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A Cartesian coordinate system (, ) in a
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
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 sign ...
that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
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
coordinate line 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 sign ...
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 * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
'', at ordered pair . The coordinates can also be defined as the positions of the 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 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 for any dimension ''n''. These coordinates are equal, up to sign, 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''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperp ...
s. The invention of Cartesian coordinates in the 17th century by René Descartes ( Latinized name: ''Cartesius'') revolutionized mathematics by providing the first systematic link between Euclidean geometry and
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
. 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 ...
s) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
of radius 2, centered at the origin of the plane, may be described as the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
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 engineer ...
, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, 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 arithm ...
,
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galax ...
, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.


History

The adjective ''Cartesian'' refers to the French mathematician and philosopher René Descartes, who published this idea in 1637 while he was resident in the Netherlands. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. The French cleric Nicole Oresme 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'' was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes's work. 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 arithm ...
by Isaac Newton and 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, 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 th ...
for the plane, and the spherical and
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
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 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 of
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
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 (''x'') coordinate and the ordinate refers to the (''y'') coordinate of a standard two-dimensional graph. The distance of a point from the y-axis, scaled with the x-axis, is called abscissa or x coo ...
'' and the '' ordinate'' of ''P'', respectively; and the point where the axes meet is called the ''origin'' of the coordinate system. The coordinates are usually written as two numbers in parentheses, in that order, separated by a comma, as in . 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 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, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
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 (with radius equal to the length unit, and center at the origin), the unit square (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 ra ...
, and so on. The two axes divide the plane into four right angles, 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 from the ''X''-axis and from the ''Y''-axis are and , respectively; where denotes the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
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 '' octants''. The octants are: \begin (+x,+y,+z) && (-x,+y,+z) && (+x,-y,+z) && (+x,+y,-z) \\ (+x,-y,-z) && (-x,+y,-z) && (-x,-y,+z) && (-x,-y,-z) \end 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''.


Higher dimensions

Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of real numbers; 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\tim ...
\R^2 = \R\times\R, where \R is the set of all real numbers. In the same way, the points in any Euclidean space of dimension ''n'' be identified with the tuples (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''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperp ...
defined by all the other axes). In such an oblique coordinate system the computations of distances and angles must be modified from that in standard Cartesian systems, and many standard formulas (such as the Pythagorean formula for the distance) do not hold (see
affine plane In 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 pl ...
).


Notations and conventions

The Cartesian coordinates of a point are usually written in parentheses 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 varies with time, the graph coordinates may be denoted ''p'' and ''t''. Each axis is usually named after the coordinate which is measured along it; so one says the ''x-axis'', the ''y-axis'', the ''t-axis'', etc. Another common convention for coordinate naming is to use subscripts, as (''x''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: by storing the coordinates of a point as an
array An array is a systematic arrangement of similar objects, usually in rows and columns. Things called an array include: {{TOC right Music * In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...
, instead of a record, the 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 (''x'') coordinate and the ordinate refers to the (''y'') coordinate of a standard two-dimensional graph. The distance of a point from the y-axis, scaled with the x-axis, is called abscissa or x coo ...
) is measured along a horizontal axis, oriented from left to right. The second coordinate (the ordinate) is then measured along a
vertical Vertical is a geometric term of location which may refer to: * Vertical direction, the direction aligned with the direction of the force of gravity, up or down * Vertical (angles), a pair of angles opposite each other, formed by two intersecting s ...
axis, usually oriented from bottom to top. Young children learning the Cartesian system, commonly learn the order to read the values before cementing the ''x''-, ''y''-, and ''z''-axis concepts, by starting with 2D mnemonics (for example, 'Walk along the hall then up the stairs' akin to straight across the ''x''-axis then up vertically along the ''y''-axis). Computer graphics and
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
, 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 fo ...
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. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of t ...
, unless specifically stated otherwise. All laws of physics and math assume this
right-handedness In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjecti ...
, 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 numerals: I (where the coordinates both have positive signs), II (where the abscissa is negative − and the ordinate is positive +), III (where both the abscissa and the ordinate are −), and IV (abscissa +, ordinate −). When the axes are drawn according to the mathematical custom, the numbering goes
counter-clockwise 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 s ...
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; for example, or . The generalization of the quadrant and octant to an arbitrary number of dimensions is the orthant, and a similar naming system applies.


