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Encyclopedia > Cartesian coordinate system
Fig. 1 - Cartesian coordinate system. Four points are marked: (2,3) in green, (-3,1) in red, (-1.5,-2.5) in blue and (0,0), the origin, in violet.

In mathematics, the Cartesian coordinate system is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point. To define the coordinates, two perpendicular directed lines (the x-axis or abscissa and the y-axis or ordinate), are specified, as well as the unit length, which is marked off on the two axes (see Figure 1). Cartesian coordinate systems are also used in space (where three coordinates are used) and in higher dimensions. Image File history File links Cartesian-coordinate-system. ... Image File history File links Cartesian-coordinate-system. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ... A number is an abstract idea used in counting and measuring. ... Fig. ... In mathematics, a directed set is a set A together with a binary relation &#8804; having the following properties: a &#8804; a for all a in A (reflexivity) if a &#8804; b and b &#8804; c, then a &#8804; c (transitivity) for any two a and b in A, there... Attempting to understand the nature of space has always been a prime occupation for philosophers and scientists. ... Higher dimension in mathematics refers to any number of dimensions greater than three. ...

Using the Cartesian coordinate system geometric shapes (such as curves) can be described by algebraic equations, namely equations satisfied by the coordinates of the points lying on the shape. For example, the circle of radius 2 may be described by the equation x² + y² = 4 (see Figure 2). Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... This article is about equations in mathematics. ...

Cartesian means relating to the French mathematician and philosopher René Descartes (Latin: Cartesius), who, among other things, worked to merge algebra and Euclidean geometry. This work was influential in the development of analytic geometry, calculus, and cartography. Leonhard Euler, one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... A philosopher is a person who thinks deeply regarding people, society, the world, and/or the universe. ... RenÃ© Descartes (French IPA: ) (March 31, 1596 â€“ February 11, 1650), also known as Renatus Cartesius (latinized form), was a highly influential French philosopher, mathematician, scientist, and writer. ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... Calculus (from Latin, pebble or little stone) is a major area in mathematics where infinitesimal data yields global information. ... Cartography or mapmaking (in Greek chartis = map and graphein = write) is the study, practice, science and art of making maps or globes. ...

The idea of this system was developed in 1637 in two writings by Descartes. In part two of his Discourse on Method Descartes introduces the new idea of specifying the position of a point or object on a surface, using two intersecting axes as measuring guides. In La Géométrie, he further explores the above-mentioned concepts. Events February 3 - Tulipmania collapses in Netherlands by government order February 15 - Ferdinand III becomes Holy Roman Emperor December 17 - Shimabara Rebellion erupts in Japan Pierre de Fermat makes a marginal claim to have proof of what would become known as Fermats last theorem. ... Illustration of a scribe writing Writing, in its most common sense, is the preservation and the preserved text on a medium, with the use of signs or symbols. ... The Discourse on Method is a philosophical and mathematical treatise published by RenÃ© Descartes in 1637. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... La GÃ©omÃ©trie was published in 1637 and written by RenÃ© Descartes. ...

See coordinates (mathematics) for other commonly used coordinate systems such as polar coordinates and coordinate systems for usage of the term in advanced mathematics. This article describes some of the common coordinate systems that appear in elementary mathematics. ... A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...

Fig. 2 - Cartesian coordinate system with the circle of radius 2 centered at the origin marked in red. The equation of the circle is x² + y² = 4.

Image File history File links Cartesian-coordinate-system-with-circle. ... Image File history File links Cartesian-coordinate-system-with-circle. ...

Fig. 3 - The four quadrants of a Cartesian coordinate system. The arrows on the axes indicate that they extend forever in their respective directions (i.e. infinitely).
Fig. 4 - Three dimensional Cartesian coordinate system with y-axis pointing away from the observer.
Fig. 5 - Three dimensional Cartesian coordinate system with the x-axis pointing towards the observer.

The point of intersection, where the axes meet, is called the origin normally labeled O. The x and y axes define a plane that is referred to as the xy plane. Given each axis, choose a unit length, and mark off each unit along the axis, forming a grid. To specify a particular point on a two dimensional coordinate system, indicate the x unit first (abscissa), followed by the y unit (ordinate) in the form (x,y), an ordered pair.

The choice of letters comes from a convention, to use the latter part of the alphabet to indicate unknown values. In contrast, the first part of the alphabet was used to designate known values.

An example of a point P on the system is indicated in Figure 3, using the coordinate (3,5). A spatial point is an entity with a location in space but no extent (volume, area or length). ...

