Case #3: Let's let a=b=1 for these cases and simply examine what happens when k is not an integer. Let's just start looking at examples:
If we consider each half leaf as one, then we have a graph with seven half leafs. This would make sense, because we let k=7/2. Let's look at another example:
Looking at this example, we see that there are fifteen half leafs. Therefore, I think one could generalize problems of this form.Let's look at anther example.
I'm not quite sure what to say about this, so let's further this example.
So, I'm making the denominator increasingly large. In other words, we're letting k approach zero. Let's go a step further:
So as we approach zero with our k value we get closer and closer to a circle. we should have expected this because cos (0)=1. So we would be left with the equation r = 1+1=2, which in polar coordinates is a circle.