A
The invention of Cartesian coordinates in the 17th century by René
Descartes (Latinized name: Cartesius) revolutionized mathematics by
providing the first systematic link between
Contents 1 History 2 Description 2.1 One dimension 2.2 Two dimensions 2.3 Three dimensions 2.4 Higher dimensions 2.5 Generalizations 3 Notations and conventions 3.1 Quadrants and octants 4 Cartesian formulae for the plane 4.1 Distance between two points 4.2 Euclidean transformations 4.2.1 Translation 4.2.2 Rotation 4.2.3 Reflection 4.2.4 Glide reflection 4.2.5 General matrix form of the transformations 4.2.6 Affine transformation 4.2.7 Scaling 4.2.8 Shearing 5 Orientation and handedness 5.1 In two dimensions 5.2 In three dimensions 6 Representing a vector in the standard basis 7 Applications 8 See also 9 References 10 Sources 11 Further reading 12 External links History[edit]
The adjective Cartesian refers to the French mathematician and
philosopher
A three dimensional Cartesian coordinate system, with origin O and axis lines X, Y and Z, oriented as shown by the arrows. The tick marks on the axes are one length unit apart. The black dot shows the point with coordinates x = 2, y = 3, and z = 4, or (2, 3, 4). Choosing a
The coordinate surfaces of the Cartesian coordinates (x, y, z). The z-axis is vertical and the x-axis is highlighted in green. Thus, the red plane shows the points with x = 1, the blue plane shows the points with z = 1, and the yellow plane shows the points with y = −1. The three surfaces intersect at the point P (shown as a black sphere) with the Cartesian coordinates (1, −1, 1). Higher dimensions[edit]
A
R 2 = R × R displaystyle mathbb R ^ 2 =mathbb R times mathbb R , where R displaystyle mathbb R is the set of all reals. 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
R n displaystyle mathbb R ^ n .
Generalizations[edit]
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 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).
Notations and conventions[edit]
The Cartesian coordinates of a point are usually written in
parentheses and separated by commas, as in (10, 5) or (3, 5, 7). 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 were 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 (x1, x2, ..., xn) for the n coordinates in an n-dimensional space,
especially when n is greater than 3 or unspecified. Some authors
prefer the numbering (x0, x1, ..., xn−1). These notations are
especially advantageous in computer programming: by storing the
coordinates of a point as an array, 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) is measured
along a horizontal axis, oriented from left to right. The second
coordinate (the ordinate) is then measured along a vertical axis,
usually oriented from bottom to top. Young children learning the
Cartesian system, commonly learn the order to read the values before
cementing the x, y, 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).[6]
The four quadrants of a Cartesian coordinate system The axes of a two-dimensional Cartesian system divide the plane into
four infinite regions, called quadrants,[5] each bounded by two
half-axes. These are often numbered from 1st to 4th and denoted by
Roman numerals: 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,[5] 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, and a similar naming system applies.
Cartesian formulae for the plane[edit]
Distance between two points[edit]
The
( x 1 , y 1 ) displaystyle (x_ 1 ,y_ 1 ) and ( x 2 , y 2 ) displaystyle (x_ 2 ,y_ 2 ) is d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . displaystyle d= sqrt (x_ 2 -x_ 1 )^ 2 +(y_ 2 -y_ 1 )^ 2 . This is the Cartesian version of Pythagoras's theorem. In three-dimensional space, the distance between points ( x 1 , y 1 , z 1 ) displaystyle (x_ 1 ,y_ 1 ,z_ 1 ) and ( x 2 , y 2 , z 2 ) displaystyle (x_ 2 ,y_ 2 ,z_ 2 ) is d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 + ( z 2 − z 1 ) 2 , displaystyle d= sqrt (x_ 2 -x_ 1 )^ 2 +(y_ 2 -y_ 1 )^ 2 +(z_ 2 -z_ 1 )^ 2 , which can be obtained by two consecutive applications of Pythagoras'
theorem.[7]
Euclidean transformations[edit]
The Euclidean transformations or Euclidean motions are the (bijective)
mappings of points of the
