Exploration of Polar Coordinates and Rectangular Coordinates
As r ranges from 0 to infinity and θ ranges from 0 to 2π, the point A specified by the polar coordinates (r,θ) covers every point in the plane. Adding 2π to θ brings us back to the same point, so if we allowed θ to range over an interval larger than 2π, each point would have multiple polar coordinates. Hence, we typically restrict θ to be in the interal 0≤θ<2π. However, even with that restriction, there still is some non-uniqueness of polar coordinates: when r=0, the point A is at the origin independent of the value of θ. The following applet allows you to explore how changing the polar coordinates r and θ and Cartesian coordinates of A (x, y) moves the point A around the plane. You can choose either polar coordinates, Cartesian coordinates, or both and move point A around and observe how the coordinates change. Notice the non-uniqueness of polar coordinates when r = 0.
Polar coordinates. When you move, point A, the blue point in the Cartesian plane directly with the mouse and observe how the polar coordinates change. The coordinate r is the length of the line segment from the point (x,y) to the origin and the coordinate θ is the angle between the line segment and the positive x-axis. Can you get θ to equal 2π?