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A Cartesian coordinate system (, ) in a
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
is a
coordinate system In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

coordinate system
that specifies each
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
uniquely by a pair of
numerical
numerical
coordinates, which are the
signed Signing may refer to: * Using sign language * Signature, placing one's name on a document * Signature (disambiguation) * Manual communication, signing as a form of communication using the hands in place of the voice * Digital signature, signing as ...
distances to the point from two fixed
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

perpendicular
oriented lines, measured in the same
unit of length A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are the metric system, metric units, used in every country globally. In the United States the U.S. c ...
. Each reference line is called a ''coordinate axis'' or just ''axis'' (plural ''axes'') of the system, and the point where they meet is its ''
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * , a Wolverine comic book mini-series published by Marvel Comics in 2002 * , a 1999 ''Buffy the Vampire Slayer'' comic book series * , a major ''Judge Dred ...
'', at ordered pair . The coordinates can also be defined as the positions of the
perpendicular projections
perpendicular projections
of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in
three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the ...
by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, ''n'' Cartesian coordinates (an element of real ''n''-space) specify the point in an ''n''-dimensional
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
for any
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

dimension
''n''. These coordinates are equal, up to
sign A sign is an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an object which does not exist at ...
, to distances from the point to ''n'' mutually perpendicular
hyperplane In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
s. The invention of Cartesian coordinates in the 17th century by
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, Mathematics, mathematician, and scientist who invented analytic geometry, linking the previously sep ...

René Descartes
(
Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s and 1930s to replace traditional writing sy ...
name: ''Cartesius'') revolutionized mathematics by providing the first systematic link between
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a sma ...
and
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
. Using the Cartesian coordinate system, geometric shapes (such as
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
s) can be described by Cartesian equations: algebraic
equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

equation
s involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates ''x'' and ''y'' satisfy the equation . Cartesian coordinates are the foundation of
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
, and provide enlightening geometric interpretations for many other branches of mathematics, such as
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
,
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
,
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
, multivariate
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
,
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
and more. A familiar example is the concept of the
graph of a function In mathematics, the graph of a Function (mathematics), function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space ...

graph of a function
. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses mathematics, phys ...
,
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

physics
,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

engineering
and many more. They are the most common coordinate system used in
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great dea ...

computer graphics
,
computer-aided geometric design Computer-aided or computer-assisted is an adjectival phrase that hints of the use of a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern computers can per ...
and other geometry-related data processing.


History

The adjective ''Cartesian'' refers to the French
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

mathematician
and
philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mi ...

philosopher
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, Mathematics, mathematician, and scientist who invented analytic geometry, linking the previously sep ...

René Descartes
, who published this idea in 1637. It was independently discovered by
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Republic (fren ...

Pierre de Fermat
, who also worked in three dimensions, although Fermat did not publish the discovery. The French cleric
Nicole Oresme Nicole Oresme (; c. 1320–1325 – July 11, 1382), also known as Nicolas Oresme, Nicholas Oresme, or Nicolas d'Oresme, was a significant philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the g ...
used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat. Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' ''
La Géométrie ''La Géométrie'' was published Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for free. Traditionally, the term refers to the distribution of printed works ...
'' was translated into Latin in 1649 by
Frans van Schooten 200px, Frans van Schooten Franciscus van Schooten (1615, Leiden Leiden (, ; in English language, English and Archaism, archaic Dutch language, Dutch also ''Leyden'') is a List of cities in the Netherlands by province, city and List of munici ...
and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work. The development of the Cartesian coordinate system would play a fundamental role in the development of the
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

Isaac Newton
and
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, ', "having learned much"; Latin Latin (, or , ...

Gottfried Wilhelm Leibniz
. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Many other coordinate systems have been developed since Descartes, such as the
polar coordinates In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

polar coordinates
for the plane, and the
spherical of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values ...

spherical
and
cylindrical coordinates 240px, A cylindrical coordinate system with origin , polar axis , and longitudinal axis . The dot is the point with radial distance , angular coordinate , and height . A cylindrical coordinate system is a three-dimensional coordinate system that s ...

cylindrical coordinates
for three-dimensional space.


Description


One dimension

Choosing a Cartesian coordinate system for a one-dimensional space—that is, for a straight line—involves choosing a point ''O'' of the line (the origin), a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by ''O'' is the positive and which is negative; we then say that the line "is oriented" (or "points") from the negative half towards the positive half. Then each point ''P'' of the line can be specified by its distance from ''O'', taken with a + or − sign depending on which half-line contains ''P''. A line with a chosen Cartesian system is called a number line. Every real number has a unique location on the line. Conversely, every point on the line can be interpreted as a
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

number
in an ordered continuum such as the real numbers.


