Mapping Cartesian Coordiantes to Polar Coordinates

Although polar functions are usually analyzed on their own terms, we can also think of them as mapping Cartesian coordinates to the Polar plane. In the left-hand pane below, we have a function plotted in Cartesian coordinates. A black "cursor" moves along the horizontal axis (), and the height of the graph above or below the axis determines the length and color of the arrow (green for positive, red for negative). In the right-hand pane, one end of the cursor is now fixed at the pole (center), and it rotates at an angle equal to . The radius is the same as the function value on the left - green if positive, red if negative.
Change f(x) as desired by typing its definition in the "f(x) = " box. The three buttons under the slider Resets to zero, Runs the animation from the current value of , and Stops the animation. In the right-hand pane, Clear clears the blue polar function graph, while the slider changes the scale (zoom) of both graphs (so that the lengths of the function value vector on the left and the radius vector on the right are the same).