Cartesian coordinate system
The term Cartesian originates from the last name of the famous French philosopher, René Descartes (Rene Descartes) in tribute to his profound system of investigation published anonymously in 1637 titled Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences. It is commonly referred to as 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; he further explores this in Geometry, book two of the volume as it was originally published. This idea provided the bridge between ancient Greek Euclidean geometry and algebra, leading to a revolution in mathematics and natural sciences. It is one of the important tools used in analytic geometry, calculus, and cartography.
The modern Cartesian 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< b>x, and the vertical axis is labeled y. In a three dimensional coordinate system, another axis, normally labeled z, is drawn orthogonally, providing a sense of a third dimension of space measurement. (Early systems allowed "oblique" axes, that is, axes that did not meet at right angles.)
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 refered to as the xy plane. To specify a particular point on a two dimensional coordinate system, you indicate the x unit first, followed by the y unit in the form (x,y). 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).
Example of two dimensional cartesian coordinate system:
http://www.wikipedia.com/images/uploads/cartesiancoordinates2D.JPG
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. Quadrant I contains all real numbers greater than 0 for both x and y measurements, quadrant II contains all numbers less than 0 for x and all numbers greater than 0 for y. (III contains only negative points for both axes, and IV contains positive numbers for x and negative for 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 1800's the third dimension of measurement was added, using the z axis. An example of a three-dimensional coordinate system is shown here: http://www.wikipedia.com/images/uploads/cartesiancoordinates3D.JPG
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. By holding up the thumb, index, and middle fingers of the right hand, you will see the orientation of the X, Y, and Z axes, respectively. 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 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 used. Attachment to visualizing each dimensions precludes understanding the many different dimensions that can be measured (time, mass, color, cost, etc.).
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.
Descartes, René . Oscamp, Paul J. (trans). Discourse on Method, Optics, Geometry, and Meteorology. 2001.
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