Cartesian coordinate system

The term 'Cartesian' is named with respect to the French mathematician Descartes in honor of his efforts to merge algebra and Euclidean geometry. This work was influential to the development of analytic geometry, calculus, and cartography.

The idea of this system was developed in 1637 in two writings by Descartes:

• Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences[?] (Discourse on Method)
  • In part two, he introduces the new idea of specifying the position of a point or object on a surface, using two intersecting axes as measuring guides.
• La Géométrie
  • further explores the above-mentioned concepts

The modern Cartesian coordinate system in two dimensions (also called a rectangular coordinate system) is commonly defined by two axes, at right angles to each other, forming a plane (an xy-plane). The horizontal axis is labeled x, and the vertical axis is labeled y. In a three dimensional coordinate system, another axis, normally labeled z, is added, providing a sense of 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.) All the points in a Cartesian coordinate system taken together form a so-called Cartesian plane.

The point of intersection, where the axes meet, is called the origin normally labeled O. With the origin labeled O, we can name the x axis Ox and the y axis Oy. The x and y axes define a plane that can be 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, you indicate the x unit first (abscissa), followed by the y unit (ordinate) in the form (x,y), an ordered pair. In three dimensions, a third z unit is added, (x,y,z).

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.

An example of a point P on the system is indicated in the picture below using the coordinate (5,2).

The arrows on the axes indicate that they extend forever in the same direction (i.e. infinitely). The intersection of the two x-y axes creates four quadrants indicated by the roman numerals I, II, III, and IV. Conventionally, the quadrants are labeled counter-clockwise starting from the northeast quadrant. In Quadrant I the values are (x,y), and II:(-x,y), III:(-x,-y) and IV:(x,-y). (see table below.)

Quadrant	x values	y values
I	> 0	> 0
II	< 0	> 0
III	< 0	< 0
IV	> 0	< 0

Three dimensional coordinate system Sometime in the early 19th century the third dimension of measurement was added, using the z axis.

A three dimensional coordinate system is usually depicted using what is called the right-hand rule, and the system is called a right-handed coordinate system (see handedness). By holding up the middle finger, index finger, and thumb of the right hand, you will see the orientation of the X, Y, and Z axes, respectively (the thumb being the Z axis). The fingers each point toward the positive direction of their representative axes. In the picture above, we see a right-handed coordinate system. Less common, but still in use (normally outside of the physical sciences) is the left-handed coordinate system.

When the z axis is depicted as pointing upward, this is sometimes called a world coordinates orientation. However, the important thing is which direction the axes point in the positive direction with respect to each other. If we drew an image in the right-handed system and then plotted the image, point for point in a left-handed system, you would have a mirror image.

The coordinates in a three dimensional system are of the form (x,y,z). An example of two points plotted in this system are in this picture, point P(5,0,2) and Q(-5,-5,10): . Notice that the axes are depicted in a world-coordinates orientation with the Z axis pointing up. With your right hand, point your thumb, index and middle finger out, tilting your hand back. Notice that your middle finger is pointing up, thumb to the right, and the index finger is point outward just like the y axis in the picture. Your thumb is pointing in the same direction as the x axes does when it moves in a positive direction. This is a right-hand coordinate system.

The three dimensional coordinate system is popular because it provides the physical dimensions of space, of height, width, and length, and this is often referred to as "the three dimensions". 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.). It is the powerful insight of Descartes that allows us to manipulate multi-dimensional object algebraically, avoiding compass and protractor for analyzing in more than three dimensions.

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. In different branches of mathematics coordinate systems can be transformed, translated, rotated, and re-defined altogether to simplify calculation and for specialized ends.

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--a trade secret. That this may have influenced Descartes is merely speculative.

References

Descartes, René. Oscamp, Paul J. (trans). Discourse on Method, Optics, Geometry, and Meteorology. 2001.

See also:

• solution point