Investigation of Polar Equations
by

Nathan Wisdom

Consider the graph of

Fig. 1 Graph with one leaf.

If a and b are constant and k is and integer, then we get the "n-leaf rose" where k is the number of leaves.

Here is a graph for k =7. Note that there are 7 leaves.

Fig 2. Graph with Seven leaves.

The behavior of this graph is predictable for any value of k. If k is not and integer, for instance let k = 7/2, the  graph will produce  three and one half leaves, (see Fig. 3)
The graph will attempt to produce fractional leaves and distort the shape of the leaves. It is easy to recognize one half leaf bur not all fractional leaves are obvious.

Fig. 3

Now suppose we replace the cos( ) with sin( )? Let us look a the graph for a, b, and k =1

Fig. 4
Compare Fig 1 and Fig 4

If k is an integer, and for various k, do you think the graph will behave as in the case with Cos ( )? Investigate..
Here is a graph for k =7. Compare with Fig. 2. Take a closer look at x =1 on all graphs... What do you see?

Fig. 5

Compare Fig 2. and Fig 5. What is the difference? Do you see a simple transformation? If so What is it?

Now look at the case when k =7/2. Will this be a division of leaves?

Fig. 6

Compare Fig. 6 and Fig 3. How are they similar? different?