Two dimensions

A Cartesian coordinate system in two dimensions (also called a rectangular coordinate system or an orthogonal coordinate system) is defined by an
ordered pair In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

ordered pair
of
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

perpendicular
lines (axes), a single
unit of length A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are the metric system, metric units, used in every country globally. In the United States the U.S. c ...
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 In common usage, the abscissa refers to the horizontal (''x'') axis and the ordinate refers to the vertical (''y'') axis of a standard two-dimensional graph. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of suc ...

abscissa
'' and the ''
ordinate In common usage, the abscissa refers to the horizontal (''x'') axis and the ordinate refers to the vertical (''y'') axis of a standard two-dimensional graph. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of suc ...
'' 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 . Thus the origin has coordinates , and the points on the positive half-axes, one unit away from the origin, have coordinates and . 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 Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great dea ...

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
with a chosen Cartesian coordinate system is called a . In a Cartesian plane one can define canonical representatives of certain geometric figures, such as the
unit circle measure. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, a ...

unit circle
(with radius equal to the length unit, and center at the origin), the
unit square In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
(whose diagonal has endpoints at and ), the
unit hyperbola In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
, and so on. The two axes divide the plane into four
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

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 , then its from the ''X''-axis and from the ''Y''-axis are , ''y'', and , ''x'', , respectively; where , ..., denotes the
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...
of a number.


Three dimensions

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 hyperplane through ''P'' perpendicular to each coordinate axis, and interprets the point where that hyperplane 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 hyperplane defined by the other two axes, with the sign determined by the orientation of the corresponding axis. Each pair of axes defines a ''coordinate hyperplane''. These hyperplanes divide space into eight trihedra, called ''octants''. The octants are: , (+x,+y,+z) , (-x,+y,+z) , (+x,+y,-z) , (-x,+y,-z) , (+x,-y,+z) , (-x,-y,+z) , (+x,-y,-z) , (-x,-y,-z) , The coordinates are usually written as three numbers (or algebraic formulas) surrounded by parentheses and separated by commas, as in or . Thus, the origin has coordinates , and the unit points on the three axes are , , and . There are no standard names for the coordinates in the three axes (however, the terms ''abscissa'', ''ordinate'' and ''applicate'' are sometimes used). The coordinates are often denoted by the letters ''X'', ''Y'', and ''Z'', or ''x'', ''y'', and ''z''. The axes may then be referred to as the ''X''-axis, ''Y''-axis, and ''Z''-axis, respectively. Then the coordinate hyperplanes 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 ; a convention that is commonly called ''the
right hand rule In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

right hand rule
''.


Higher dimensions

Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s; that is with the
Cartesian product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
\R^2 = \R\times\R, where \R is the set of all real numbers. In the same way, the points in any
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
of dimension ''n'' be identified with the
tuple In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s (lists) of ''n'' real numbers, that is, with the Cartesian product \R^n.


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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
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 planeIn geometry, an affine plane is a two-dimensional affine space. Examples Typical examples of affine planes are *Euclidean planes, which are affine planes over the real number, reals, equipped with a metric (mathematics), metric, the Euclidean distan ...
).


Notations and conventions

The Cartesian coordinates of a point are usually written in
parentheses A bracket is either of two tall fore- or back-facing punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding ...
and separated by commas, as in or . 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 Pressure (symbol: ''p'' or ''P'') is the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

pressure
varies with
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

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''1, ''x''2, ..., ''x''''n'') for the ''n'' coordinates in an ''n''-dimensional space, especially when ''n'' is greater than 3 or unspecified. Some authors prefer the numbering (''x''0, ''x''1, ..., ''x''''n''−1). These notations are especially advantageous in
computer programming Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, generating algorithms, Profilin ...
: by storing the coordinates of a point as an
array ARRAY, also known as ARRAY Now, is an independent distribution company launched by film maker and former publicist Ava DuVernay Ava Marie DuVernay (; born August 24, 1972) is an American filmmaker. She won the directing award in the U.S. dram ...
, instead of a record, the
subscript A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the , while supersc ...

