Cartesian coordinate system
From Wikipedia, the free encyclopedia
In mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the xcoordinate or abscissa and the ycoordinate or ordinate of the point. To define the coordinates, two perpendicular directed lines (the xaxis, and the yaxis), 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.
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 x^{2} + y^{2} = 4 (see Figure 2).
Contents[hide] 
[edit] History
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.
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 abovementioned concepts.
[edit] Twodimensional coordinate system
A Cartesian coordinate system in two dimensions is commonly defined by two axes, at right angles to each other, forming a plane (an xyplane). The horizontal axis is normally labeled x, and the vertical axis is normally labeled y. In a three dimensional coordinate system, another axis, normally labeled z, is added, providing a third dimension of space measurement. The axes are commonly defined as mutually orthogonal to each other (each at a right angle to the other). (Early systems allowed "oblique" axes, that is, axes that did not meet at right angles, and such systems are occasionally used today, although mostly as theoretical exercises.) All the points in a Cartesian coordinate system taken together form a socalled Cartesian plane. Equations that use the Cartesian coordinate system are called Cartesian equations.
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).
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 counterclockwise starting from the upper right ("northeast") quadrant. In the first quadrant, both coordinates are positive, in the second quadrant xcoordinates are negative and ycoordinates positive, in the third quadrant both coordinates are negative and in the fourth quadrant, xcoordinates are positive and ycoordinates negative (see table below.)
[edit] Threedimensional coordinate system
The three dimensional Cartesian coordinate system provides the three physical dimensions of space — length, width, and height. Figures 4 and 5, show two common ways of representing it.
The three Cartesian axes defining the system are perpendicular to each other. The relevant coordinates are of the form (x,y,z). As an example, figure 4 shows two points plotted in a threedimensional Cartesian coordinate system: P(3,0,5) and Q(−5,−5,7). The axes are depicted in a "worldcoordinates" orientation with the zaxis pointing up.
The x, y, and zcoordinates of a point can also be taken as the distances from the yzplane, xzplane, and xyplane respectively. Figure 5 shows the distances of point P from the planes.
The xy, yz, and xzplanes divide the threedimensional space into eight subdivisions known as octants, similar to the quadrants of 2D space. While conventions have been established for the labelling of the four quadrants of the xy plane, only the first octant of three dimensional space is labelled. It contains all of the points whose x, y, and z coordinates are positive.
The zcoordinate is also called applicate.
[edit] Orientation and handedness
 see also: righthand rule
[edit] In two dimensions
Fixing or choosing the xaxis determines the yaxis up to direction. Namely, the yaxis is necessarily the perpendicular to the xaxis through the point marked 0 on the xaxis. 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 xaxis pointing right and the positive yaxis pointing up (and the xaxis being the "first" and the yaxis the "second" axis) is considered the positive or standard orientation, also called the righthanded 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 xaxis to the yaxis, 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.
[edit] In three dimensions
Once the x and yaxes are specified, they determine the line along which the zaxis should lie, but there are two possible directions on this line. The two possible coordinate systems which result are called 'righthanded' and 'lefthanded'. The standard orientation, where the xyplane is horizontal and the zaxis points up (and the x and the yaxis form a positively oriented twodimensional coordinate system in the xyplane if observed from above the xyplane) is called righthanded or positive.
The name derives from the righthand 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 zaxes in a righthanded system. The thumb indicates the xaxis, the index finger the yaxis and the middle finger the zaxis. Conversely, if the same is done with the left hand, a lefthanded system results.
Different disciplines use different variations of the coordinate systems. For example, mathematicians typically use a righthanded coordinate system with the yaxis pointing up, while engineers typically use a lefthanded coordinate system with the zaxis 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 righthanded coordinate system. Because a threedimensional object is represented on the twodimensional 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 xyplane and indicates rotation from the xaxis to the yaxis (in both cases). Hence the red arrow passes in front of the zaxis.
Figure 8 is another attempt at depicting a righthanded coordinate
system. Again, there is an ambiguity caused by projecting the
threedimensional 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 lefthanded
coordinate system. Thus the "correct" way to view Figure 8 is to
imagine the xaxis as pointing towards the observer and thus seeing a concave corner.
[edit] Representing a vector in the standard basis
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 . In three dimensions, the vector from the origin to the point with Cartesian coordinates (x,y,z) is sometimes written as^{[1]}:
where , , and are unit vectors that point the same direction as the x, y, and z axes, respectively. This is the quaternion representation of the vector, and was introduced by Sir William Rowan Hamilton. The unit vectors , , and are called the versors of the coordinate system, and are the vectors of the standard basis in threedimensions.
[edit] Applications
Cartesian coordinates are often used to represent two or three dimensions of space, but they can also be used to represent many other quantities (such as mass, time, force, etc.). In such cases the coordinate axes will typically be labelled with other letters (such as m, t, F, etc.) in place of x, y, and z. Each axis may also have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). It is also possible to define coordinate systems with more than three dimensions to represent relationships between more than three quantities. Although four and higherdimensional 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 higherdimensional spaces, which can become rather complicated.) 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 (perhaps two or three of many) nonspatial variables.
[edit] Further notes
In computational 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 and physicists often use spherical coordinates, a type of threedimensional polar coordinate system.
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.)
[edit] See also
 Other coordinate systems
 Two dimensional orthogonal coordinate systems

 Cartesian coordinate system
 Polar coordinate system
 Parabolic coordinate system
 Bipolar coordinates
 Three dimensional orthogonal coordinate systems
 History
 Related topics
[edit] References
Descartes, René. Oscamp, Paul J. (trans). Discourse on Method, Optics, Geometry, and Meteorology. 2001.
 ^ David J. Griffith (1999). Introduction to Electromagnetics. Prentice Hall. ISBN 013805326X.