Cartesian formulae for the plane


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 o ...
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, a bijection, also known as a 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 ...
) mappings of points of the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
to themselves which preserve distances between points. There are four types of these mappings (also called isometries): translations, rotations, 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 reflecti ...
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 ''transla ...
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 Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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 s ...
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 \begin x' &= x \cos \theta - y \sin \theta \\ y' &= x \sin \theta + y \cos \theta . \end 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 reflection 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 \begin x' &= x \cos 2\theta + y \sin 2\theta \\ y' &= x \sin 2\theta - y \cos 2\theta . \end 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

All
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
s of the plane can be described in a uniform way by using matrices. For this purpose the coordinates (x,y) of a point are commonly represented as the
column matrix In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
\beginx\\y\end. The result (x', y') of applying an affine transformation to a point (x,y) is given by the formula \beginx'\\y'\end = A \beginx\\y\end + b, where A = \begin A_ & A_ \\ A_ & A_ \end is a 2×2
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
and b=\beginb_1\\b_2\end is a column matrix. That is, \begin x' &= x A_ + y A_ + b_ \\ y' &= x A_ + y A_ + b_. \end Among the affine transformations, the Euclidean transformations are characterized by the fact that the matrix A is
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
; that is, its columns are orthogonal vectors of Euclidean norm one, or, explicitly, A_ A_ + A_ A_ = 0 and A_^2 + A_^2 = A_^2 + A_^2 = 1. This is equivalent to saying that times its transpose is the identity matrix. If these conditions do not hold, the formula describes a more general
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
. The transformation is a translation if and only if is the identity matrix. The transformation is a rotation around some point if and only if is a rotation matrix, meaning that it is orthogonal and A_ A_ - A_ A_ = 1 . A reflection or glide reflection is obtained when, A_ A_ - A_ A_ = -1 . Assuming that translations are not used (that is, b_1=b_2=0) transformations can be composed by simply multiplying the associated transformation matrices. In the general case, it is useful to use the
augmented matrix In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices. Given the matrices and , where A ...
of the transformation; that is, to rewrite the transformation formula \beginx'\\y'\\1\end = A' \beginx\\y\\1\end, where A' = \begin A_ & A_&b_1 \\ A_ & A_&b_2\\0&0&1 \end. With this trick, the composition of affine transformations is obtained by multiplying the augmented matrices.


Affine transformation

Affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
s of the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
are transformations that map lines to lines, but may change distances and angles. As said in the preceding section, they can be represented with augmented matrices: \begin A_ & A_ & b_ \\ A_ & A_ & b_ \\ 0 & 0 & 1 \end \begin x \\ y \\ 1 \end = \begin x' \\ y' \\ 1 \end. The Euclidean transformations are the affine transformations such that the 2×2 matrix of the A_ is
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
. The augmented matrix that represents the
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of two affine transformations is obtained by multiplying their augmented matrices. Some affine transformations that are not Euclidean transformations have received specific names.


Scaling

An example of an affine transformation which is not Euclidean 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 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, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
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. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of t ...
''. 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 one axis 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 most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
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. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of t ...
. If the
index finger The index finger (also referred to as forefinger, first finger, second finger, pointer finger, trigger finger, digitus secundus, digitus II, and many other terms) is the second digit of a human hand. It is located between the thumb and the mid ...
of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the
thumb The thumb is the first digit of the hand, next to the index finger. 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 thum ...
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 polyto ...
cube and a
concave Concave or concavity may refer to: Science and technology * Concave lens * Concave mirror Mathematics * Concave function, the negative of a convex function * Concave polygon, a polygon which is not convex * Concave set In geometry, a subset o ...
"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, 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 vec ...
in the direction of the ''x''-axis and ''y''-axis respectively, generally referred to as the ''
standard basis In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the ...
'' (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 The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. 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 ...
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. # Units of distance must be decided defining the spatial size represented by the numbers used as coordinates. # An origin must be assigned to a specific spatial location or landmark, and # 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 (that is, geospatial 3D). Kilometers are a good choice of units, since the original definition of the kilometer was geospatial, with equaling the surface distance from the equator to the North Pole. Based on symmetry, the gravitational center of the Earth suggests a natural placement of the origin (which can be sensed via satellite orbits). The axis of Earth's rotation provides a natural orientation for the ''X'', ''Y'', and ''Z'' axes, strongly associated with "up vs. down", so positive ''Z'' can adopt the direction from the geocenter to the 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 the geocenter out to longitude, 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 longitude, latitude. From a longitude of , a latitude , and Earth radius of 40,000/2''π'' km, and transforming from spherical to Cartesian coordinates, one can estimate the geocentric coordinates of the Empire State Building, . 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 previously was no such coordinate system 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 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.


See also

*
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 hor ...
*
Jones diagram A Jones diagram is a type of Cartesian graph developed by Loyd A. Jones in the 1940s, where each axis represents a different variable. In a Jones diagram opposite directions of an axis represent different quantities, unlike in a Cartesian graph ...
, which plots four variables rather than two *
Orthogonal coordinates In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
* Polar coordinate system * Regular grid *
Spherical coordinate system In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...


References


Sources

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Further reading

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External links


Cartesian Coordinate System




* ttp://www.mathopenref.com/coordpoint.html Coordinates of a pointInteractive 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 Three-dimensional coordinate systems fi:Koordinaatisto#Suorakulmainen koordinaatisto