Assignment 11:

Polar Equations

by

Jenny Johnson

What are polar equations?

First, we need to understand that polar equations are graphed on the polar coordinate system, which is a two-dimensional coordinate system wherein a point on the plane (r, θ) is determined by the distance r from the origin and the angle θ from the positive x-axis, measured counter-clockwise.   An example of a point z on the polar coordinate system is shown below.

Clearly, polar coordinates (r, θ) of a point are related to the x- and y-coordinates of a point.  Considering the picture above, we can see the red lines form a right triangle.  Using basic trigonometry, we see that the length of the leg of the right triangle on the x-axis would  be x = r cos θ and the length of the leg parallel to the y-axis would be  y = r sin θ.

Polar equations are algebraic curves expressed in polar coordinates.  For polar equations in this exploration we will define r as a function of θ.    The curves (graphs of the polar functions r) will consist of points in the form (r(θ), θ).

First, letÕs consider the graph of r = 1.  Based on our definition, this is the set of points a constant distance r from the origin (r does not depend on θ).  So, this would be the unit circle.

Similarly, all graphs of r = a where a is a constant will be circles centered at the origin with radius a.

What does the graph of r = a θ look like?

LetÕs consider the graph of the polar equation r = a θ when a = 1 as θ ranges from 0 to 2¹.

It is interesting to note that the curve crosses the x-axis at (-¹, 0) and (2 ¹, 0).  This makes sense since r will be a distance of ¹  from the origin (since r = θ ) when θ  is ¹ radians from the x-axis, and r will equal 2¹ when r is 2¹ radians from the origin.  Here are some other curves of the form r = a θ.

Notice that when a is negative, the curve is a reflection in the x-axis and the y-axis of the curve when a is positive.  Also, the curve of r = 0.5 θ crosses the x-axis at (-¹/2, 0) and (¹,0) since r will be half the length of the value of θ.  So when θ is ¹ radians from the origin, r will be a distance of ¹/2 from the origin and when θ is 2¹ radians from the origin (back on the positive x-axis), then r will be a distance of ¹ from the origin.

Let us consider the curves of a few other polar equations.

This is a line.  What if we substituted 2θ for each θ in the equation above?

What if we substituted θ/2 for 2θ in this equation?

All of these curves have been very different.  Watch a movie of the curves of the following form as a ranges from -10 to 10.

What does the graph of r = a cos θ look like?

When a = 1, and θ ranges from 0 to 2 , we get the following graph.

The graph is a circle with radius a/2 centered at (a/2, 0).  What if a = 2?

This graph is also a circle with radius a/2 with a center of (a/2, 0).  Now letÕs look at the algebraic curve of r = 2 cos θ – 1.

This is an interesting curve.  We can watch a movie of the curve r = a cos θ - 1 to see to see the curve as a ranges from -10 to 10.

What if we were to graph the curve of r = 2 cos (θ – 1)?

It looks like the same size circle as r = 2 cos θ, but shifted up and to the left.

What does the graph of r = a/ θ look like?

First, we observe the graph when a = 1.

This graph is a curve whose maximum y-value is a (when we extend out the x-axis, the curve never gets above 1).  Will this same thing occur when a = 2?

Yes.  The graph of r = 2/ θ is a curve with a maximum y-value of a.  Watch this movie that shows the curve of r = a/ θ as a ranges from -10 to 10.

Let us explore a few more graphs of polar equations.

These two curves are very similar.

Now letÕs examine the graph from above in a movie by changing the value of 10 to n and allowing it to range from -10 to 10.