Polar Equation Exploration


Chad Crumley


 In a rectangular coordinate system, every point in the plane can be identified by a unique ordered pair (x, y) representing the point’s 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 we’ll let a=1 and remind you of some things below.


θ (in degrees)





















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 , 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 ? Below are graphs for b = 2, 3, and 4.


Below are the graphs of r = sin 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.


Limaćons of Pascal

Dimpled Limaćon


Limaćon 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 limaćon 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.