A coordinate system is a method by which a set of numbers is used to locate the position of a point. The numbers are called the point's coordinates. In a coordinate system, a single point corresponds to each set of coordinates. Coordinate systems are used in analytic geometry to study properties of geometric objects with algebraic techniques.

When an object having a finite number of degrees of freedom is considered among all the objects of that kind, the object in question can be conveniently characterized and distinguished from the other objects by a set of coordinatesÑthat is, a set of numbers, one for each degree of freedom. For example, a point in a plane has two degrees of freedom, so that the point has two coordinates with respect to any coordinate system of the plane.

There are many different coordinate systems. Usually the geometry and symmetry of a problem will suggest an appropriate coordinate system. Common coordinate systems are Cartesian (after RenŽ Descartes) coordinates and polar coordinates in two -dimensional space and Cartesian, spherical, and cylindrical coordinates in three-dimensional space.

   Coordinate Systems in Two Dimensions

Through an arbitrary point O in the plane, two mutually perpendicular lines, usually horizontal and vertical, are drawn. The x-axis is taken to be horizontal, the y-axis is vertical, and point O is called the origin. The portion of the x-axis to the right of the origin is the positive x-axis, and the part of the y-axis above the origin is called the positive y-axis. The two axes (coordinate axes) divide the plane into four quadrants: the upper right (first), the upper left (second), the lower left (third), and the lower right (fourth). The x-coordinate, or abscissa, of a point P in the plane is the perpendicular distance of P from the y-axis. It is positive if P is to the right of the y-axis, negative if P is to the left, and zero if P is on the y-axis. The y-coordinate, or ordinate, of P is analogously the perpendicular distance of P from the x- axis. It is, respectively, positive, negative, or zero if P is above, below, or on the x-axis. The ordered pair (x, y) represents the coordinates of P in the coordinate system thus defined. The point P with coordinates (x, y) is symbolically represented as P (x, y). This system is called a two- dimensional, or plane, Cartesian coordinate system.

A polar coordinate system in two dimensions is a system determined by a fixed point O, called the pole, and an axis through it, called the initial line. A point P in the plane can then be fixed by specifying two quantities: (1) the angle < through which the axis must be rotated in the counterclockwise direction so as to pass through P, and (2) the positive distance r of the point P from the pole. The notation P (r, <) is used to represent P in polar coordinates r and <.

If P (x, y) is the Cartesian representation of P, and P (r, <) is the polar coordinate representation of the same point, and if the origin and x-axis of the Cartesian system coincide, respectively, with the pole and the initial line of the polar coordinate system, then the two systems are related by x = r cos <, y = r sin <, and r = the square root of x6 + y6, tan < = y/x. These are the equations of transformation from one system to another.

   Coordinate Systems in Three Dimensions

Three mutually perpendicular lines (the coordinate axes) are drawn through an arbitrary point O, the origin, in space. The axes are called the x-axis, y-axis, and z-axis. The plane containing the x-axis and the y-axis is the xy-plane (a coordinate plane) and the z-axis is a normal (a line that is perpendicular) to this plane. The other two coordinate planes are defined likewise. The x-coordinate of a point P is the perpendicular distance of P from the yz-plane. The y-coordinate and the z-coordinate are defined similarly. The three coordinate planes divide all space into octants. If P is a point in the first octant, all the coordinates of P are positive.

The system thus described is a three-dimensional Cartesian system, and P (x, y, z) is the Cartesian representation of P with coordinates x, y, z with respect to a fixed frame of reference. For every point there corresponds uniquely a set of three real numbers, and vice versa.

The spherical coordinate system in space is a system that locates a point P by its distance from a fixed point O (the pole), and by two angles that describe the orientation of the segment OP. The coordinate system is fixed by two perpendicular half-lines through O. One of these is the polar axis. The plane that contains the two half-lines is called the initial meridian plane. The spherical coordinates of P are (r, <, ñ), where r is the length of OP, < is the angle from the initial meridian plane to the plane through the polar axis and OP, and ñ is the angle from the polar axis to OP. The spherical system is usually aligned with a Cartesian system in which the pole is the origin. The polar axis coincides with the z-axis, and the initial meridian plane with the xz-plane. The equations x = r sin ñ cos <, y = r sin <, and z = r cos ñ express the relation between the two systems.

A coordinate system consisting of a plane with polar coordinates and a z-axis through the pole, or origin, perpendicular to the plane is called a cylindrical polar coordinate system. A point P is located in space by its distance z from the plane and the polar coordinates (r, <) of the foot of the perpendicular from P to the plane. The two systems are related by the equations x = r cos <, y = sin <, and z = z.

V. K. Balakrishnan

Bibliography: Fuller, Gordon, Analytic Geometry (1993); Leithold, Louis, Before Calculus, 2d ed. (1990).