Cartesian coordinate system: Difference between revisions
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{{Short description| | {{Short description|Coordinate system using perpendicular axes}} | ||
{{Use dmy dates|date=December 2022}} | {{Use dmy dates|date=December 2022}} | ||
[[File:Cartesian-coordinate-system.svg|thumb|Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: {{nowrap|(2, 3)}} in green, {{nowrap|(−3, 1)}} in red, {{nowrap|(−1.5, −2.5)}} in blue, and the origin {{nowrap|(0, 0)}} in purple.]] | [[File:Cartesian-coordinate-system.svg|thumb|Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: {{nowrap|(2, 3)}} in green, {{nowrap|(−3, 1)}} in red, {{nowrap|(−1.5, −2.5)}} in blue, and the origin {{nowrap|(0, 0)}} in purple.]] | ||
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[[File:Cartesian-coordinate-system-with-circle.svg|thumb|Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is {{nowrap|1=(''x'' − ''a'')<sup>2</sup> + (''y'' − ''b'')<sup>2</sup> = ''r''<sup>2</sup>}} where ''a'' and ''b'' are the coordinates of the center {{nowrap|(''a'', ''b'')}} and ''r'' is the radius.]] | [[File:Cartesian-coordinate-system-with-circle.svg|thumb|Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is {{nowrap|1=(''x'' − ''a'')<sup>2</sup> + (''y'' − ''b'')<sup>2</sup> = ''r''<sup>2</sup>}} where ''a'' and ''b'' are the coordinates of the center {{nowrap|(''a'', ''b'')}} and ''r'' is the radius.]] | ||
Cartesian coordinates are named for [[René Descartes]], whose invention | Cartesian coordinates are named for [[René Descartes]], whose invention thereof in the 17th century revolutionized mathematics by allowing the expression of problems of geometry in terms of [[algebra]] and [[calculus]]. Using the Cartesian coordinate system, geometric shapes (such as [[curve]]s) can be described by [[equation]]s involving the coordinates of points of the shape. For example, a [[circle]] of radius 2, centered at the origin of the plane, may be described as the [[set (mathematics)|set]] of all points whose coordinates {{math|''x''}} and {{math|''y''}} satisfy the equation {{math|1=''x''<sup>2</sup> + ''y''<sup>2</sup> = 4}}; the [[area]], the [[perimeter]] and the [[tangent line]] at any point can be computed from this equation by using [[integral]]s and [[derivative]]s, in a way that can be applied to any curve. | ||
Cartesian coordinates are the foundation of [[analytic geometry]], and provide enlightening geometric interpretations for many other branches of mathematics, such as [[linear algebra]], [[complex analysis]], [[differential geometry]], multivariate [[calculus]], [[group theory]] and more. A familiar example is the concept of the [[graph of a function]]. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including [[astronomy]], [[physics]], [[engineering]] and many more. They are the most common coordinate system used in [[computer graphics]], [[computer-aided geometric design]] and other [[computational geometry|geometry-related data processing]]. | Cartesian coordinates are the foundation of [[analytic geometry]], and provide enlightening geometric interpretations for many other branches of mathematics, such as [[linear algebra]], [[complex analysis]], [[differential geometry]], multivariate [[calculus]], [[group theory]] and more. A familiar example is the concept of the [[graph of a function]]. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including [[astronomy]], [[physics]], [[engineering]] and many more. They are the most common coordinate system used in [[computer graphics]], [[computer-aided geometric design]] and other [[computational geometry|geometry-related data processing]]. | ||
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The adjective ''Cartesian'' refers to the French [[mathematician]] and [[philosopher]] [[René Descartes]], who published this idea in 1637 while he was resident in the Netherlands. It was independently discovered by [[Pierre de Fermat]], who also worked in three dimensions, although Fermat did not publish the discovery.<ref>{{Cite web|url=https://www.britannica.com/topic/analytic-geometry|title=Analytic geometry|last1=Bix|first1=Robert A.|last2=D'Souza|first2=Harry J.|website=Encyclopædia Britannica|access-date=2017-08-06}}</ref> The French cleric [[Nicole Oresme#Mathematics|Nicole Oresme]] used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat.<ref>{{harvnb|Kent|Vujakovic|2017|loc=See [https://books.google.com/books?id=EVRSDwAAQBAJ&q=Nicole+Oresme+coordinate&pg=PT307 here]}}</ref> | The adjective ''Cartesian'' refers to the French [[mathematician]] and [[philosopher]] [[René Descartes]], who published this idea in 1637 while he was resident in the Netherlands. It was independently discovered by [[Pierre de Fermat]], who also worked in three dimensions, although Fermat did not publish the discovery.<ref>{{Cite web|url=https://www.britannica.com/topic/analytic-geometry|title=Analytic geometry|last1=Bix|first1=Robert A.|last2=D'Souza|first2=Harry J.|website=Encyclopædia Britannica|access-date=2017-08-06}}</ref> The French cleric [[Nicole Oresme#Mathematics|Nicole Oresme]] used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat.<ref>{{harvnb|Kent|Vujakovic|2017|loc=See [https://books.google.com/books?id=EVRSDwAAQBAJ&q=Nicole+Oresme+coordinate&pg=PT307 here]}}</ref> | ||
Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis.<ref>{{Cite book |last=Katz |first=Victor J. |url=https://www.worldcat.org/title/71006826 |title=A history of mathematics: an introduction |date=2009 |publisher=Addison-Wesley |isbn=978-0-321-38700-4 |edition=3rd |location=Boston |pages=484 |oclc=71006826}}</ref> The concept of using a pair of axes was introduced later, after Descartes' ''[[La Géométrie]]'' was translated into Latin in 1649 by [[Frans van Schooten]] and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes's work.<ref>{{harvnb|Burton|2011|loc=p. 374}}.</ref> | Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis.<ref>{{Cite book |last=Katz |first=Victor J. |author-link=Victor J. Katz |url=https://www.worldcat.org/title/71006826 |title=A history of mathematics: an introduction |date=2009 |publisher=Addison-Wesley |isbn=978-0-321-38700-4 |edition=3rd |location=Boston |pages=484 |oclc=71006826}}</ref> The concept of using a pair of axes was introduced later, after Descartes' ''[[La Géométrie]]'' was translated into Latin in 1649 by [[Frans van Schooten]] and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes's work.<ref>{{harvnb|Burton|2011|loc=p. 374}}.</ref> | ||
The development of the Cartesian coordinate system | The development of the Cartesian coordinate system played a fundamental role in the development of the [[calculus]] by [[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]].<ref>{{harvnb|Berlinski|2011}}</ref> The two-coordinate description of the plane was later generalized into the concept of [[vector spaces]].<ref>{{harvnb|Axler|2015|p=1}}</ref> | ||
Many other coordinate systems have been developed since Descartes, such as the [[Polar coordinate system|polar coordinates]] for the plane, and the [[Spherical coordinate system|spherical]] and [[Cylindrical coordinate system|cylindrical coordinates]] for three-dimensional space. | Many other coordinate systems have been developed since Descartes, such as the [[Polar coordinate system|polar coordinates]] for the plane, and the [[Spherical coordinate system|spherical]] and [[Cylindrical coordinate system|cylindrical coordinates]] for three-dimensional space. | ||
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{{Further|Two-dimensional space}} | {{Further|Two-dimensional space}} | ||
A Cartesian coordinate system in two dimensions (also called a '''rectangular coordinate system''' or | A Cartesian coordinate system in two dimensions (also called a '''rectangular coordinate system''' or a '''Cartesian 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)}}. | 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)}}. | ||
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{{Anchor|Cartesian coordinates in three dimensions}} | {{Anchor|Cartesian coordinates in three dimensions}} | ||
===Three dimensions=== | ===Three dimensions=== | ||
{{Further|Three-dimensional space}} | {{Further|Three-dimensional space}} | ||
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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. | 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 | 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]]. | 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]]. | ||
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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 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">{{ | 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">{{SpringerEOM|title=Cartesian orthogonal coordinate system|oldid=31381|first=A. B.|last=Ivanov}}</ref> | ||
===Quadrants and octants=== | ===Quadrants and octants=== | ||
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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. | 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 '' | Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or ''[[Octant (solid geometry)|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. | ||
==Cartesian formulae for the plane== | ==Cartesian formulae for the plane== | ||
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[[File:Right hand cartesian.svg|thumb|Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes]] | [[File:Right hand cartesian.svg|thumb|Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes]] | ||
Once the ''x''- and ''y''-axes are specified, they determine the [[line (geometry)|line]] along which the ''z''-axis should lie, but there are two possible orientations for this line. The two possible coordinate systems | Once the ''x''- and ''y''-axes are specified, they determine the [[line (geometry)|line]] along which the ''z''-axis should lie, but there are two possible orientations for this line. The two possible coordinate systems which result are called 'right-handed' and 'left-handed'.<ref>{{harvnb|Anton|Bivens|Davis|2021|p=[https://books.google.com/books?id=001EEAAAQBAJ&pg=PA657 657]}}</ref> The standard orientation, where the ''xy''-plane is horizontal and the ''z''-axis points up (and the ''x''- and the ''y''-axis form a positively oriented two-dimensional coordinate system in the ''xy''-plane if observed from ''above'' the ''xy''-plane) is called '''right-handed''' or '''positive'''. | ||
[[File:3D Cartesian Coodinate Handedness.jpg|thumb|3D Cartesian coordinate handedness]] | [[File:3D Cartesian Coodinate Handedness.jpg|thumb|3D Cartesian coordinate handedness]] | ||
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* [[Horizontal and vertical]] | * [[Horizontal and vertical]] | ||
* [[Jones diagram]], which plots four variables rather than two | * [[Jones diagram]], which plots four variables rather than two | ||
* [[Orthogonal coordinates]] | * [[Orthogonal coordinates]], a kind of curvilinear coordinate system | ||
* [[Polar coordinate system]] | * [[Polar coordinate system]] | ||
* [[Regular grid]] | * [[Regular grid]] | ||
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| last1 = Brannan | first1 = David A. | | last1 = Brannan | first1 = David A. | ||
| last2 = Esplen | first2 = Matthew F. | | last2 = Esplen | first2 = Matthew F. | ||
| last3 = Gray | first3 = Jeremy J. | | last3 = Gray | first3 = Jeremy J. |author-link3=Jeremy Gray (mathematician) | ||
| title= Geometry | | title= Geometry | ||
| publisher = Cambridge University Press | | publisher = Cambridge University Press | ||