Editing Cartesian coordinate system (section)

==Notations and conventions==
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The Cartesian coordinates of a point are usually written in [[parentheses]] and separated by commas, as in {{nowrap|(10, 5)}} or {{nowrap|(3, 5, 7)}}. The origin is often labelled with the capital letter ''O''. In analytic geometry, unknown or generic coordinates are often denoted by the letters (''x'', ''y'') in the plane, and (''x'', ''y'', ''z'') in three-dimensional space. This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as the coordinates of points in many geometric problems), and letters near the beginning for given quantities.

These conventional names are often used in other domains, such as physics and engineering, although other letters may be used. For example, in a graph showing how a [[pressure]] varies with [[time]], the graph coordinates may be denoted ''p'' and ''t''. Each axis is usually named after the coordinate which is measured along it; so one says the ''x-axis'', the ''y-axis'', the ''t-axis'', etc.

Another common convention for coordinate naming is to use subscripts, as (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) for the ''n'' coordinates in an ''n''-dimensional space, especially when ''n'' is greater than 3 or unspecified. Some authors prefer the numbering (''x''<sub>0</sub>, ''x''<sub>1</sub>, ..., ''x''<sub>''n''−1</sub>). These notations are especially advantageous in [[computer programming]]: by storing the coordinates of a point as an [[Array data type|array]], instead of a [[record (computer science)|record]], the [[subscript]] can serve to index the coordinates.

In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called the [[abscissa]]) is measured along a [[horizontal plane|horizontal]] axis, oriented from left to right. The second coordinate (the [[ordinate]]) is then measured along a [[vertical direction|vertical]] axis, usually oriented from bottom to top. Young children learning the Cartesian system, commonly learn the order to read the values before cementing the ''x''-, ''y''-, and ''z''-axis concepts, by starting with 2D mnemonics (for example, 'Walk along the hall then up the stairs' akin to straight across the ''x''-axis then up vertically along the ''y''-axis).

Computer graphics and [[image processing]], however, often use a coordinate system with the ''y''-axis oriented downwards on the computer display. This convention developed in the 1960s (or earlier) from the way that images were originally stored in [[framebuffer|display buffers]].

For three-dimensional systems, a convention is to portray the ''xy''-plane horizontally, with the ''z''-axis added to represent height (positive up). Furthermore, there is a convention to orient the ''x''-axis toward the viewer, biased either to the right or left. If a diagram ([[3D projection]] or [[Perspective (graphical)|2D perspective drawing]]) shows the ''x''- and ''y''-axis horizontally and vertically, respectively, then the ''z''-axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the ''z''-axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera [[Perspective (graphical)|perspective]]. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the [[right-hand rule]], unless specifically stated otherwise. All laws of physics and math assume this [[#Orientation and handedness|right-handedness]], which ensures consistency.

For 3D diagrams, the names "abscissa" and "ordinate" are rarely used for ''x'' and ''y'', respectively. When they are, the ''z''-coordinate is sometimes called the '''applicate'''. The words ''abscissa'', ''ordinate'' and ''applicate'' are sometimes used to refer to coordinate axes rather than the coordinate values.<ref name=":0">{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Cartesian_orthogonal_coordinate_system|title=Cartesian orthogonal coordinate system|website=Encyclopedia of Mathematics|language=en|access-date=2017-08-06}}</ref>

===Quadrants and octants===
{{Main|Octant (solid geometry)|Quadrant (plane geometry)}}
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[[File:Cartesian coordinates 2D.svg|thumb|The four quadrants of a Cartesian coordinate system]]

The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called ''quadrants'',<ref name=":0" /> each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by [[Roman numeral]]s: I (where the coordinates both have positive signs), II (where the abscissa is negative − and the ordinate is positive +), III (where both the abscissa and the ordinate are −), and IV (abscissa +, ordinate −). When the axes are drawn according to the mathematical custom, the numbering goes [[clockwise|counter-clockwise]] starting from the upper right ("north-east") quadrant.

Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or '''octants''',<ref name=":0" /> according to the signs of the coordinates of the points. The convention used for naming a specific octant is to list its signs; for example, {{nowrap|(+ + +)}} or {{nowrap|(− + −)}}. The generalization of the quadrant and octant to an arbitrary number of dimensions is the '''[[orthant]]''', and a similar naming system applies.