Coordinate System, ThreeDimensional
Coordinate System, ThreeDimensional
The threedimensional coordinate system is an extension of the twodimensional coordinate system invented in 1637 by French mathematician René Descartes (1596–1650). Soon after Descartes wrote about his twodimensional coordinate system, other mathematicians took Descartes's idea and expanded it from a twodimensional plane to threedimensional space. This new development greatly enlarged the uses of the coordinate system.
With the enhancement of the twodimensional coordinate system to three dimensions, it was possible to locate any object in threedimensional space. For example, with Descartes's twodimensional coordinate system, you could describe the location of a coin on the floor of a room by referring to how many feet the coin was located from a front wall and a side wall. However, in a twodimensional coordinate system, it is impossible to describe the location of an object that is off the plane of the floor, such as the location of a crystal on a chandelier that is hanging from the ceiling. In a threedimensional coordinate system, it is possible to refer to the location of a chandelier crystal with reference to the two walls of the room and also the floor.
Understanding ThreeDimensional Coordinates
A twodimensional coordinate system is formed by two perpendicular lines designated as an x axis and a y axis. The two lines meet at a point called the origin. With a threedimensional coordinate system, a third axis line is added. The third axis line intersects the other two axis lines at the origin, forming right angles with each of them. The x axis shows locations of points in space lying in front and back of the origin. The y axis shows the location of points in space to the left and to the right of the origin. The z axis shows points in space lying above and below the origin.
One way to visualize a threedimensional coordinate system is to imagine yourself standing in a room near one of its corners, and facing the corner. The x axis extends along the base of the wall coming towards you. The y axis extends along the base of the other wall (either extending to your left or right depending on which side of the corner you are facing). The z axis extends upward to the ceiling.
In the sketch above and its threedimensional coordinate system shown below, suppose you are standing at point A. The coordinates of the soles of your feet would be (4, 3, 0), meaning that you are standing four units in front of the origin along the x axis; three units to the right of the origin along the y axis; and are neither above nor below the origin with respect to the z axis. The coordinates of the top of your head might be (4, 3, 5), meaning that your head is 5 units above the origin. If your identical twin were standing in the next room directly "across" from you (through the wall) and at mirrored distances from both walls, the coordinates of his feet and head would be (4, 3, 0) and (4, 3, 5).
This diagram gives a limited illustration of how objects can be located using a threedimensional coordinate system, because it shows only the four quadrants lying "toward" the observer or "in front of" the origin relative to the x axis. There are four complementary quadrants lying "away from" the observer or "behind" the origin relative to the x axis. Hence, the threedimensional coordinate system can be used to represent any point in space relative to the three axis lines.
The threedimensional coordinate system may also be used to determine the distance between any two points of a threedimensional object. In the sketch above, the distance between Point B and the origin may be calculated by using the following formula:
In this formula x _{1}, y _{1}, z _{1} represents the coordinates of Point B and x _{2}, y _{2}, z _{2} represents the coordinates of the origin. Thus the distance between Point B and the origin can be calculated as shown below.
see also Descartes and His Coordinate System; Computer Animation; Coordinate System, Polar; Dimensions.
Arthur V. Johnson II
Bibliography
Bellman, Allan. Advanced Algebra: Tools for a Changing World. Needham, MA: Prentice Hall, 1998.
Clapham, Christopher. A Concise Oxford Dictionary of Mathematics. New York: Oxford University Press, 1990.
USING A THREEDIMENSIONAL COORDINATE SYSTEM
Graphic artists, fashion designers, tool and die makers, and blueprint designers are examples of people who use a threedimensional coordinate system to map threedimensional objects—in this case, on a twodimensional computer monitor or a drawing board. A threedimensional coordinate system helps the artist draw the object in proper perspective. This system also insures that any object drawn by the artist can be properly interpreted by anyone familiar with threedimensional coordinates.
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