TheInfoList

A Cartesian coordinate system (, ) in a plane is a
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is signif ...
that specifies each
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points, ...
uniquely by a pair of
numerical Numerical may refer to: * Number * Numerical digit * Numerical analysis {{disambig ...
coordinates, which are the signed distances to the point from two fixed
perpendicular In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects. A line is said to be perpend ...
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 units, used in every country globally. In the United States the U.S. customary units ...
. 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, origins, or original may refer to: Arts, entertainment, and media Comics and manga * ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * Origins (''Judge Dredd'' story), a major ''Judge Dredd'' ...
'', at ordered pair . The coordinates can also be defined as the positions of the 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 t ...
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 classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-di ...
for any
dimension thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to s ...
''n''. These coordinates are equal, up to
sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or med ...
, to distances from the point to ''n'' mutually perpendicular
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dim ...
s. The invention of Cartesian coordinates in the 17th century by
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French-born philosopher, mathematician, and scientist who spent a large portion of his working life in the Dutch Republic, initially serving the ...
( Latinized name: ''Cartesius'') revolutionized mathematics by providing the first systematic link between
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's method consists in assuming a small set of intuitively appealing axioms, ...
and
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the broad areas of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical ...
. 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, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that app ...
s) can be described by Cartesian equations: algebraic
equation In mathematics, an equation is a statement that asserts the equality of two expressions, which are connected by the equals sign "=". The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in Fre ...
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 using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering ...
, and provide enlightening geometric interpretations for many other branches of mathematics, such as
linear algebra#REDIRECT Linear algebra#REDIRECT Linear algebra {{R from other capitalisation ...
{{R from other capitalisation ...
,
complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex n ...
,
differential geometry Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in th ...
, multivariate
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic ...
,
group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation groups. See Rubik's Cube group. In mathematics and abstract algebra, group theory studies the algebraic structures known ...
and more. A familiar example is the concept of the
graph of a function In mathematics, the graph of a function is the set of ordered pairs , where . In the common case where and are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and thus form a subset of this plane. In t ...
. 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 celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain t ...
,
physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its motion and behavior through space and time, and the related ent ...
,
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 specializ ...
and many more. They are the most common coordinate system used in
computer graphics#REDIRECT Computer graphics#REDIRECT Computer graphics {{R from other capitalisation ...
{{R from other capitalisation ...
,
computer-aided geometric design Computer-aided or computer-assisted is an adjectival phrase that hints of the use of a computer as an indispensable tool in a certain field, usually derived from more traditional fields of science and engineering. Instead of the phrase computer-aide ...
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 in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of ...
and
philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek thinker Pythagoras (6th ...
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French-born philosopher, mathematician, and scientist who spent a large portion of his working life in the Dutch Republic, initially serving the ...
, 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 lawyer at the ''Parlement'' of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, incl ...

, 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 of the later Middle Ages. He wrote influential works on economics, mathematics, physics, astrolog ...
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 in 1637 as an appendix to ''Discours de la méthode'' (''Discourse on the Method''), written by René Descartes. In the ''Discourse'', he presents his method for obtaining clarity on any subject. ''La Géométrie' ...
'' was translated into Latin in 1649 by
Frans van Schooten200px, Frans van Schooten Franciscus van Schooten (1615, Leiden – 29 May 1660, Leiden) was a Dutch mathematician who is most known for popularizing the analytic geometry of René Descartes. Life Van Schooten's father was a professor of mathematics ...
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 mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic ...
by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, theologian, and author (described in his time as a "natural philosopher") who is widely recognised as one of the greatest math ...

and
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "1666–1676" section. (; or ; – 14 November 1716) was a prominent German polymath and one of the most important logicians, mathematicians and natural philoso ...

. The two-coordinate description of the plane was later generalized into the concept of
vector spaces#REDIRECT Vector space#REDIRECT Vector space#REDIRECT Vector space {{Redirect category shell, 1= {{R for alternate capitalisation ... {{Redirect category shell, 1= {{R for alternate capitalisation ...
{{Redirect category shell, 1= {{R for alt ...
. Many other coordinate systems have been developed since Descartes, such as the
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
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 that is the surface of a ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its " ...

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 spe ...
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 used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be re ...
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, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In contr ...
of
perpendicular In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects. A line is said to be perpend ...
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 units, used in every country globally. In the United States the U.S. customary units ...
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, the abscissa (; plural ''abscissae'' or ''abscissæ'' or ''abscissas'') ...
'' 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, the abscissa (; plural ''abscissae'' or ''abscissæ'' or ''abscissas'') ...
'' 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#REDIRECT Computer graphics#REDIRECT Computer graphics {{R from other capitalisation ...
{{R from other capitalisation ...
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, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of som ...
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, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in th ...

