Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates:(2, 3) in green,(−3, 1) in red,(−1.5, −2.5) in blue, and the origin(0, 0) in purple.
Similarly, the position of any point inthree-dimensional space can be specified by threeCartesian coordinates, which are the signed distances from the point to three mutually perpendicular planes. More generally,n Cartesian coordinates specify the point in ann-dimensionalEuclidean space for anydimensionn. These coordinates are the signed distances from the point ton mutually perpendicular fixedhyperplanes.
Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is(x −a)2 + (y −b)2 =r2 wherea andb are the coordinates of the center(a,b) andr is the radius.
Cartesian coordinates are named forRené Descartes, whose invention of them in the 17th century revolutionized mathematics by allowing the expression of problems of geometry in terms ofalgebra andcalculus. Using the Cartesian coordinate system, geometric shapes (such ascurves) can be described byequations involving the coordinates of points of the shape. For example, acircle of radius 2, centered at the origin of the plane, may be described as theset of all points whose coordinatesx andy satisfy the equationx2 +y2 = 4; thearea, theperimeter and thetangent line at any point can be computed from this equation by usingintegrals andderivatives, in a way that can be applied to any curve.
The adjectiveCartesian refers to the Frenchmathematician andphilosopherRené Descartes, who published this idea in 1637 while he was resident in the Netherlands. It was independently discovered byPierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery.[1] The French clericNicole Oresme used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat.[2]
Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis.[3] The concept of using a pair of axes was introduced later, after Descartes'La Géométrie was translated into Latin in 1649 byFrans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes's work.[4]
The development of the Cartesian coordinate system would play a fundamental role in the development of thecalculus byIsaac Newton andGottfried Wilhelm Leibniz.[5] The two-coordinate description of the plane was later generalized into the concept ofvector spaces.[6]
Anaffine line with a chosen Cartesian coordinate system is called anumber line. Every point on the line has a real-number coordinate, and every real number represents some point on the line.
There are twodegrees of freedom in the choice of Cartesian coordinate system for a line, which can be specified by choosing two distinct points along the line and assigning them to two distinctreal numbers (most commonly zero and one). Other points can then be uniquely assigned to numbers bylinear interpolation. Equivalently, one point can be assigned to a specific real number, for instance anorigin point corresponding to zero, and anoriented length along the line can be chosen as a unit, with the orientation indicating the correspondence between directions along the line and positive or negative numbers.[7] Each point corresponds to its signed distance from the origin (a number with an absolute value equal to the distance and a+ or− sign chosen based on direction).
Ageometric transformation of the line can be represented by afunction of a real variable, for exampletranslation of the line corresponds to addition, andscaling the line corresponds to multiplication. Any two Cartesian coordinate systems on the line can be related to each-other by alinear function (function of the form) taking a specific point's coordinate in one system to its coordinate in the other system. Choosing a coordinate system for each of two different lines establishes anaffine map from one line to the other taking each point on one line to the point on the other line with the same coordinate.
A Cartesian coordinate system in two dimensions (also called arectangular coordinate system or anorthogonal coordinate system[8]) is defined by anordered pair ofperpendicular lines (axes), a singleunit of length for both axes, and an orientation for each axis. The point where the axes meet is taken as the origin for both, thus turning each axis into anumber line. For any pointP, a line is drawn throughP 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 theCartesian coordinates ofP. The reverse construction allows one to determine the pointP given its coordinates.
The first and second coordinates are called theabscissa and theordinate ofP, respectively; and the point where the axes meet is called theorigin of the coordinate system. The coordinates are usually written as two numbers in parentheses, in that order, separated by a comma, as in(3, −10.5). Thus the origin has coordinates(0, 0), and the points on the positive half-axes, one unit away from the origin, have coordinates(1, 0) and(0, 1).
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 somecomputer graphics contexts, the ordinate axis may be oriented downwards.) The origin is often labeledO, and the two coordinates are often denoted by the lettersX andY, orx andy. The axes may then be referred to as theX-axis andY-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.
AEuclidean plane with a chosen Cartesian coordinate system is called aCartesian plane. In a Cartesian plane, one can define canonical representatives of certain geometric figures, such as theunit circle (with radius equal to the length unit, and center at the origin), theunit square (whose diagonal has endpoints at(0, 0) and(1, 1)), theunit hyperbola, and so on.
The two axes divide the plane into fourright angles, calledquadrants. The quadrants may be named or numbered in various ways, but the quadrant where all coordinates are positive is usually called thefirst quadrant.
If the coordinates of a point are(x,y), then itsdistances from theX-axis and from theY-axis are |y| and |x|, respectively; where | · | denotes theabsolute value of a number.
