Write-up #11

Graphing Polar Equations

by

Jim Meneguzzo

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.