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
°
°
.