(with radius equal to the length unit, and center at the origin), the
unit square 300px, The unit square in the real plane In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian ...
(whose diagonal has endpoints at and ), the
unit hyperbola In geometry, the unit hyperbola is the set of points (''x,y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative radial l ...
, and so on. The two axes divide the plane into four
right angle In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a ...
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
distances Distance is a numerical measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). The distance from a point A to ...

from the ''X''-axis and from the ''Y''-axis are , ''y'', and , ''x'', , respectively; where , ..., denotes the
absolute value of the absolute value function for real numbers In mathematics, the absolute value or modulus of a real number , denoted , is the non-negative value of  without regard to its sign. Namely, if is positive, and if is negative (in whi ...
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 and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. Most of the various left-hand and right-hand rules arise from the fact that the three axes of three-dimensiona ...
''.

## Higher dimensions

Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of
real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R\$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
s; that is with the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\times ...
$\R^2 = \R\times\R$, 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 classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-di ...
of dimension ''n'' be identified with the
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defin ...
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, a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dim ...
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 reals, equipped with a metric, the Euclidean distance. In other words, an affine plane ...
).

# 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 marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a ''left'' or ...
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 applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and ev ...
varies with
time Time is the indefinite continued progress of existence and events that occur in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence event ...
, 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 specific task. Programming involves tasks such as: analysis, generating algorithms, profiling ...
: 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 in 2010 under the name African-American Film Festival Releasing Movement (AFFRM). In 2015 the company rebranded itself ...
, instead of a record, the
subscript Pro; the size of the subscript is about 62% of the original characters, dropped below the baseline by about 16%. The second typeface is Myriad Pro; the superscript is about 60% of the original characters, raised by about 44% above the baseline.) ...
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, the abscissa (; plural ''abscissae'' or ''abscissæ'' or ''abscissas'') ...
) 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, the abscissa (; plural ''abscissae'' or ''abscissæ'' or ''abscissas'') ...
) is then measured along a
vertical Vertical may refer to: * Vertical direction, the direction aligned with the direction of the force of gravity,up or down, as materialized with a plumb line * Vertical (angles), a pair of angles sharing the same vertex and bounded by the same pair of ...
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 to process digital images through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing. It allows a ...
, 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 design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspect analysis to project a complex object for v ...
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 and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. Most of the various left-hand and right-hand rules arise from the fact that the three axes of three-dimensiona ...
, 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-han ...
, 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.

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 numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin ...
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 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, 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, an orthant or hyperoctant is the analogue in ''n''-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions. In general an orthant in ''n''-dimensions can be considered the intersection of ''n'' mutually ...
, and a similar naming system applies.

# Cartesian formulae for the plane

## Distance between two points

The
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occas ...
between two points of the plane with Cartesian coordinates $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ is :$d = \sqrt.$ This is the Cartesian version of
Pythagoras's theorem In mathematics, the Pythagorean theorem, or Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
. In three-dimensional space, the distance between points $\left(x_1,y_1,z_1\right)$ and $\left(x_2,y_2,z_2\right)$ 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, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element ...
) mappings of points of the
Euclidean plane In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of som ...
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 ''translating' ...
,
rotations A rotation is a circular movement of an object around a center (or point) of rotation. The geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center and perpendicular to ...
, reflections and
glide reflection In 2-dimensional geometry, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. The intermediate step between reflection an ...

s.

### Translation

Translating 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 ''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 :$\left(x\text{'}, y\text{'}\right) = \left(x + a, y + b\right) .$

### Rotation

To
rotate A rotation is a circular movement of an object around a center (or point) of rotation. The geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center and perpendicular to ...
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 ...
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 :$x\text{'}=x \cos \theta - y \sin \theta$ :$y\text{'}=x \sin \theta + y \cos \theta .$ Thus: $\left(x\text{'},y\text{'}\right) = \left(\left(x \cos \theta - y \sin \theta\,\right) , \left(x \sin \theta + y \cos \theta\,\right)\right) .$

### 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 :$x\text{'}=x \cos 2\theta + y \sin 2\theta$ :$y\text{'}=x \sin 2\theta - y \cos 2\theta .$ Thus: $\left(x\text{'},y\text{'}\right) = \left(\left(x \cos 2\theta + y \sin 2\theta\,\right) , \left(x \sin 2\theta - y \cos 2\theta\,\right)\right) .$