The intersection of the two axes creates four regions, called quadrants, indicated by the Roman numerals I (+,+), II (−,+), III (−,−), and IV (+,−). Conventionally, the quadrants are labeled counter-clockwise starting from the upper right ("northeast") quadrant. In the first quadrant, both coordinates are positive, in the second quadrant x-coordinates are negative and y-coordinates positive, in the third quadrant both coordinates are negative and in the fourth quadrant, x-coordinates are positive and y-coordinates negative (see table below.)

## Three-dimensional coordinate system

The three dimensional coordinate system provides the three physical dimensions of space — length, width, and height. Figures 4 and 5, below, show two common ways of representing the three-dimensional coordinate system.

The coordinates in a three dimensional system are of the form (x, y,z). As an example, two points are plotted in this system in Figure 4, points P(3,0,5) and Q(−5,−5,7). The axes are depicted in a world-coordinates orientation with the z-axis pointing up. A spatial point is an entity with a location in space but no extent (volume, area or length). ...

The x-, y-, and z-coordinates of a point can also be taken as the distances from the yz-plane, xz-plane, and xy-plane respectively. Figure 5 shows the distances of point P from the planes.

The xy-, yz-, and xz-planes divide the three-dimensional space into eight subdivisions known as octants, similar to the quadrants of 2D space. While conventions have been established for the labeling of the four quadrants of the x-y plane, only the first octant of three dimensional space is labeled. It contains all of the points whose x, y, and z coordinates are positive. Can refer to a region of Euclidean 3-space with a specific sign for x, y and z coordinates. ...

## Orientation and handedness

In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ... The left-handed orientation is shown on the left, and the right-handed on the right. ...

### 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. Fig. ...

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.

Regardless of the rule used to orient the axes, rotating the coordinate system will preserve the orientation. Switching the role of x and y will reverse the orientation.

### In three dimensions

Fig. 6 - A rendition of the octants in the three-dimensional coordinate system
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 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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links Cartesian_coordinate_system_handedness. ... Image File history File links Cartesian_coordinate_system_handedness. ... Image File history File links Right_hand_cartesian. ... Image File history File links Right_hand_cartesian. ... A representation of one line Three lines â€” the red and blue lines have same slope, while the red and green ones have same y-intercept. ...

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. The left-handed orientation is shown on the left, and the right-handed on the right. ... The Index finger The index finger, pointer finger or forefinger is the second digit of a human hand, located between the thumb and the middle finger. ... This article is about the vulgar gesture. ... // This digit is one of the five fingers (though the word finger can also refer exclusively to the non-thumb digits). ...

Different disciplines use different variations of the coordinate systems. For example, mathematicians typically use a right-handed coordinate system with the y-axis pointing up, while engineers typically use a left-handed coordinate system with the z-axis pointing up. This has the potential to lead to confusion when engineers and mathematicians work on the same project.

Figure 7 is an attempt at depicting 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.

## In physics

The above discussion applies to Cartesian coordinate systems in mathematics, where it is common to not use any units of measurement. In physics it is important to note that a dimension is simply a measure of something, and that, for each class of features to be measured, another dimension can be added. Attachment to visualizing the dimensions precludes understanding the many different dimensions that can be measured (time, mass, color, cost, etc.). Multi-dimensional objects can be calculated and manipulated algebraically.

## Representation as a Vector

A a point in space in a Cartesian coordinate system may also be represented by a vector; which can be thought of as an arrow pointing from the origin of the coordinate system to the point. If the coordinates represent spatial positions (displacements) it is common to represent the vector from the origin to the point of interest as $vec r$. In Cartesian coordinates the vector from the origin to the point (x,y,z) can be written as[1]: Look up vector in Wiktionary, the free dictionary. ...

$vec r = x hatimath + y hatjmath + z hat k$

Where $hatimath$, $hatjmath$, and are unit vectors that point the same direction as the x, y, and z axes, respectively. In mathematics, a unit vector in a normed vector space is a vector (most commonly a spatial vector) whose length is 1. ...

## Further notes

In analytic geometry the Cartesian coordinate system is the foundation for the algebraic manipulation of geometrical shapes. Many other coordinate systems have been developed since Descartes. One common set of systems use polar coordinates; astronomers often use spherical coordinates, a type of polar coordinate system. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...

