Figure 1. Planets follow elliptical paths as they orbit around the Sun. (credit: modification of work by NASA/JPL-Caltech)

The planets move through space in elliptical, periodic orbits about the sun, as shown in Figure 1. They are in constant motion, so fixing an exact position of any planet is valid only for a moment. In other words, we can fix only a planet’s instantaneous position. This is one application of polar coordinates, represented as [latex]\left(r,\theta \right)[/latex]. We interpret [latex]r[/latex] as the distance from the sun and [latex]\theta[/latex] as the planet’s angular bearing, or its direction from a fixed point on the sun. In this section, we will focus on the polar system and the graphs that are generated directly from polar coordinates.

Testing Polar Equations for Symmetry

Just as a rectangular equation such as [latex]y={x}^{2}[/latex] describes the relationship between [latex]x[/latex] and [latex]y[/latex] on a Cartesian grid, a polar equation describes a relationship between [latex]r[/latex] and [latex]\theta[/latex] on a polar grid. Recall that the coordinate pair [latex]\left(r,\theta \right)[/latex] indicates that we move counterclockwise from the polar axis (positive x-axis) by an angle of [latex]\theta[/latex], and extend a ray from the pole (origin) [latex]r[/latex] units in the direction of [latex]\theta[/latex]. All points that satisfy the polar equation are on the graph.

Symmetry is a property that helps us recognize and plot the graph of any equation. If an equation has a graph that is symmetric with respect to an axis, it means that if we folded the graph in half over that axis, the portion of the graph on one side would coincide with the portion on the other side. By performing three tests, we will see how to apply the properties of symmetry to polar equations. Further, we will use symmetry (in addition to plotting key points, zeros, and maximums of [latex]r[/latex])
to determine the graph of a polar equation.

In the first test, we consider symmetry with respect to the line [latex]\theta =\frac{\pi }{2}[/latex] (y-axis). We replace [latex]\left(r,\theta \right)[/latex] with [latex]\left(-r,-\theta \right)[/latex] to determine if the new equation is equivalent to the original equation. For example, suppose we are given the equation [latex]r=2\sin \theta[/latex];

This equation exhibits symmetry with respect to the line [latex]\theta =\frac{\pi }{2}[/latex].

In the second test, we consider symmetry with respect to the polar axis ( [latex]x[/latex] -axis). We replace [latex]\left(r,\theta \right)[/latex] with [latex]\left(r,-\theta \right)[/latex] or [latex]\left(-r,\pi -\theta \right)[/latex] to determine equivalency between the tested equation and the original. For example, suppose we are given the equation [latex]r=1 - 2\cos \theta[/latex].

The graph of this equation exhibits symmetry with respect to the polar axis.

In the third test, we consider symmetry with respect to the pole (origin). We replace [latex]\left(r,\theta \right)[/latex] with [latex]\left(-r,\theta \right)[/latex] to determine if the tested equation is equivalent to the original equation. For example, suppose we are given the equation [latex]r=2\sin \left(3\theta \right)[/latex].

The equation has failed the symmetry test, but that does not mean that it is not symmetric with respect to the pole. Passing one or more of the symmetry tests verifies that symmetry will be exhibited in a graph. However, failing the symmetry tests does not necessarily indicate that a graph will not be symmetric about the line [latex]\theta =\frac{\pi }{2}[/latex], the polar axis, or the pole. In these instances, we can confirm that symmetry exists by plotting reflecting points across the apparent axis of symmetry or the pole. Testing for symmetry is a technique that simplifies the graphing of polar equations, but its application is not perfect.

A General Note: Symmetry Tests

A polar equation describes a curve on the polar grid. The graph of a polar equation can be evaluated for three types of symmetry, as shown in Figure 2.