### 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, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformations in ...
s of the plane can all be described in a uniform way by using matrices. The result $\left(x\text{'}, y\text{'}\right)$ of applying a Euclidean transformation to a point $\left(x,y\right)$ is given by the formula :$\left(x\text{'},y\text{'}\right) = \left(x,y\right) A + b$ 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, :$x\text{'} = x A_ + y A_ + b_$ :$y\text{'} = x A_ + y A_ + b_,$ 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, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements ''u'' and ''v'' of a vector space with bilinear form ''B'' are orthogonal when . Depending on the bilin ...
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, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tran ...
must be the
identity matrix In linear algebra, the identity matrix (sometimes ambiguously called a unit matrix) of size ''n'' is the ''n'' × ''n'' square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by ''I'n'', or simply by ''I'' if the size is ...
. If these conditions do not hold, the formula describes a more general
affine transformation In Euclidean geometry, an affine transformation, or an affinity (from the Latin, ''affinis'', "connected with"), is a geometric transformation that preserves lines and parallelism (but not necessarily distances and angles). More generally, an ''a ...
of the plane provided that the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero i ...
of ''A'' is not zero. The formula defines a translation
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondit ...
''A'' is the
identity matrix In linear algebra, the identity matrix (sometimes ambiguously called a unit matrix) of size ''n'' is the ''n'' × ''n'' square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by ''I'n'', or simply by ''I'' if the size is ...
. The transformation is a rotation around some point if and only if ''A'' is a
rotation matrixIn linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end ...
, 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, an affine transformation, or an affinity (from the Latin, ''affinis'', "connected with"), is a geometric transformation that preserves lines and parallelism (but not necessarily distances and angles). More generally, an ''a ...
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: ::$\begin A_ & A_ & b_ \\ A_ & A_ & b_ \\ 0 & 0 & 1 \end \begin x \\ y \\ 1 \end = \begin x\text{'} \\ y\text{'} \\ 1 \end.$ /nowiki>Note the matrix ''A'' from above was transposed. The matrix is on the left and column vectors for point coordinates are used./nowiki> 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 :$\left(x\text{'},y\text{'}\right) = \left(m x, m y\right).$ If ''m'' is greater than 1, the figure becomes larger; if ''m'' is between 0 and 1, it becomes smaller.

### Shearing

A shearing transformation will push the top of a square sideways to form a parallelogram. Horizontal shearing is defined by: :$\left(x\text{'},y\text{'}\right) = \left(x+y s, y\right)$ Shearing can also be applied vertically: :$\left(x\text{'},y\text{'}\right) = \left(x, x s+y\right)$

# 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, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects. A line is said to be perpend ...
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 and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. Most of the various left-hand and right-hand rules arise from the fact that the three axes of three-dimensiona ...
''. 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 Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Literatu ...
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 and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. Most of the various left-hand and right-hand rules arise from the fact that the three axes of three-dimensiona ...
. If the
index finger The index finger (also referred to as forefinger, first finger, pointer finger, trigger finger, digitus secundus, digitus II, and many other terms) is the second finger of a human hand. It is located between the first and third digits, between th ...
of the right hand is pointed forward, the
middle finger The middle finger, long finger, or tall finger is the third digit of the human hand, located between the index finger and the ring finger. It is typically the longest finger. In anatomy, it is also called ''the third finger'', ''digitus medius ...
bent inward at a right angle to it, and the
thumb The thumb is the first digit of the hand. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thumb is pollex (compare ''hallu ...
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
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope, ...
cube and a
concave Concave means curving in or hollowed inward, as opposed to convex. Concave may refer to: * Concave function, the negative of a convex function * Concave lens * Concave mirror * Concave polygon, a polygon which is not convex * Concave set See also ...
"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
vector Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
, 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 mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vect ...
in the direction of the ''x''-axis and ''y''-axis respectively, generally referred to as the ''
standard basis Standard may refer to: Flags * Colours, standards and guidons * Standard (flag), a type of flag used for personal identification Norm, convention or requirement * Standard (metrology), an object that bears a defined relationship to a unit of m ...
'' (in some application areas these may also be referred to as
versor In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). Each versor has the form :q = \exp(a\mathbf) = \cos a + \mathbf \sin a, \quad \mathbf^2 = -1, \quad a \in ,\pi where the r2 = −1 condition means that r is a unit-leng ...
s). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates $\left(x,y,z\right)$ 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 number In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this equation, was called ...
s 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 The imaginary unit or unit imaginary number () is a solution to the quadratic equation . Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and multipl ...
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
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion a ...
s.

# 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 A prime meridian is the meridian (a line of longitude) in a geographic coordinate system at which longitude is defined to be 0°. Together, a prime meridian and its anti-meridian (the 180th meridian in a 360°-system) form a great circle. This g ...

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 A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multipl ...
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 is the set of ordered pairs , where . In the common case where and are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and thus form a subset of this plane. In t ...
or 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.

*
Horizontal and vertical In astronomy, geography, and related sciences and contexts, a ''direction'' or ''plane'' passing by a given point is said to be vertical if it contains the local gravity direction at that point. Conversely, a direction or plane is said to be hori ...
* Jones diagram, which plots four variables rather than two *
Orthogonal coordinatesIn mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate surfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate surface fo ...
*
Polar coordinate system In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
*
Regular grid A regular grid is a tessellation of ''n''-dimensional Euclidean space by congruent parallelotopes (e.g. bricks). Grids of this type appear on graph paper and may be used in finite element analysis, finite volume methods, finite difference methods, ...

*
Spherical coordinate system File:3D Spherical 2.svg, 240px, Spherical coordinates as often used in ''mathematics'': radial distance , azimuthal angle , and polar angle . The meanings of and have been swapped compared to the physics convention. As in physics, (rho) is ...

* * *

* * * * * *