Cartesian coordinate system

A Cartesian coordinate system (UK: /kɑːˈtiːzjən/, US: /kɑːrˈtiʒən/) is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0). 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 hyperplanes.

Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (xa)2 + (yb)2 = r2 where a and b are the coordinates of the center (a, b) and r is the radius.

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. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory 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, 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.

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.


The adjective Cartesian refers to the French mathematician and philosopher René Descartes, who published this idea in 1637. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery.[1] The French cleric Nicole 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. 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' work.[3]

The development of the Cartesian coordinate system would play a fundamental role in the development of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz.[4] The two-coordinate description of the plane was later generalized into the concept of vector spaces.[5]

Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space.


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[6]) is defined by an ordered pair of perpendicular lines (axes), a single unit 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 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 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 (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 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 with a chosen Cartesian coordinate system is called a Cartesian plane. 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 (0, 0) and (1, 1)), the unit hyperbola, and so on.

The two axes divide the plane into four right angles, called quadrant. 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 (x, y), then its distances from the X-axis and from the Y-axis are |y| and |x|, respectively; where |...| denotes the absolute value of a number.

Three dimensions

Coord system CA 0
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).

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; a 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 plane through P perpendicular to each coordinate axis, and interprets the point where that plane 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 plane defined by the other two axes, with the sign determined by the orientation of the corresponding axis.

Each pair of axes defines a coordinate plane. These planes divide space into eight trihedra, called octants.

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).

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 planes 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 (0, 0, 1); a convention that is commonly called the right hand rule.

Cartesian coordinate surfaces
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

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 , where 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 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 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

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-, and z-axis concepts, by starting with 2D mnemonics (e.g. 'Walk along the hall then up the stairs' akin to straight across the x-axis then up vertically along the y-axis).[7]

Computer graphics and image processing, 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 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, unless specifically stated otherwise. All laws of physics and math assume this right-handedness, 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.[6]

Quadrants and octants

Cartesian coordinates 2D
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,[6] 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,[6] 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

Distance between two points

The Euclidean distance between two points of the plane with Cartesian coordinates and is

This is the Cartesian version of Pythagoras's theorem. In three-dimensional space, the distance between points and is

which can be obtained by two consecutive applications of Pythagoras' theorem.[8]

Euclidean transformations

The Euclidean transformations or Euclidean motions are the (bijective) mappings of points of the Euclidean plane to themselves which preserve distances between points. There are four types of these mappings (also called isometries): translations, rotations, reflections and glide reflections.[9]


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


To rotate a figure counterclockwise 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 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 with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where


Glide reflection

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

General matrix form of the transformations

These Euclidean transformations of the plane can all be described in a uniform way by using matrices. The result of applying a Euclidean transformation to a point is given by the formula

where A is a 2×2 orthogonal matrix and b = (b1, b2) is an arbitrary ordered pair of numbers;[10] that is,


[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,


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 reflection or glide reflection is obtained when,

Assuming that translation is not used transformations can be combined by simply multiplying the associated transformation matrices.

Affine transformation

Another way to represent coordinate transformations in Cartesian coordinates is through affine 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:

[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.


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

If m is greater than 1, the figure becomes larger; if m is between 0 and 1, it becomes smaller.


A shearing transformation will push the top of a square sideways to form a parallelogram. Horizontal shearing is defined by:

Shearing can also be applied vertically:

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 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. Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the x-axis to the y-axis, in a positively oriented coordinate system.

The other way of orienting the plane is following the left hand rule, placing the left hand on the plane with the thumb pointing up.

When pointing the thumb away from the origin along an axis towards positive, the curvature of the fingers indicates a positive rotation along that axis.

Regardless of the rule used to orient the plane, rotating the coordinate system will preserve the orientation. Switching any two axes will reverse the orientation, but switching both will leave the orientation unchanged.

In three dimensions

Cartesian coordinate system handedness
Fig. 7 – The left-handed orientation is shown on the left, and the right-handed on the right.
Right hand cartesian
Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes.

Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible orientation for this line. The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.

3D Cartesian Coodinate Handedness
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 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 cube and a concave "corner". This corresponds to the two possible orientations of the space. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

Representing a vector in the standard basis

A point in space in a Cartesian coordinate system may also be represented by a position vector, which can be thought of as an arrow pointing from the origin of the coordinate system to the point.[11] 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 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 can be written as:[12]

where 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.


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? The axis of Earth's spin provides a natural orientation strongly associated with "up vs. down", so positive Z can adopt the direction from geocenter to North Pole. A location on the Equator is needed to define the X-axis, and the prime meridian stands out as a reference orientation, so the X-axis takes the orientation from geocenter out to [ 0 degrees longitude, 0 degrees latitude ]. Note that with 3 dimensions, and two perpendicular axes orientations pinned down for X and Z, the Y-axis is determined by the first two choices. In order to obey the right-hand rule, the Y-axis must point out from the geocenter to [ 90 degrees longitude, 0 degrees latitude ]. So what are the geocentric coordinates of the Empire State Building in New York City? 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


  1. ^ Bix, Robert A.; D'Souza, Harry J. "Analytic geometry". Encyclopædia Britannica. Retrieved 2017-08-06.
  2. ^ Kent, Alexander J.; Vujakovic, Peter (2017-10-04). The Routledge Handbook of Mapping and Cartography. Routledge. ISBN 9781317568216.
  3. ^ Burton 2011, p. 374
  4. ^ A Tour of the Calculus, David Berlinski
  5. ^ Axler, Sheldon (2015). Linear Algebra Done Right - Springer. Undergraduate Texts in Mathematics. p. 1. doi:10.1007/978-3-319-11080-6. ISBN 978-3-319-11079-0.
  6. ^ a b c d "Cartesian orthogonal coordinate system". Encyclopedia of Mathematics. Retrieved 2017-08-06.
  7. ^ "Charts and Graphs: Choosing the Right Format". Retrieved 2017-08-29.
  8. ^ Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2013). Calculus : Single and Multivariable (6 ed.). John wiley. ISBN 978-0470-88861-2.
  9. ^ Smart 1998, Chap. 2
  10. ^ Brannan, Esplen & Gray 1998, pg. 49
  11. ^ Brannan, Esplen & Gray 1998, Appendix 2, pp. 377–382
  12. ^ David J. Griffiths (1999). Introduction to Electrodynamics. Prentice Hall. ISBN 978-0-13-805326-0.


  • Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998), Geometry, Cambridge: Cambridge University Press, ISBN 978-0-521-59787-6
  • 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 978-0-534-35188-5

Further reading

  • Descartes, René (2001). Discourse on Method, Optics, Geometry, and Meteorology. Translated by Paul J. Oscamp (Revised ed.). Indianapolis, IN: Hackett Publishing. ISBN 978-0-87220-567-3. 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 Physics and Chemistry. New York: D. van Nostrand. LCCN 55-10911.
  • 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). ISBN 978-0-387-18430-2.
  • Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. ISBN 978-0-07-043316-8. LCCN 52-11515.
  • Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. LCCN 67-25285.

External links

Abscissa and ordinate

In mathematics, the abscissa (/æbˈsɪs.ə/; plural abscissae or abscissæ or abscissas) and the ordinate are respectively the first and second coordinate of a point in a coordinate system.

The abscissa of a point is the signed measure of its projection on the primary axis, whose absolute value is the distance between the projection and the origin of the axis, and whose sign is given by the location on the projection relative to the origin (before: negative; after: positive).

The ordinate of a point is the signed measure of its projection on the secondary axis, whose absolute value is the distance between the projection and the origin of the axis, and whose sign is given by the location on the projection relative to the origin (before: negative; after: positive).

