**Polar Equation Exploration**

**by**

**Chad**** Crumley**

In a rectangular
coordinate system, every point in the plane can be identified by a unique
ordered pair (x, y) representing the points distances and direction from two
perpendicular axes. In a polar coordinate
system, a pair of numbers [r, θ] represent a unique
point. Here r or r is the
distance, but θ is the magnitude of rotation measured in degrees or radians.

Recall that *x = a* forms
a vertical line (where *a*
is a constant) and *y = b* forms
a horizontal line (where *b* is a
constant). What would *r = a* form? *Θ
= b*?

Below are the graphs of θ = - pi / 3 (in blue) and r=3 (in red).

Results: *r = a* forms a circle centered at the
origin with radius *a *and
*Θ = b *is a line where *b *is the angle formed between the
positive x-axis and the line in a counter-clockwise direction.

What about r = a sin θ and r = a cos
θ? First well let a=1 and remind
you of some things below.

θ (in degrees) |
0 |
30 |
45 |
60 |
90 |
120 |
135 |
150 |
180 |

r |
1 |
0.866 |
0.707 |
0.5 |
0 |
-0.5 |
-0.707 |
-0.866 |
-1 |

Graphing the points in the table for r = cos
θ:

The results are a circle centered at (o.5, 0) with radius length
0.5; since a = 1 then the diameter is between (0, 0) and (1, 0).

The graph below is r = a cos θ for a =
0.5 (red), 1 (blue), 2 (purple), and 3 (green).

What about r = a sin θ?

Conclusions: The
center (0, a/2) for r = sin θ is on the y-axis with a diameter between (0, 0)
and (0, a).

**Rose Curves:**

Recall that y = cos bθ,
where b is a positive integer, are sine waves with amplitude 1 and period 2 pi
/ b. What about the graphs of polar
equations in the form r = cos bθ?
Below are graphs for b = 2, 3, and 4.

Below are the graphs of r = sin bθ for b
=2, 3, and 4.

These graphs are part of a family of polar graphs called rose
curves or petal curves.

**Archimedean Spiral**

The graph above is for
θ between 0 and 4 pi.

**Limaons**** of Pascal**

Dimpled Limaon

Limaon with an inner loop

For the graph above, a
=1.5 and b = 3. Generally, to get
the inner loop, b>a where a and b are positive
values.

Cardioid (a limaon with a cusp)

The graph above is for
a = 5 and θ between 0 and 2 pi.

**Cissoid**** of Diocles**

a=1, θ is between 0
and 2 pi

**Folium of Descartes**

a=1, θ is between 0
and 2 pi

These are just a few famous polar graphs. Patterns could be discovered by exploring
different values of a (or b) in the above graphs. Other graphs that we could possibly
explore are the cochleoid, strophoid,
lemniscate, lituus, and
many more.

** **