Logo

 Michelle E. Chung

 
* EMAT6680 Assignment 11: Polar Equations
 

 

1. Investigate

Eq1

Note:

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

    Eq2

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

  •  
 

First, let's think about the graph of    PolarEq1 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. Graph1_1
When k=2 The graph has two leaves when k=2. Graph1_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|.

Graph1_3
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.

Graph1_4
When k=-5.3 Also, as you see, the graphs are exactly same when k has same value of absolute vlaues. Graph1_5


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    second (Purple) when a=0 and b=1.

When k=1 The graph has one leaf that is a circle when k=1. Image004
When k=2

The graph has four leaves when k=2.

So, it looks like that the graph of second has two times of |k| leaves when k is even.

image005_N
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  Eq4 comparing to Eq5.

When k=2

The graph of r=a+bcos2 has two leaves and the leaves are bigger than the ones from r=a+bcostheita. Image006_N
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 equ and equhas three leave respectively, but the leaves of equ is about two times bigger than the one of equ.

Graph1_3
When k=-5.3

The graph of is BLUE and the graph of is PURPLE here.

While the graph of equ has only five complete leaves, the graph of equ has ten complete leaves.

So, when k is odd, two graphs have same number of leaves, but when k is even, the graph of equ has two times of k leaves while the graph of equ has only as same number as k leaves.

Graph1_5

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    PolarEq

When k=1

When k=1, the graph of PolarEq is a circle that has (0,0.5) as its center and 0.5 as its radius. image
When k=2

The graph of PolarEq is RED and the graph of Eq4 is BLUE.

Both of the graphs have four leaves; however, the graph of PolarEq is 45 degrees rotated toward the origin.

Image

When k=-2

The graph of polarEq is GREEN.

Both graphs have four leaves; however, the graph of PolarEq is 45 degrees rotated toward the origin.

Graph1_3
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.

 

image


The graphs of   comparing to   PolarEq 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 equ 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 equ 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).

image

When k=-2

When k=-2, the graph is same as when k=2.

 

image
When k=-3

When k=-3, the graph of equ has three big leaves and three small leaves, which are inside the bigger ones.

So, the graph of equ 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.

image
When k=-1.7

Now, let's think about k is not integer.

When k=-1.7, the graph of equ 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. image


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

 

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

The graph of when k is in [-10, 10] with 200 steps.

Seventh, now let's think about the graph of     when a=1 and b=-2.

When k=1

 

The graph of equ is reflection of the graph of equ toward y-axis.

The graph of when k is in [-10, 10] with 200 steps.

Button Go Back to Top
Button Go Back to Michelle's Main page
Button Go Back to EMAT 6680 Homepage
Copyright @ Michelle E. Chung