Usually these are the horizontal and vertical coordinates of a point in a two-dimensional rectangular Cartesian coordinate system. The terms can also refer to the horizontal and vertical axes respectively (typically x-axis and y–axis) of a two-dimensional graph. An ordered pair consists of two terms—the abscissa (horizontal, usually x) and the ordinate (vertical, usually y)—which define the location of a point in two-dimensional rectangular space.

Bipolar cylindrical coordinates

Bipolar cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional bipolar coordinate system in the perpendicular -direction. The two lines of foci and of the projected Apollonian circles are generally taken to be defined by and , respectively, (and by ) in the Cartesian coordinate system.

The term "bipolar" is often used to describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term bipolar coordinates is never used to describe coordinates associated with those curves, e.g., elliptic coordinates.

Cantor–Dedekind axiom

In mathematical logic, the Cantor–Dedekind axiom is the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one-to-one correspondence between real numbers and points on a line.

This axiom is the cornerstone of analytic geometry. The Cartesian coordinate system developed by René Descartes implicitly assumes this axiom by blending the distinct concepts of real number system with the geometric line or plane into a conceptual metaphor. This is sometimes referred to as the real number line blend.A consequence of this axiom is that Alfred Tarski's proof of the decidability of first-order theories of the real numbers could be seen as an algorithm to solve any first-order problem in Euclidean geometry.

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 significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.

Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or value for each member of the domain. Conversely, the set of values the function takes on as output is termed the image of the function, which is sometimes also referred to as the range of the function.

For instance, the domain of cosine is the set of all real numbers, while the domain of the square root consists only of numbers greater than or equal to 0 (ignoring complex numbers in both cases).

If the domain of a function is a subset of the real numbers and the function is represented in a Cartesian coordinate system, then the domain is represented on the x-axis.

Fundamental plane (spherical coordinates)

The fundamental plane in a spherical coordinate system is a plane of reference that divides the sphere into two hemispheres. The latitude of a point is then the angle between the fundamental plane and the line joining the point to the centre of the sphere.For a geographic coordinate system of the Earth, the fundamental plane is the Equator. Celestial coordinate systems have varying fundamental planes:

The horizontal coordinate system uses the observer's horizon.

The Besselian coordinate system uses Earth's terminator (day/night boundary). This is a Cartesian coordinate system (x, y, z).

The equatorial coordinate system uses the celestial equator.

The ecliptic coordinate system uses the ecliptic.

The galactic coordinate system uses the Milky Way's galactic equator.

Identity line

In a 2-dimensional Cartesian coordinate system, with x representing the abscissa and y the ordinate, the identity line or line of equality is the y = x line. The line, sometimes called the 1:1 line, has a slope of 1. When the abscissa and ordinate are on the same scale, the identity line forms a 45° angle with the abscissa, and is thus also, informally, called the 45° line. The line is often used as a reference in a 2-dimensional scatter plot comparing two sets of data expected to be identical under ideal conditions. When the corresponding data points from the two data sets are equal to each other, the corresponding scatters fall exactly on the identity line.In economics, an identity line is used in the Keynesian cross diagram to identify equilibrium, as only on the identity line does aggregate demand equal aggregate supply.

Minimum bounding box

In geometry, the minimum or smallest bounding or enclosing box for a point set (S) in N dimensions is the box with the smallest measure (area, volume, or hypervolume in higher dimensions) within which all the points lie. When other kinds of measure are used, the minimum box is usually called accordingly, e.g., "minimum-perimeter bounding box".

The minimum bounding box of a point set is the same as the minimum bounding box of its convex hull, a fact which may be used heuristically to speed up computation.The term "box"/"hyperrectangle" comes from its usage in the Cartesian coordinate system, where it is indeed visualized as a rectangle (two-dimensional case), rectangular parallelepiped (three-dimensional case), etc.

In the two-dimensional case it is called the minimum bounding rectangle.

Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space.

In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry.

Quadrant (plane geometry)

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 signs of the (x;y) 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 ("northeast") quadrant.

