate.

{{Anchor|Cartesian coordinates in two dimensions}}

===Two dimensions===
{{Further|Two-dimensional space}}

A Cartesian coordinate system in two dimensions (also called a '''rectangular coordinate system''' or an '''orthogonal coordinate system'''<ref name=":0" />) is defined by an [[ordered pair]] of [[perpendicular]] lines (axes), a single [[unit of length]] for both axes, and an orientation for each axis. The point where the axes meet is taken as the origin for both, thus turning each axis into a [[number line]]. For any point ''P'', a line is drawn through ''P'' perpendicular to each axis, and the position where it meets the axis is interpreted as a number. The two numbers, in that chosen order, are the ''Cartesian coordinates'' of ''P''. The reverse construction allows one to determine the point ''P'' given its coordinates.

The first and second coordinates are called the ''[[abscissa]]'' and the ''[[ordinate]]'' of ''P'', respectively; and the point where the axes meet is called the ''origin'' of the coordinate system. The coordinates are usually written as two numbers in parentheses, in that order, separated by a comma, as in {{nowrap|(3, −10.5)}}. Thus the origin has coordinates {{nowrap|(0, 0)}}, and the points on the positive half-axes, one unit away from the origin, have coordinates {{nowrap|(1, 0)}} and {{nowrap|(0, 1)}}.

In mathematics, physics, and engineering, the first axis is usually defined or depicted as horizontal and oriented to the right, and the second axis is vertical and oriented upwards. (However, in some [[computer graphics]] contexts, the ordinate axis may be oriented downwards.) The origin is often labeled ''O'', and the two coordinates are often denoted by the letters ''X'' and ''Y'', or ''x'' and ''y''. The axes may then be referred to as the ''X''-axis and ''Y''-axis. The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values.

A [[Euclidean plane]] with a chosen Cartesian coordinate system is called a '''{{vanchor|Cartesian plane}}'''. In a Cartesian plane, one can define canonical representatives of certain geometric figures, such as the [[unit circle]] (with radius equal to the length unit, and center at the origin), the [[unit square]] (whose diagonal has endpoints at {{nowrap|(0, 0)}} and {{nowrap|(1, 1)}}), the [[unit hyperbola]], and so on.

The two axes divide the plane into four [[right angle]]s, called ''quadrants''. The quadrants may be named or numbered in various ways, but the quadrant where all coordinates are positive is usually called the ''first quadrant''.

If the coordinates of a point are {{nowrap|(''x'', ''y'')}}, then its [[distance from a point to a line|distances]] from the ''X''-axis and from the ''Y''-axis are {{abs|''y''}} and {{abs|''x''}}, respectively; where {{abs}} denotes the [[absolute value (algebra)|absolute value]] of a number.

{{Anchor|Cartesian coordinates in three dimensions}}
===Three dimensions===
{{Further|Three-dimensional space}}
[[File:Coord system CA 0.svg|thumb|A three dimensional Cartesian coordinate system, with origin ''O'' and axis lines ''X'', ''Y'' and ''Z'', oriented as shown by the arrows. The tick marks on the axes are one length unit apart. The black dot shows the point with coordinates {{math|1=''x'' = 2}}, {{math|1=''y'' = 3}}, and {{math|1=''z'' = 4}}, or {{math|(2, 3, 4)}}.]]

A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines (the ''axes'') that go through a common point (the ''origin''), and are pair-wise perpendicular; an orientation for each axis; and a single unit of length for all three axes. As in the two-dimensional case, each axis becomes a number line. For any point ''P'' of space, one considers a plane through ''P'' perpendicular to each coordinate axis, and interprets the point where that plane cuts the axis as a number. The Cartesian coordinates of ''P'' are those three numbers, in the chosen order. The reverse construction determines the point ''P'' given its three coordinates.

Alternatively, each coordinate of a point ''P'' can be taken as the distance from ''P'' to the plane defined by the other two axes, with the sign determined by the orientation of the corresponding axis.

Each pair of axes defines a ''coordinate plane''. These planes divide space into eight ''[[octant (solid geometry)|octants]]''. The octants are:

<math display=block>
\begin{align}
(+x,+y,+z) && (-x,+y,+z) && (+x,-y,+z) && (+x,+y,-z) \\
(+x,-y,-z) && (-x,+y,-z) && (-x,-y,+z) && (-x,-y,-z)
\end{align}
</math>

The coordinates are usually written as three numbers (or algebraic formulas) surrounded by parentheses and separated by commas, as in {{math|(3, −2.5, 1)}} or {{math|(''t'', ''u'' + ''v'', ''π''/2)}}. Thus, the origin has coordinates {{math|(0, 0, 0)}}, and the unit points on the three axes are {{math|(1, 0, 0)}}, {{math|(0, 1, 0)}}, and {{math|(0, 0, 1)}}.

Standard names for the coordinates in the three axes are ''abscissa'', ''ordinate'' and ''applicate''.<ref>{{Cite web |title=Cartesian coordinates |url=https://planetmath.org/cartesiancoordinates |access-date=2024-08-25 |website=planetmath.org}}</ref> The coordinates are often denoted by the letters ''x'', ''y'', and ''z''. The axes may then be referred to as the ''x''-axis, ''y''-axis, and ''z''-axis, respectively. Then the coordinate planes can be referred to as the ''xy''-plane, ''yz''-plane, and ''xz''-plane.

In mathematics, physics, and engineering contexts, the first two axes are often defined or depicted as horizontal, with the third axis pointing up. In that case the third coordinate may be called ''height'' or ''altitude''. The orientation is usually chosen so that the 90-degree angle from the first axis to the second axis looks counter-clockwise when seen from the point {{math|(0, 0, 1)}}; a convention that is commonly called ''the [[right-hand rule]]''.

[[File:Cartesian coordinate surfaces.png|thumb| The [[Coordinate system#Coordinate surface|coordinate surfaces]] of the Cartesian coordinates {{math|(''x'', ''y'', ''z'')}}. The ''z''-axis is vertical and the ''x''-axis is highlighted in green. Thus, the red plane shows the points with {{math|1=''x'' = 1}}, the blue plane shows the points with {{math|1=''z'' = 1}}, and the yellow plane shows the points with {{math|1=''y'' = −1}}. The three surfaces intersect at the point ''P'' (shown as a black sphere) with the Cartesian coordinates {{math|(1, −1, 1}}).]]

===Higher dimensions===
Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of [[real number]]s; that is, with the [[Cartesian product]] <math>\R^2 = \R\times\R</math>, where <math>\R</math> is the set of all real numbers. In the same way, the points in any [[Euclidean space]] of dimension ''n'' be identified with the [[tuple]]s (lists) of ''n'' real numbers; that is, with the Cartesian product <math>\R^n</math>.

===Generalizations===
The concept of Cartesian coordinates generalizes to allow axes that are not perpendicular to each other, and/or different units along each axis. In that case, each coordinate is obtained by projecting the point onto one axis along a direction that is parallel to the other axis (or, in general, to the [[hyperplane]] defined by all the other axes). In such an ''[[oblique coordinate system]]'' the computations of distances and angles must be modified from that in standard Cartesian systems, and many standard formulas (such as the Pythagorean formula for the distance) do not hold (see [[affine plane]]).