Graphing Lemniscates
Graph a Lemniscate of the form by first graphing the rectangular graph .
You can drag point with the mouse and the points on the polar graph will move with it.
Note that the rectangular graph is not exactly the corresponding function. In order to get the radius for the polar function, since here is playing the role of , we need to take , which, if will yield an imaginary number. This means that there will be nothing on the polar graph, as we cannot use an imaginary radius. This is why the negative portions of the rectangular graph are dashed, since they do not correspond to any points on the polar graph.
Animate . Notice that as the polar graph is drawn, the black portion shows us , while the red portion represents .