subscript
can serve to index the coordinates. In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called the
abscissa In common usage, the abscissa refers to the horizontal (''x'') axis and the ordinate refers to the vertical (''y'') axis of a standard two-dimensional graph. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of suc ...

abscissa
) is measured along a horizontal axis, oriented from left to right. The second coordinate (the
ordinate In common usage, the abscissa refers to the horizontal (''x'') axis and the ordinate refers to the vertical (''y'') axis of a standard two-dimensional graph. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of suc ...
) is then measured along a
vertical Vertical may refer to: * Vertical direction In astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects ...
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 (e.g. '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 Digital image processing is the use of a digital computer A computer is a machine A machine is a man-made device that uses power to apply forces and control movement to perform an action. Machines can be driven by animals and people ...
, 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 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 A 3D projection (or graphical projection) is a used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspect analysis to a complex object for viewing capability on a ...
or 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. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

right-hand rule
, unless specifically stated otherwise. All laws of physics and math assume this
right-handedness In human biology, handedness is the better, faster, or more precise performance or individual preference for use of a hand, known as the dominant hand. The incapable, less capable or less preferred hand is called the non-dominant hand. Right-ha ...
, 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.


Quadrants and octants

The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by
Roman numeral Roman numerals are a that originated in and remained the usual way of writing numbers throughout Europe well into the . Numbers in this system are represented by combinations of letters from the . Modern style uses seven symbols, each with a ...
s: I (where the signs of the two coordinates are I (+,+), II (−,+), III (−,−), and IV (+,−). When the axes are drawn according to the mathematical custom, the numbering goes starting from the upper right ("north-east") quadrant. Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or octants, according to the signs of the coordinates of the points. The convention used for naming a specific octant is to list its signs, e.g. or . The generalization of the quadrant and octant to an arbitrary number of dimensions is the
orthant In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
, and a similar naming system applies.


Cartesian formulae for the plane


Distance between two points

The
Euclidean distance In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
between two points of the plane with Cartesian coordinates (x_1, y_1) and (x_2, y_2) is :d = \sqrt. This is the Cartesian version of Pythagoras's theorem. In three-dimensional space, the distance between points (x_1,y_1,z_1) and (x_2,y_2,z_2) is :d = \sqrt , which can be obtained by two consecutive applications of Pythagoras' theorem.


Euclidean transformations

The Euclidean transformations or Euclidean motions are the (
bijective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
) mappings of points of the
Euclidean plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
to themselves which preserve distances between points. There are four types of these mappings (also called isometries):
translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
,
rotations A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...
, reflections and
glide reflection In 2-dimensional geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with propert ...

glide reflection
s.


Translation

Translating a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers to the Cartesian coordinates of every point in the set. That is, if the original coordinates of a point are , after the translation they will be :(x', y') = (x + a, y + b) .


Rotation

To
rotate A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...
a figure
counterclockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite sen ...

counterclockwise
around the origin by some angle \theta is equivalent to replacing every point with coordinates (''x'',''y'') by the point with coordinates (''x''',''y'''), where :x'=x \cos \theta - y \sin \theta :y'=x \sin \theta + y \cos \theta . Thus: (x',y') = ((x \cos \theta - y \sin \theta\,) , (x \sin \theta + y \cos \theta\,)) .


Reflection

If are the Cartesian coordinates of a point, then are the coordinates of its
reflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal r ...
across the second coordinate axis (the y-axis), as if that line were a mirror. Likewise, are the coordinates of its reflection across the first coordinate axis (the x-axis). In more generality, reflection across a line through the origin making an angle \theta with the x-axis, is equivalent to replacing every point with coordinates by the point with coordinates , where :x'=x \cos 2\theta + y \sin 2\theta :y'=x \sin 2\theta - y \cos 2\theta . Thus: (x',y') = ((x \cos 2\theta + y \sin 2\theta\,) , (x \sin 2\theta - y \cos 2\theta\,)) .


Glide reflection

A glide reflection is the composition of a reflection across a line followed by a translation in the direction of that line. It can be seen that the order of these operations does not matter (the translation can come first, followed by the reflection).