A three dimensional Cartesian coordinate system, with originO and axis linesX,Y andZ, oriented as shown by the arrows. The tick marks on the axes are one length unit apart. The black dot shows the point with coordinatesx = 2,y = 3, andz = 4, or(2, 3, 4).
A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines (theaxes) that go through a common point (theorigin), 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 pointP of space, one considers a plane throughP perpendicular to each coordinate axis, and interprets the point where that plane cuts the axis as a number. The Cartesian coordinates ofP are those three numbers, in the chosen order. The reverse construction determines the pointP given its three coordinates.
Alternatively, each coordinate of a pointP can be taken as the distance fromP to the plane defined by the other two axes, with the sign determined by the orientation of the corresponding axis.
Each pair of axes defines acoordinate plane. These planes divide space into eightoctants. The octants are:
The coordinates are usually written as three numbers (or algebraic formulas) surrounded by parentheses and separated by commas, as in(3, −2.5, 1) or(t,u +v,π/2). Thus, the origin has coordinates(0, 0, 0), and the unit points on the three axes are(1, 0, 0),(0, 1, 0), and(0, 0, 1).
Standard names for the coordinates in the three axes areabscissa,ordinate andapplicate.[9] The coordinates are often denoted by the lettersx,y, andz. The axes may then be referred to as thex-axis,y-axis, andz-axis, respectively. Then the coordinate planes can be referred to as thexy-plane,yz-plane, andxz-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 calledheight oraltitude. 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(0, 0, 1); a convention that is commonly calledtheright-hand rule.
Thecoordinate surfaces of the Cartesian coordinates(x,y,z). Thez-axis is vertical and thex-axis is highlighted in green. Thus, the red plane shows the points withx = 1, the blue plane shows the points withz = 1, and the yellow plane shows the points withy = −1. The three surfaces intersect at the pointP (shown as a black sphere) with the Cartesian coordinates(1, −1, 1).
Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs ofreal numbers; that is, with theCartesian product, where is the set of all real numbers. In the same way, the points in anyEuclidean space of dimensionn be identified with thetuples (lists) ofn real numbers; that is, with the Cartesian product.
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 thehyperplane defined by all the other axes). In such anoblique 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 (seeaffine plane).
The Cartesian coordinates of a point are usually written inparentheses and separated by commas, as in(10, 5) or(3, 5, 7). The origin is often labelled with the capital letterO. 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 apressure varies withtime, the graph coordinates may be denotedp andt. Each axis is usually named after the coordinate which is measured along it; so one says thex-axis, they-axis, thet-axis, etc.
Another common convention for coordinate naming is to use subscripts, as (x1,x2, ...,xn) for then coordinates in ann-dimensional space, especially whenn is greater than 3 or unspecified. Some authors prefer the numbering (x0,x1, ...,xn−1). These notations are especially advantageous incomputer programming: by storing the coordinates of a point as anarray, instead of arecord, thesubscript can serve to index the coordinates.
In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called theabscissa) is measured along ahorizontal axis, oriented from left to right. The second coordinate (theordinate) is then measured along avertical axis, usually oriented from bottom to top. Young children learning the Cartesian system, commonly learn the order to read the values before cementing thex-,y-, andz-axis concepts, by starting with 2D mnemonics (for example, 'Walk along the hall then up the stairs' akin to straight across thex-axis then up vertically along they-axis).
Computer graphics andimage processing, however, often use a coordinate system with they-axis oriented downwards on the computer display. This convention developed in the 1960s (or earlier) from the way that images were originally stored indisplay buffers.
For three-dimensional systems, a convention is to portray thexy-plane horizontally, with thez-axis added to represent height (positive up). Furthermore, there is a convention to orient thex-axis toward the viewer, biased either to the right or left. If a diagram (3D projection or2D perspective drawing) shows thex- andy-axis horizontally and vertically, respectively, then thez-axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, thez-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 cameraperspective. 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 theright-hand rule, unless specifically stated otherwise. All laws of physics and math assume thisright-handedness, which ensures consistency.
For 3D diagrams, the names "abscissa" and "ordinate" are rarely used forx andy, respectively. When they are, thez-coordinate is sometimes called theapplicate. The wordsabscissa,ordinate andapplicate are sometimes used to refer to coordinate axes rather than the coordinate values.[8]
The four quadrants of a Cartesian coordinate system
The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, calledquadrants,[8] each bounded by two half-axes. These are often numbered from 1st to 4th and denoted byRoman 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 goescounter-clockwise starting from the upper right ("north-east") quadrant.
Similarly, a three-dimensional Cartesian system defines a division of space into eight regions oroctants,[8] 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 theorthant, and a similar naming system applies.
