Coordinate System (mathematics).

Coordinate System (mathematics). Coordinate System (mathematics), system for identifying elements in a set of points by labeling them with numbers. The numbers are called coordinates and can be thought of as giving the position of a point within the set. The system of latitude and longitude is an example of a coordinate system that uses two coordinates to specify the position of a point on the surface of the earth. The system of Cartesian coordinates is the most commonly used coordinate system. In two dimensions, this system consists of a pair of lines on a flat surface, or plane, that intersect at right angles. Each of the lines is called an axis and the point at which they intersect is called the origin. The axes are usually drawn horizontally and vertically and are usually referred to as the x and y axes, respectively. In Cartesian coordinates, a point on the plane whose coordinates are (2,3) is 2 units to the right of the y axis and 3 units up from the x axis, as shown in Figure 1. In three-dimensional Cartesian coordinates, the z axis is added so that there are three axes all perpendicular to each other. In the polar coordinate system, each point in the plane is assigned coordinates (r, ? ) with respect to a fixed line in the plane called the axis and a point on that line called the pole. For a point in the plane, the r coordinate is the distance from the point to the pole and the ? coordinate is the counterclockwise angle between the axis and a line joining the origin to the point, as shown in Figure 2. For example, the point with polar coordinates (1, ) is one unit from the pole and makes an angle of radians, or 90 degrees, with the axis. Cylindrical coordinates and spherical coordinates are two different extensions of polar coordinates to three dimensions. The coordinates of a point or set of points in one coordinate system usually can be translated into coordinates in a different coordinate system. For example, if the axis and pole of a polar coordinate system correspond to the x axis and origin of a Cartesian coordinate system, respectively, then the point with polar coordinates (1, is one unit directly above the origin, making its Cartesian coordinates (0,1). Similarly, the point with polar coordinates (Ã ) is the point with Cartesian coordinates (- 1,1). Polar coordinates are particularly useful for graphing functions that are defined in terms of their distance from a fixed point. For example, the equation of a circle of radius d is given in Cartesian coordinates by x2 + y2 = d 2; in polar coordinates a circle of radius d has the simpler form r = d. Contributed By: William James Ralph Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved. )