Michelle E. Chung

* EMAT6680 Assignment 11: Polar Equations

1. Investigate

Note:

When a and b are equal, and k is an integer, this is one textbook version of the “n-leaf rose.”
Compare with

for various k. What if … cos( ) is replaced with sin( )?

First, let's think about the graph of when a=b=1 and k is in [-10, 10].

* Since I set a=b=1, these are the graphs of 'n-leaf rose'.
When k=1	The graph has only one leaf when k=1.
When k=2	The graph has two leaves when k=2.
When k=-3	The graph has three leaves when k=-3. So, it looks like that the graph has same number of leaves as \|k\|.
When k=5.3	Now, let's think about k is not integer. The graph has four complete leaves and two-incomplete leaves when k=5.3. So, it looks like that the last leaf isbeing drawn when k is between two consecutive integers.
When k=-5.3	Also, as you see, the graphs are exactly same when k has same value of absolute vlaues.

When k is in [-10, 10] with 200 steps.

From this movie, you can check the facts we assume previously are true.

Second, let's think about the graph of (Purple) when a=0 and b=1.

When k=1	The graph has one leaf that is a circle when k=1.
When k=2	The graph has four leaves when k=2. So, it looks like that the graph of has two times of \|k\| leaves when k is even.
When k=-3	The graph has three-leaves when k=-3.
When k=5.3	Now, let's think about k is not integer. The graph has eight complete leaves and one incomplete leaves when k=5.3. So, it looks like that the last leaf isbeing drawn when k is between two consecutive integers.
When k=-5.3	Also, as you see, the graphs are exactly same when k has same value of absolute vlaues.

When k is in [-10, 10] with 200 steps.

Third, let's think about the graph of comparing to .

When k=2	The graph of r=a+bcos2 has two leaves and the leaves are bigger than the ones from r=a+bcos.
When k=-2	The graph is as same as when k=-2.
When k=-3	The graph of is RED and the graph of is PURPLE here. The graphs of and has three leave respectively, but the leaves of is about two times bigger than the one of .
When k=-5.3	The graph of is BLUE and the graph of is PURPLE here. While the graph of has only five complete leaves, the graph of has ten complete leaves. So, when k is odd, two graphs have same number of leaves, but when k is even, the graph of has two times of k leaves while the graph of has only as same number as k leaves.

When k is in [-10, 10] with 200 steps.

From this movie, you can check the facts we assume previously are true.

Fourth, let's think about the graph of

When k=1	When k=1, the graph of is a circle that has (0,0.5) as its center and 0.5 as its radius.
When k=2	The graph of is RED and the graph of is BLUE. Both of the graphs have four leaves; however, the graph of is 45 degrees rotated toward the origin.
When k=-2	The graph of is GREEN. Both graphs have four leaves; however, the graph of is 45 degrees rotated toward the origin.
When k=-3	When k=-3, the graph has three leaves.
When k=5.3	Now, let's think about k is not integer. When k=5.3, both graphs have eight complete leaves.

The graphs of comparing to when interval of k is [-10, 10] with 200 steps.

Fifth, let's think about the graph of when a=1, b=2.

When k=1	The graph of has a different shape of leaf. When k=1, it has one big leaf and one small leaf, which is inside the big leaf.
When k=2	When k=2, the graph of has four leaves, which are two big leaves and two small leaves. Also, the big leaves pass through (-3,0), (0,0), and (3,0), and the small leaves pass through (0,-1), (0,0), and (0,1).
When k=-2	When k=-2, the graph is same as when k=2.
When k=-3	When k=-3, the graph of has three big leaves and three small leaves, which are inside the bigger ones. So, the graph of has two times of k leaves, which are big and small. The number of big and small leaves are same. Also, when k is odd, the small leaves are inside the big leaves.
When k=-1.7	Now, let's think about k is not integer. When k=-1.7, the graph of has three complete leaves and half of the last one. Since we know that it has four leaves when k=2, we can say that we have complete leaves when k is an integer. Also, we can see that the small leaves are coming out from the big leaves.
When k=-5.3	Almost same as above, but because k is closer to odd number, the small leaves are inside the big leaves.