Translating a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers(a,b) to the Cartesian coordinates of every point in the set. That is, if the original coordinates of a point are(x,y), after the translation they will be
Torotate a figurecounterclockwise around the origin by some angle is equivalent to replacing every point with coordinates (x,y) by the point with coordinates (x',y'), where
If(x,y) are the Cartesian coordinates of a point, then(−x,y) are the coordinates of itsreflection across the second coordinate axis (the y-axis), as if that line were a mirror. Likewise,(x, −y) 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 with the x-axis, is equivalent to replacing every point with coordinates(x,y) by the point with coordinates(x′,y′), where
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).
Allaffine transformations of the plane can be described in a uniform way by using matrices. For this purpose, the coordinates of a point are commonly represented as thecolumn matrix The result of applying an affine transformation to a point is given by the formulawhereis a 2×2matrix and is a column matrix.[12] That is,
The transformation is a translationif and only ifA is theidentity matrix. The transformation is a rotation around some point if and only ifA is arotation matrix, meaning that it is orthogonal and
A reflection or glide reflection is obtained when,
Assuming that translations are not used (that is,) transformations can becomposed by simply multiplying the associated transformation matrices. In the general case, it is useful to use theaugmented matrix of the transformation; that is, to rewrite the transformation formulawhereWith this trick, the composition of affine transformations is obtained by multiplying the augmented matrices.
Effect of applying various 2D affine transformation matrices on a unit square (reflections are special cases of scaling)
Affine transformations of theEuclidean plane 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:
The Euclidean transformations are the affine transformations such that the 2×2 matrix of the isorthogonal.
The augmented matrix that represents thecomposition of two affine transformations is obtained by multiplying their augmented matrices.
Some affine transformations that are not Euclidean transformations have received specific names.
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 numberm. If(x,y) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates
Ifm is greater than 1, the figure becomes larger; ifm is between 0 and 1, it becomes smaller.
Fixing or choosing thex-axis determines they-axis up to direction. Namely, they-axis is necessarily theperpendicular to thex-axis through the point marked 0 on thex-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 calledhandedness) of the Cartesian plane.
The usual way of orienting the plane, with the positivex-axis pointing right and the positivey-axis pointing up (and thex-axis being the "first" and they-axis the "second" axis), is considered thepositive orstandard orientation, also called theright-handed orientation.
A commonly used mnemonic for defining the positive orientation is theright-hand rule. Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from thex-axis to they-axis, in a positively oriented coordinate system.
The other way of orienting the plane is following theleft-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.
Fig. 7 – The left-handed orientation is shown on the left, and the right-handed on the right.Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes
Once thex- andy-axes are specified, they determine theline along which thez-axis should lie, but there are two possible orientations for this line. The two possible coordinate systems, which result are called 'right-handed' and 'left-handed'.[13] The standard orientation, where thexy-plane is horizontal and thez-axis points up (and thex- and they-axis form a positively oriented two-dimensional coordinate system in thexy-plane if observed fromabove thexy-plane) is calledright-handed orpositive.
3D Cartesian coordinate handedness
The name derives from theright-hand rule. If theindex finger of the right hand is pointed forward, themiddle finger bent inward at a right angle to it, and thethumb placed at a right angle to both, the three fingers indicate the relative orientation of thex-,y-, andz-axes in aright-handed system. The thumb indicates thex-axis, the index finger they-axis and the middle finger thez-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 pointtowards the observer, whereas the "middle"-axis is meant to pointaway from the observer. The red circle isparallel to the horizontalxy-plane and indicates rotation from thex-axis to they-axis (in both cases). Hence the red arrow passesin front of thez-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 aconvex cube and aconcave "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 thex-axis as pointingtowards the observer and thus seeing a concave corner.
A point in space in a Cartesian coordinate system may also be represented by a positionvector, which can be thought of as an arrow pointing from the origin of the coordinate system to the point.[14] If the coordinates represent spatial positions (displacements), it is common to represent the vector from the origin to the point of interest as. In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as:
where and areunit vectors in the direction of thex-axis andy-axis respectively, generally referred to as thestandard basis (in some application areas these may also be referred to asversors). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates can be written as:[15]
where and
There is nonatural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to usecomplex numbers to provide such a multiplication. In a two-dimensional cartesian plane, identify the point with coordinates(x,y) with the complex numberz =x +iy. Here,i is theimaginary unit and is identified with the point with coordinates(0, 1), so it isnot the unit vector in the direction of thex-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 thequaternions.
^Consider the tworays or half-lines resulting from splitting the line at the origin. One of the half-lines can be assigned to positive numbers, and the other half-line to negative numbers.
Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2013).Calculus: Single and Multivariable (6th ed.).John Wiley & Sons.ISBN978-0470-88861-2.
Moon P, Spencer DE (1988). "Rectangular Coordinates (x, y, z)".Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd, 3rd print ed.). New York: Springer-Verlag. pp. 9–11 (Table 1.01).ISBN978-0-387-18430-2.