( x ′ , y ′ ) = ( x + a , y + b ) . displaystyle (x',y')=(x+a,y+b). Rotation[edit] To rotate a figure counterclockwise around the origin by some angle θ displaystyle theta is equivalent to replacing every point with coordinates (x,y) by the point with coordinates (x',y'), where x ′ = x cos θ − y sin θ displaystyle x'=xcos theta -ysin theta y ′ = x sin θ + y cos θ . displaystyle y'=xsin theta +ycos theta . Thus: ( x ′ , y ′ ) = ( ( x cos θ − y sin θ ) , ( x sin θ + y cos θ ) ) . displaystyle (x',y')=((xcos theta -ysin theta ,),(xsin theta +ycos theta ,)). Reflection[edit] If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection 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 θ displaystyle theta with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where x ′ = x cos 2 θ + y sin 2 θ displaystyle x'=xcos 2theta +ysin 2theta y ′ = x sin 2 θ − y cos 2 θ . displaystyle y'=xsin 2theta -ycos 2theta . Thus: ( x ′ , y ′ ) = ( ( x cos 2 θ + y sin 2 θ ) , ( x sin 2 θ − y cos 2 θ ) ) . displaystyle (x',y')=((xcos 2theta +ysin 2theta ,),(xsin 2theta -ycos 2theta ,)). Glide reflection[edit] 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[edit] These Euclidean transformations of the plane can all be described in a uniform way by using matrices. The result ( x ′ , y ′ ) displaystyle (x',y') of applying a
( x , y ) displaystyle (x,y) is given by the formula ( x ′ , y ′ ) = ( x , y ) A + b displaystyle (x',y')=(x,y)A+b where A is a 2×2 orthogonal matrix and b = (b1, b2) is an arbitrary ordered pair of numbers;[9] that is, x ′ = x A 11 + y A 21 + b 1 displaystyle x'=xA_ 11 +yA_ 21 +b_ 1 y ′ = x A 12 + y A 22 + b 2 , displaystyle y'=xA_ 12 +yA_ 22 +b_ 2 , where A = ( A 11 A 12 A 21 A 22 ) . displaystyle A= begin pmatrix A_ 11 &A_ 12 \A_ 21 &A_ 22 end pmatrix . [Note the use of row vectors for point coordinates and that the matrix is written on the right.] To be orthogonal, the matrix A must have orthogonal rows with same Euclidean length of one, that is, A 11 A 21 + A 12 A 22 = 0 displaystyle A_ 11 A_ 21 +A_ 12 A_ 22 =0 and A 11 2 + A 12 2 = A 21 2 + A 22 2 = 1. displaystyle A_ 11 ^ 2 +A_ 12 ^ 2 =A_ 21 ^ 2 +A_ 22 ^ 2 =1. This is equivalent to saying that A times its transpose must be the identity matrix. If these conditions do not hold, the formula describes a more general affine transformation of the plane provided that the determinant of A is not zero. The formula defines a translation if and only if A is the identity matrix. The transformation is a rotation around some point if and only if A is a rotation matrix, meaning that A 11 A 22 − A 21 A 12 = 1. displaystyle A_ 11 A_ 22 -A_ 21 A_ 12 =1. A reflection or glide reflection is obtained when, A 11 A 22 − A 21 A 12 = − 1. displaystyle A_ 11 A_ 22 -A_ 21 A_ 12 =-1. Assuming that translation is not used transformations can be combined by simply multiplying the associated transformation matrices. Affine transformation[edit] Another way to represent coordinate transformations in Cartesian coordinates is through affine transformations. 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: ( A 11 A 21 b 1 A 12 A 22 b 2 0 0 1 ) ( x y 1 ) = ( x ′ y ′ 1 ) . displaystyle begin pmatrix A_ 11 &A_ 21 &b_ 1 \A_ 12 &A_ 22 &b_ 2 \0&0&1end pmatrix begin pmatrix x\y\1end pmatrix = begin pmatrix x'\y'\1end pmatrix . [Note the matrix A from above was transposed. The matrix is on the left and column vectors for point coordinates are used.] Using affine transformations multiple different euclidean transformations including translation can be combined by simply multiplying the corresponding matrices. Scaling[edit] 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 (x, y) 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 ) . displaystyle (x',y')=(mx,my). If m is greater than 1, the figure becomes larger; if m is between 0 and 1, it becomes smaller. Shearing[edit] 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 ) displaystyle (x',y')=(x+ys,y) Shearing can also be applied vertically: ( x ′ , y ′ ) = ( x , x s + y ) displaystyle (x',y')=(x,xs+y) Orientation and handedness[edit] Main article: orientation (mathematics) See also: right-hand rule and axes conventions In two dimensions[edit] The right-hand rule Fixing or choosing the x-axis determines the y-axis up to direction. Namely, the y-axis is necessarily the perpendicular to the x-axis through the point marked 0 on the x-axis. But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative. Each of these two choices determines a different orientation (also called handedness) of the Cartesian plane. The usual way of orienting the axes, 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. 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 axes 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 axes, 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[edit] Fig. 7 – The left-handed orientation is shown on the left, and the right-handed on the right. Fig. 8 – The right-handed
Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on 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. 3D Cartesian coordinate handedness The name derives from the right-hand rule. If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative directions 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 cube and a concave "corner". This corresponds to the two possible orientations of the coordinate system. 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[edit]
A point in space in a
r displaystyle mathbf r . In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as: r = x i + y j displaystyle mathbf r =xmathbf i +ymathbf j where i = ( 1 0 ) displaystyle mathbf i = begin pmatrix 1\0end pmatrix , and j = ( 0 1 ) displaystyle mathbf j = begin pmatrix 0\1end pmatrix are unit vectors in the direction of the x-axis and y-axis respectively, generally referred to as the standard basis (in some application areas these may also be referred to as versors). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates ( x , y , z ) displaystyle (x,y,z) can be written as:[11] r = x i + y j + z k displaystyle mathbf r =xmathbf i +ymathbf j +zmathbf k where k = ( 0 0 1 ) displaystyle mathbf k = begin pmatrix 0\0\1end pmatrix is the unit vector in the direction of the z-axis. 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 (x, y) with the complex number z = x + iy. Here, i is the imaginary unit and is identified with the point with coordinates (0, 1), 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[edit] 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 equalling 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 directions? The axis of Earth's spin provides a natural direction strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the prime meridian stands out as a reference direction, so the X-axis takes the direction from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions, and two perpendicular axes directions 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? Using [ longitude = −73.985656, latitude = 40.748433 ], Earth radius = 40,000/2π, and transforming from spherical --> 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' ingenious thinking. While spatial apps employ identical units along all axes, in business and scientific apps, 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 (x, y), where y = f(x) is the graph of the function f. For a function g of two variables, the set of all points (x, y, z), where z = g(x, y) 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[edit] Horizontal and vertical
Jones diagram, which plots four variables rather than two
References[edit] ^ Bix, Robert A.; D'Souza, Harry J. "Analytic geometry". Encyclopædia
Britannica. Retrieved 2017-08-06.
^ Burton 2011, p. 374
^ A Tour of the Calculus, David Berlinski
^ Axler, Sheldon. Linear
Sources[edit] Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998), Geometry, Cambridge: Cambridge University Press, ISBN 0-521-59787-0 Burton, David M. (2011), The History of Mathematics/An Introduction (7th ed.), New York: McGraw-Hill, ISBN 978-0-07-338315-6 Smart, James R. (1998), Modern Geometries (5th ed.), Pacific Grove: Brooks/Cole, ISBN 0-534-35188-3 Further reading[edit] Descartes, René (2001). Discourse on Method, Optics, Geometry, and
Meteorology. Translated by Paul J. Oscamp (Revised ed.). Indianapolis,
IN: Hackett Publishing. ISBN 0-87220-567-3.
OCLC 488633510.
Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and
Engineers (1st ed.). New York: McGraw-Hill. pp. 55–79.
LCCN 59-14456. OCLC 19959906.
Margenau H, Murphy GM (1956). The Mathematics of
External links[edit] Cartesian Coordinate System "Cartesian coordinates". PlanetMath. MathWorld description of Cartesian coordinates Coordinate Converter – converts between polar, Cartesian and spherical coordinates Coordinates of a point Interactive tool to explore coordinates of a point open source JavaScript class for 2D/3D Cartesian coordinate system manipulation v t e
Two dimensional Cartesian coordinate system Polar coordinate system Parabolic coordinate system Bipolar coordinates Elliptic coordinates Three dimensional Cartesian coordinate system Cylindrical coordinate system Spherical coordinate system Parabolic cylindrical coordinates Paraboloidal coordinates Oblate spheroidal coordinates Prolate spheroidal coordinates Ellipsoidal coordinates Elliptic cylindrical coordinates Toroidal coordinates Bispherical coordinates Bipolar cylindrical coordinates Conical coordinates 6-sphere coordinates Flat-ring cyclide coordinates Flat-disk cyclide coordinates Bi-cyclide coordinates Cap-cyclide coordinates Concave bi-sinusoidal single-centered coordinates Concave bi-sinusoidal double-centered coordinates Convex inverted-sinusoidal spherically aligned coordinates Quasi-random-intersection car |