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 coordinates—that 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 René 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.