Exploration 11: Polar Equations

by Elizabeth Gieseking

Polar equations give us a different mathematical perspective on graphing. In rectangular coordinates, we use two axes which meet at the origin and are perpendicular to one another. In polar coordinates, we start with a fixed point, O, called the pole or origin and then we construct an initial ray called the polar axis. Then each point P in the plane is assigned polar coordinates where

 from O to P

 counterclockwise from the polar axis to segment

 

 

If we wish to graph a straight line through the origin in polar coordinates, we can set tangent θ equal to the desired slope, as shown below.

If we wish to graph a circle about the origin, we set r equal to the radius of the desired circle.

We can use what we know about the radius and angle to find the relationship between polar and rectangular coordinates. Looking at the figure, we can use trigonometry to see that

Now we will consider some more interesting equations. We will start by considering equations of the form and as shown below.

When we observe this graph, we see that both the sine and cosine graphs have a radius of 1 or . The sine graph has its diameter along the y-axis from (0, 0) to (0, 2) and the cosine graph has its diameter along the x-axis from (0, 0) to (2, 0). Let's look at these graphs a little bit at a time to see what is happening.

From these graphs, we see that the cosine graph and sine graph start at (2, 0) and (0, 0), respectively, and when they have each completed  of the circle, while the graph of r = 2 has only completed  of the circle. Similarly, when , both the sine graph and the cosine graph have completed an entire circle, while the graph of has only completed a semicircle. These graphs complete the circle again during the interval . We can also look at the angle and radius in the following table.

θ	2 sin θ	2 cos θ
0	0	2
	2	0
	0
		0
	0	2

 

 

We note in our definition that r is a directed distance, thus when r is negative, the point plotted is in the direction opposite to the indicated angle. If we take the absolute value to keep r positive, each of these graphs forms two circles instead of one.

Now we will examine two special types of polar graphs - limaçons and rose curves.  Limaçons have the general form  or .  In the graph below, we have limaçons with the equations , where b is varied from 1 to 5.

If we select negative values for b, the resulting graph is a reflection of this graph over the y-axis.

If we choose to use rather than , our axis of symmetry is over the y-axis rather than the x-axis.

The ratio,  determines the shape of the limaçon.  If  we get a convex limaçon which looks like a circle which has been flattened on one side. 

If  the resulting shape is a dimpled limaçon.

If the resulting shape is called a cardioid because it is somewhat heart shaped.

Finally, if  then we get a limaçon with an inner loop.  The size of the loop increases as this ratio decreases.

Let’s consider what is happening as we change the ratio, .  When  which implies   When  or with the result that   We see that in all of the graphs with  pass through the points  and  in rectangular coordinates which correspond to  and  in polar coordinates.  When which implies   This leads to the different shapes of graphs shown above.  When  the left hand side of the graph goes from rounded to flattened to indented.  When , we get  at , resulting in the cardioid shape.  When  then we get , which results in the inner loop.  We also see that no matter the shape of the limaçon, the distance between the points when  and  is always

The equations of rose curves can be written as  or .  On the graphs below, we set  and varied n from 2 to 5.  For all of these graphs,

When we replace cosine with sine we get the following graphs.

We see that the sine graphs are a rotation of the cosine graphs.  The cosine rose curves all have one petal aligned with the positive x-axis.  The even sine curves have all their petals oriented between the axes and the odd sine curves have one of their petals aligned with the positive or negative y-axis.  We also notice something somewhat unusual in these graphs.  When n is odd, the rose has n petals, but when n is even we have 2n petals.  We can see why this is happening by stepping through  a little at a time.  First we will examine the equation

In each  interval, half of a petal is completed and the graph does not retrace itself in the interval    Next we will examine the graph of  in the same way.

In this case, half of a petal is completed in each  interval, however, the greatest difference is that the graph traces over itself during the interval  rather than completing additional petals as does the  graph.   We can again look at a table of the points.  On the  curve, the values of the radius from  are repeated over the interval , resulting in a reflection of the graph over the origin.  On the  curve, the values over the interval  are opposite in sign to those from the interval , resulting in a retracing of the curve.

θ	2 cos 2θ	2 cos 3θ
0	2	2
	-2	0
	2
	0
		0
	0
	2	2

 

We can make a modification to our equations of the rose curves to produce curves which always have n petals.  Instead of graphing  we can graph .  This will have the same maximum radius as the previous graph but will always have n petals.  These petals are not, however, the same shape as the petals in the original graph.

Through this exploration, we have seen several useful applications of polar graphs.  Polar coordinates make it easy to define lines through the origin or circles centered at the origin.  Polar coordinates also give much simpler equations for certain figures such as the limaçon and rose curves in which the distance from the origin is related to the angle.

 

 

 

 

 

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