Sinje J. Butler

# Write-up # 11

## Polar Equations

We begin by looking at the definition of polar coordinates.  The position of point P is given by

().

The position of a point is described with reference to a fixed point, called the origin, and a fixed positively directed line called the polar axis.

The following will investigate various polar equations in the form

.

Lets begin by taking a look at the simple case of

where

a = 1

b =1

and

k = 1.

This graph is called a cardiod.  Notice what takes place when the value of b is changed to higher numbers.  Thus, we have

for

a = 1

b =1, 2, and 3.

and

k =1.

As k increases, it appears that the inner loop is increasing and the outer loop is increasing.

When the following equation,

°

with

a = 1

b = 1

k = 1

is changed to

°

with

a = 1

b = 1

k = 1

we have the following graph.

When cos is changed to sin in the equation, the cardiod appears to rotate counter clockwise around the origin 90 degrees.

The graph below is

for

a = 1

b = 1

and

k = 2, 3, and 4.

This is what is referred to as the “n-leaf rose”.  As k is increased the flower like graph will have a number of petals that is equal to k. Therefore, if k is increased to 25, the graph will have 25 petals and looks as follows.

If

°

is changed to

°

the petals appear to change positions.

Let us take a look at when

is changed to

If k is an even number it appears that the number of petals double.

The graphs below are

°

°.

If k is an odd number it appears that the number of petals remain the same.  The following graphs are

°

°.