Polar Coordinates of Points in the Euclidean Plane .

We assign coordinates to points in so that we can locate them. Let P be a point in , and be the line segment between P and the origin. Let r be the distance between P and the origin. (So r is the length of .) Let be the angle, measured in the counterclockwise direction, that makes with the x-axis. (See the figure below.)

The point P lies on the circle of radius r, centered at the origin, given by the equation . We can verify that and satisfy this equation.

are called the polar coordinates for the point P.

polar.epsTwo sets of polar coordinates and describe precisely the same point when one of the following cases occurs.

1. r=r'=0, and and are arbitrary.
2. , r=r', and , where k is an integer.
3. , r'=-r, and , where k is an integer.