It may be interesting to note that some have indicated that the master artists of the Renaissance used a grid, in the form of a wire mesh, as a tool for breaking up the component parts of their subjects they painted. That this may have influenced Descartes is merely speculative.[citation needed] (See perspective, projective geometry.) The Renaissance (French for rebirth, or Rinascimento in Italian), was a cultural movement in Italy (and in Europe in general) that began in the late Middle Ages, and spanned roughly the 14th through the 17th century. ... A cube in two-point perspective. ... Projective geometry is a non-metrical form of geometry. ...

This is a list of canonical coordinate transformations. ... relation graph theory In mathematics, the graph of a function f is the collection of all ordered pairs (x,f(x)). In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc. ... Point plotting is an elementary skill required in analytic geometry. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... A representation of one line Three lines â€” the red and blue lines have same slope, while the red and green ones have same y-intercept. ... Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... The left-handed orientation is shown on the left, and the right-handed on the right. ... In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ... Example of regular grid. ... In mathematics, The n-dimensional integer lattice, or cubic lattice, denoted Zn, is the lattice in the standard n-dimensional real inner product space, where the inner product is the dot product, and where the lattice points are n-tuples of integers. ...

### Other coordinate systems

This article describes some of the common coordinate systems that appear in elementary mathematics. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ... In mathematics, orthogonal coordinates are defined as a set of coordinates that have no off-diagonal terms in their metric tensor, i. ... Manhattan versus Euclidean distance: The red, blue, and yellow lines representing the Manhattan distance all have the same length (12), whereas the green line representing the Euclidian distance has length 6Ã—âˆš2 â‰ˆ 8. ... Curvilinear coordinates are a coordinate system based on some transformation of the standard coordinate system. ... In mathematics, hyperbolic coordinates are a useful method of locating points in Quadrant I of the Cartesian plane {(x,y) : x > 0, y > 0} = Q. Hyperbolic coordinates take values in HP = {(u,v) : u &#8712; R, v > 0 }. For (x,y) in Q take u = &#8722;1/2 log(y... Stereographic projection of a circle of radius R onto the x axis. ... The multidimensional system of Parallel coordinates [1] is a common way of studying high-dimensional geometry and visualizing multivariate problems in applications such as statistics. ... Coordinates centered on the earths system. ...

### History

RenÃ© Descartes (French IPA: ) (March 31, 1596 â€“ February 11, 1650), also known as Renatus Cartesius (latinized form), was a highly influential French philosopher, mathematician, scientist, and writer. ... The Discourse on Method is a philosophical and mathematical treatise published by RenÃ© Descartes in 1637. ... La GÃ©omÃ©trie was published in 1637 and written by RenÃ© Descartes. ...

### Examples

Circle illustration This article is about the shape and mathematical concept of circle. ... Illustration of a unit circle. ... For other uses, see Ellipse (disambiguation). ... The derivation of the cartesian form for an ellipse is simple and instructive. ... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... A lemniscate In mathematics, a lemniscate is a type of curve described by a Cartesian equation of the form: Graphing this equation produces a curve similar to . ... A hyperbolic spiral is a transcendental plane curve also known as a reciprocal spiral. ... The kappa curve has two vertical asymptotes. ... A Spring A left-handed and a right-handed spring. ... A superellipse is a geometrical figure which in a cartesian coordinate system can be described as the set of all points (x, y) with where and and are the radii of the oval shape. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ... 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ...

### Context

In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications. ... An address is a code and abstract concept expressing the fixed location of a home, business or other building on the earths surface. ... Notation systems are an interpreted system of tokens having a syntax and a semantics. ... ISO 31-1 is the part of international standard ISO 31 that defines names and symbols for quantities and units related to space and time. ... Mathematics and architecture have always enjoyed a close association with each other, not only in the sense that the latter is informed by the former, but also in that both share the search for order and beauty, the former in nature and the latter in buildings. ... Graph paper or quad-ruled paper is writing paper that is printed with fine lines making up a regular grid. ...

## References

Descartes, René. Oscamp, Paul J. (trans). Discourse on Method, Optics, Geometry, and Meteorology. 2001. Year 2001 (MMI) was a common year starting on Monday of the Gregorian calendar. ...

1. ^ David J. Griffith (1999). Introduction to Electromagnetics. Prentice Hall. ISBN 0-13-805326-X.

Year 1999 (MCMXCIX) was a common year starting on Friday (link will display full 1999 Gregorian calendar). ...

Results from FactBites:

 Coordinate system - Wikipedia, the free encyclopedia (1119 words) The system of assigning longitude and latitude to geographical locations is a coordinate system. Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves. Cylindrical coordinate system represents a point in space by an angle, a distance from the origin and a height.
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