Right conoid

In geometry, a right conoid is a ruled surface generated by a family of straight lines that all intersect perpendicularly to a fixed straight line, called the axis of the right conoid.

Using a Cartesian coordinate system in three-dimensional space, if we take the z-axis to be the axis of a right conoid, then the right conoid can be represented by the parametric equations:

where h(u) is some function for representing the height of the moving line.

Rotation of axes

In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x'y'-Cartesian coordinate system in which the origin is kept fixed and the x' and y' axes are obtained by rotating the x and y axes counterclockwise through an angle . A point P has coordinates (x, y) with respect to the original system and coordinates (x', y') with respect to the new system. In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly. A rotation of axes is a linear map and a rigid transformation.

SK-42 reference system

The SK-42 reference system also known as the Krasovsky 1940 ellipsoid, is a coordinate system established in the Soviet Union in 1942 as Systema koordinat (Russian: Система координат 1942 года), and provides parameters which are linked to the geocentric Cartesian coordinate system PZ-90 . It was used in geodetic calculations, notably in military mapping and determining state borders.

The coordinate system SK-42 served as a foundation for developing the SK-63 reference system which was created and used primarily for civilian and industrial development purposes..

The Krasovsky 1940 ellipsoid uses a semi-major axis (equatorial radius) a of 6,378,245 m, and an inverse flattening 1/f of 298.3

Translation of axes

In mathematics, a translation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x'y'-Cartesian coordinate system in which the x' axis is parallel to the x axis and k units away, and the y' axis is parallel to the y axis and h units away. This means that the origin O' of the new coordinate system has coordinates (h, k) in the original system. The positive x' and y' directions are taken to be the same as the positive x and y directions. A point P has coordinates (x, y) with respect to the original system and coordinates (x', y') with respect to the new system, where

or equivalently

In the new coordinate system, the point P will appear to have been translated in the opposite direction. For example, if the xy-system is translated a distance h to the right and a distance k upward, then P will appear to have been translated a distance h to the left and a distance k downward in the x'y'-system . A translation of axes in more than two dimensions is defined similarly. A translation of axes is a rigid transformation, but not a linear map. (See Affine transformation.)

Unit circle

In mathematics, a unit circle is a circle with unit radius. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1; the generalization to higher dimensions is the unit sphere.

If (x, y) is a point on the unit circle's circumference, then |x| and |y| are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation

Since x2 = (−x)2 for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant.

The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk.

One may also use other notions of "distance" to define other "unit circles", such as the Riemannian circle; see the article on mathematical norms for additional examples.

Versor (physics)

In geometry and physics, the versor of an axis or of a vector is a unit vector indicating its direction.

The versor of a Cartesian axis is also known as a standard basis vector. The versor of a vector is also known as a normalized vector.

Vertical and horizontal

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 horizontal if it is perpendicular to the vertical direction.

The usage of the inter-related terms horizontal and vertical as well as their symmetries and asymmetries vary with context (e.g. two vs. three dimensions).

In general, something that is vertical can be drawn from up to down (or down to up), such as the y-axis in the Cartesian coordinate system.

Vertical translation

In geometry, a vertical translation (also known as vertical shift) is a translation of a geometric object in a direction parallel to the vertical axis of the Cartesian coordinate system.

Often, vertical translations are considered for the graph of a function. If f is any function of x, then the graph of the function f(x) + c (whose values are given by adding a constant c to the values of f) may be obtained by a vertical translation of the graph of f(x) by distance c. For this reason the function f(x) + c is sometimes called a vertical translate of f(x). For instance, the antiderivatives of a function all differ from each other by a constant of integration and are therefore vertical translates of each other.


WIRIS is a set of proprietary HTML-based JavaScript tools which can author and edit mathematical formulas, execute mathematical problems and show mathematical graphics on the Cartesian coordinate system. WIRIS equation editor is a native browser application, with a light server-side, that supports both MathML and LaTeX.

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