General matrix form of the transformations

These
Euclidean transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s of the plane can all be described in a uniform way by using matrices. The result (x', y') of applying a Euclidean transformation to a point (x,y) is given by the formula :(x',y') = (x,y) A + b where ''A'' is a 2×2 orthogonal
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
and is an arbitrary ordered pair of numbers; that is, :x' = x A_ + y A_ + b_ :y' = x A_ + y A_ + b_, where ::A = \begin A_ & A_ \\ A_ & A_ \end. /nowiki>The row vectors are used for point coordinates, and the matrix is written on the right./nowiki> To be ''orthogonal'', the matrix ''A'' must have
orthogonal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
rows with same Euclidean length of one, that is, :A_ A_ + A_ A_ = 0 and :A_^2 + A_^2 = A_^2 + A_^2 = 1. This is equivalent to saying that ''A'' times its
transpose In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...

transpose
must be the
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

identity matrix
. If these conditions do not hold, the formula describes a more general
affine transformation In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...
of the plane provided that the determinant of ''A'' is not zero. The formula defines a translation if and only if ''A'' is the
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

identity matrix
. The transformation is a rotation around some point if and only if ''A'' is a rotation matrix, meaning that : A_ A_ - A_ A_ = 1 . A reflection or glide reflection is obtained when, : A_ A_ - A_ A_ = -1 . Assuming that translation is not used transformations can be combined by simply multiplying the associated transformation matrices.


Affine transformation

Another way to represent coordinate transformations in Cartesian coordinates is through
affine transformation In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...
s. In affine transformations an extra dimension is added and all points are given a value of 1 for this extra dimension. The advantage of doing this is that point translations can be specified in the final column of matrix ''A''. In this way, all of the euclidean transformations become transactable as matrix point multiplications. The affine transformation is given by: ::\begin A_ & A_ & b_ \\ A_ & A_ & b_ \\ 0 & 0 & 1 \end \begin x \\ y \\ 1 \end = \begin x' \\ y' \\ 1 \end. [Note the matrix ''A'' from above was transposed. The matrix is on the left and column vectors for point coordinates are used.] Using affine transformations multiple different euclidean transformations including translation can be combined by simply multiplying the corresponding matrices.


Scaling

An example of an affine transformation which is not a Euclidean motion is given by scaling. To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number ''m''. If are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates :(x',y') = (m x, m y). If ''m'' is greater than 1, the figure becomes larger; if ''m'' is between 0 and 1, it becomes smaller.


Shearing

A shear mapping, shearing transformation will push the top of a square sideways to form a parallelogram. Horizontal shearing is defined by: :(x',y') = (x+y s, y) Shearing can also be applied vertically: :(x',y') = (x, x s+y)


Orientation and handedness


In two dimensions

Fixing or choosing the ''x''-axis determines the ''y''-axis up to direction. Namely, the ''y''-axis is necessarily the
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

perpendicular
to the ''x''-axis through the point marked 0 on the ''x''-axis. But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative. Each of these two choices determines a different orientation (also called ''handedness'') of the Cartesian plane. The usual way of orienting the plane, with the positive ''x''-axis pointing right and the positive ''y''-axis pointing up (and the ''x''-axis being the "first" and the ''y''-axis the "second" axis), is considered the ''positive'' or ''standard'' orientation, also called the ''right-handed'' orientation. A commonly used mnemonic for defining the positive orientation is the ''
right-hand rule In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

right-hand rule
''. Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the ''x''-axis to the ''y''-axis, in a positively oriented coordinate system. The other way of orienting the plane is following the ''left hand rule'', placing the left hand on the plane with the thumb pointing up. When pointing the thumb away from the origin along an axis towards positive, the curvature of the fingers indicates a positive rotation along that axis. Regardless of the rule used to orient the plane, rotating the coordinate system will preserve the orientation. Switching any two axes will reverse the orientation, but switching both will leave the orientation unchanged.


In three dimensions

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 orientation for this line. The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. 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. The name derives from the
right-hand rule In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

right-hand rule
. If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative orientation of the ''x''-, ''y''-, and ''z''-axes in a ''right-handed'' system. The thumb indicates the ''x''-axis, the index finger the ''y''-axis and the middle finger the ''z''-axis. Conversely, if the same is done with the left hand, a left-handed system results. Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point ''towards'' the observer, whereas the "middle"-axis is meant to point ''away'' from the observer. The red circle is ''parallel'' to the horizontal ''xy''-plane and indicates rotation from the ''x''-axis to the ''y''-axis (in both cases). Hence the red arrow passes ''in front of'' the ''z''-axis. Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a wikt:convex, convex cube and a wikt:concave, concave "corner". This corresponds to the two possible orientations of the space. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the ''x''-axis as pointing ''towards'' the observer and thus seeing a concave corner.


