Assignment 11

Exploring Polar Equations

by

Behnaz Rouhani

In this assignment we are going to investigate the following polar equation cases.

Case I:

for various b and k.

Case II:

for various a, b, and k.

Case III: Special investigation

Case I:

We will explore the above polar equation for various b and k values.

### Part A: Varying b when k = even number

First we will consider k = 2. In this instance the polar equation will produce the following graphs for sine and cosine functions. Different values of b are represented in the following graphs.

b = 1 : red

b = 2: blue

b = 3: green

b = 6: magenta

Notice that multiplying the above two graphs by a factor of b causes the graphs to become  larger by a factor of b. In these graphs the sine polar curve is symmetrical about the vertical line, whereas the cosine polar curve is symmetrical about the polar axis (horizontal axis). When k = 2, we have a 4-leaved rose.

Let's see what happens when k = 4.

In the above sine and cosine graphs when k = 4, we have 8 petals. Based on the above graphs we could conclude that:

• The sine polar curve remains symmetrical about the vertical axis, and the cosine polar curve remains symmetrical about the polar axis.
• Varying b will expand the graph by a factor of b.
• When k is even the number of rose petals will be 2k.

### Part B: Varying b when k = odd number

Now,  we will consider k = 3. Different values of b are represented in the following graphs of sine and cosine functions.

b = 1 : red

b = 2: blue

b = 3: green

b = 6: magenta

From the above graphs we could conclude the following:

• Varying b will expand the graph by a factor of b
• When k is odd  the number of petals remain as k

Case II: In this case we will explore the graphs of

### Part A: Varying k, a = b

Different values of a and b are represented in the following graph.

a = b = 1: red

a = b = 3: blue

a = b = 6: magenta

Consider k=1 ----this is graph of Cardioid.

This curve is called Cardioid because it is shaped like a heart. This polar curve is symmetrical about the polar axis (horizontal axis).

Consider k = 2

This polar curve is symmetrical about the vertical axis.

Consider k = 3

From the above we can make the following observations:

• The different values of a = b will cause the graph to be larger
• The number of petals is same as k

### Part B: Varying k, and a = b

Different values of a and b are considered in the following graphs.

a = b = 1 : red

a = b = 3: blue

a = b = 6: magenta

Let k =1 --- case of a Cardioid

This polar curve is symmetrical about the vertical axis.

Let k = 2

This polar curve is symmetrical about the pole.

Let k = 4

As we vary a = b, the graph just gets larger by that factor. For instance when a = b = 3, then the graph gets larger by a factor of 3. In this case the number of petals is same as k.

Following are graphs of a Limacon.

In all the above cases whether k is odd or even we get 2k petals half of which are small and half are large. The only difference is that when k is odd the small petals are inside the large ones, but when k is even the small petals are outside the large ones.

Case III: Special investigation

This following is the graph of a spiral.