Just as in the case of rectangular coordinates, it is possible to use relations between the polar coordinates to specify points in the plane. The process of sketching their graphs is very similar to that used to sketch graphs of functions in rectangular coordinates. I will only consider the case where the relation between the polar coordinates is of the form, r = f
and then graph the resulting points and connect them up with a curve. The following picture illustrates how to begin this process.
To obtain the point in the plane which goes with the pair
Example 12.1.1 Graph the polar equation r = 1 + cosθ.
Using a computer algebra system, here is the graph of this cardioid.
You can also see just from your knowledge of the trig. functions that the graph should look something like this. When θ = 0,r = 2 and then as θ increases to π∕2, you see that cosθ decreases to 0. Thus the line from the origin to the point on the curve should get shorter as θ goes from 0 to π∕2. Then from π∕2 to π, cosθ gets negative eventually equaling −1 at θ = π. Thus r = 0 at this point. Viewing the graph, you see this is exactly what happens. The above function is called a cardioid.
Here is another example. This is the graph obtained from r = 3 + sin
In polar coordinates people sometimes allow r to be negative. When this happens, it means that to obtain the point in the plane, you go in the opposite direction along the ray which starts at the origin and makes an angle of θ with the positive x axis. I do not believe the fussiness occasioned by this extra generality is justified by any sufficiently interesting application so no more will be said about this. It is mainly a fun way to obtain pretty pictures. Here is such an example.