Let us investigate a polar equation of the form:

Let us first look at where **b** and **k**
are 1 and we change the values of **a** as such:

We can see from the graph that for **a**=1,
we get the one-leaf stem and as we increase **a, **the graphs
start to become circles and their diameter is related to **a,**
that is the diameter of the circles become twice as much as **a**.

Now let's look at what happens when we let
**a **and **b **be 1 and we vary **k**. Here are some
example graphs:

These equations produce leaf-rose graphs where
**k** controls the number of leafs. If we look at the same
graph except replace cos with sin, we still get leaf-rose graphs
except they are rotated:

Let's investigate what happens to the n-leaf
rose graphs when you change **a** and **b**:

It seems clear that **a **and **b** just
magnify the length of the stem at a ratio of 2 to 1, similar to
what happened with the circle graphs, that is the length of the
stem is twice the value of **a** or **b** which must be
equal to produce the n-leaf rose graph.

Now let's investigate the polar equation:

Let's first investigate what happens when we
vary **k**:

We still get the n-leaf rose graph with the
number of stems controlled by **k **as before and it would
seem to appear that **b **will control the length of the stem
as **a** and **b** did in the prior polar equations, let's
graph and see if this holds to be true:

Our conviction seems to hold true, it appears
that **b** controls the length of the leaf. However, it is
interesting to note that the lenght of the leaf and the value
of **b** is in a 1 to 1 ratio which is different than the prior
polar equations which leads one to think that adding **a**
to the equation is what increases the length of the leaf to the
value of **b** to a 2 to 1 ratio as seen in the previous graphs.