Representing a vector in the standard basis

A point in space in a Cartesian coordinate system may also be represented by a position Euclidean vector, vector, which can be thought of as an arrow pointing from the origin of the coordinate system to the point. If the coordinates represent spatial positions (displacements), it is common to represent the vector from the origin to the point of interest as \mathbf. In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as: : \mathbf = x \mathbf + y \mathbf, where \mathbf = \begin 1 \\ 0 \end and \mathbf = \begin 0 \\ 1 \end are unit vectors in the direction of the ''x''-axis and ''y''-axis respectively, generally referred to as the ''standard basis'' (in some application areas these may also be referred to as versors). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates (x,y,z) can be written as: : \mathbf = x \mathbf + y \mathbf + z \mathbf, where \mathbf = \begin 1 \\ 0 \\ 0 \end, \mathbf = \begin 0 \\ 1 \\ 0 \end, and \mathbf = \begin 0 \\ 0 \\ 1 \end. There is no ''natural'' interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. In a two-dimensional cartesian plane, identify the point with coordinates with the complex number . Here, ''i'' is the imaginary unit and is identified with the point with coordinates , so it is ''not'' the unit vector in the direction of the ''x''-axis. Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three-dimensional cartesian space a similar identification can be made with a subset of the quaternions.


Applications

Cartesian coordinates are an abstraction that have a multitude of possible applications in the real world. However, three constructive steps are involved in superimposing coordinates on a problem application. 1) Units of distance must be decided defining the spatial size represented by the numbers used as coordinates. 2) An origin must be assigned to a specific spatial location or landmark, and 3) the orientation of the axes must be defined using available directional cues for all but one axis. Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (i.e. geospatial 3D). What units make sense? Kilometers are a good choice, since the original definition of the kilometer was geospatial—10 000 km equaling the surface distance from the Equator to the North Pole. Where to place the origin? Based on symmetry, the gravitational center of the Earth suggests a natural landmark (which can be sensed via satellite orbits). Finally, how to orient X-, Y- and Z-axis? The axis of Earth's rotation provides a natural orientation strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the prime meridian stands out as a reference orientation, so the X-axis takes the orientation from geocenter out to 0 degrees longitude, 0 degrees latitude. Note that with three dimensions, and two perpendicular axes orientations pinned down for X and Z, the Y-axis is determined by the first two choices. In order to obey the right-hand rule, the Y-axis must point out from the geocenter to 90 degrees longitude, 0 degrees latitude. So what are the geocentric coordinates of the Empire State Building in New York City? From a longitude of −73.985656 degrees, a latitude 40.748433 degrees, and Earth radius of 40,000/2π km, and transforming from spherical to Cartesian coordinates, you can estimate the geocentric coordinates of the Empire State Building, (''x'', ''y'', ''z'') = (1330.53 km, –4635.75 km, 4155.46 km). GPS navigation relies on such geocentric coordinates. In engineering projects, agreement on the definition of coordinates is a crucial foundation. One cannot assume that coordinates come predefined for a novel application, so knowledge of how to erect a coordinate system where there is none is essential to applying René Descartes' thinking. While spatial applications employ identical units along all axes, in business and scientific applications, each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables. The
graph of a function In mathematics, the graph of a Function (mathematics), function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space ...

graph of a function
or relation (mathematics), relation is the set of all points satisfying that function or relation. For a function of one variable, ''f'', the set of all points , where is the graph of the function ''f''. For a function ''g'' of two variables, the set of all points , where is the graph of the function ''g''. A sketch of the graph of such a function or relation would consist of all the salient parts of the function or relation which would include its relative extrema, its concavity and points of inflection, any points of discontinuity and its end behavior. All of these terms are more fully defined in calculus. Such graphs are useful in calculus to understand the nature and behavior of a function or relation.


See also

* Horizontal and vertical * Jones diagram, which plots four variables rather than two * Orthogonal coordinates * Polar coordinate system * Regular grid * Spherical coordinate system


References


Sources

* * *


Further reading

* * * * * *


External links


Cartesian Coordinate System




* [http://www.mathopenref.com/coordpoint.html Coordinates of a point] Interactive tool to explore coordinates of a point
open source JavaScript class for 2D/3D Cartesian coordinate system manipulation
{{DEFAULTSORT:Cartesian Coordinate System Orthogonal coordinate systems Elementary mathematics René Descartes Analytic geometry fi:Koordinaatisto#Suorakulmainen koordinaatisto