Figure 2. (a) A graph is symmetric with respect to the line [latex]\theta =\frac{\pi }{2}[/latex] (y-axis) if replacing [latex]\left(r,\theta \right)[/latex] with [latex]\left(-r,-\theta \right)[/latex] yields an equivalent equation. (b) A graph is symmetric with respect to the polar axis (x-axis) if replacing [latex]\left(r,\theta \right)[/latex] with [latex]\left(r,-\theta \right)[/latex] or [latex]\left(-r,\mathrm{\pi -}\theta \right)[/latex] yields an equivalent equation. (c) A graph is symmetric with respect to the pole (origin) if replacing [latex]\left(r,\theta \right)[/latex] with [latex]\left(-r,\theta \right)[/latex] yields an equivalent equation.

Example 1: Testing a Polar Equation for Symmetry

Test the equation [latex]r=2\sin \theta[/latex] for symmetry.

Show Solution

Test for each of the three types of symmetry.

1) Replacing [latex]\left(r,\theta \right)[/latex] with [latex]\left(-r,-\theta \right)[/latex] yields the same result. Thus, the graph is symmetric with respect to the line [latex]\theta =\frac{\pi }{2}[/latex].	[latex]\begin{align}&-r=2\sin \left(-\theta \right) \\ &-r=-2\sin \theta&& \text{Even-odd identity} \\ &r=2\sin \theta&& \text{Multiply}\text{by}-1 \\ &\text{Passed} \end{align}[/latex]
2) Replacing [latex]\theta[/latex] with [latex]-\theta[/latex] does not yield the same equation. Therefore, the graph fails the test and may or may not be symmetric with respect to the polar axis.	[latex]\begin{align}&r=2\sin \left(-\theta \right) \\ r&=-2\sin \theta && \text{Even-odd identity} \\ &r=-2\sin \theta \ne 2\sin \theta \\ &\text{Failed} \end{align}[/latex]
3) Replacing [latex]r[/latex] with [latex]-r[/latex] changes the equation and fails the test. The graph may or may not be symmetric with respect to the pole.	[latex]\begin{align}&-r=2\sin \theta \\ &r=-2\sin \theta \ne 2\sin \theta \\ &\text{Failed} \end{align}[/latex]

Analysis of the Solution

Using a graphing calculator, we can see that the equation [latex]r=2\sin \theta[/latex] is a circle centered at [latex]\left(0,1\right)[/latex] with radius [latex]r=1[/latex] and is indeed symmetric to the line [latex]\theta =\frac{\pi }{2}[/latex]. We can also see that the graph is not symmetric with the polar axis or the pole. See Figure 3.

Figure 3

 Graphing Polar Equations by Plotting Points

To graph in the rectangular coordinate system we construct a table of [latex]x[/latex] and [latex]y[/latex] values. To graph in the polar coordinate system we construct a table of [latex]\theta[/latex] and [latex]r[/latex] values. We enter values of [latex]\theta[/latex] into a polar equation and calculate [latex]r[/latex]. However, using the properties of symmetry and finding key values of [latex]\theta[/latex] and [latex]r[/latex] means fewer calculations will be needed.

Finding Zeros and Maxima

To find the zeros of a polar equation, we solve for the values of [latex]\theta[/latex] that result in [latex]r=0[/latex]. Recall that, to find the zeros of polynomial functions, we set the equation equal to zero and then solve for [latex]x[/latex]. We use the same process for polar equations. Set [latex]r=0[/latex], and solve for [latex]\theta[/latex].

For many of the forms we will encounter, the maximum value of a polar equation is found by substituting those values of [latex]\theta[/latex] into the equation that result in the maximum value of the trigonometric functions. Consider [latex]r=5\cos \theta[/latex]; the maximum distance between the curve and the pole is 5 units. The maximum value of the cosine function is 1 when [latex]\theta =0[/latex], so our polar equation is [latex]5\cos \theta[/latex], and the value [latex]\theta =0[/latex] will yield the maximum [latex]|r|[/latex].

Similarly, the maximum value of the sine function is 1 when [latex]\theta =\frac{\pi }{2}[/latex], and if our polar equation is [latex]r=5\sin \theta[/latex], the value [latex]\theta =\frac{\pi }{2}[/latex] will yield the maximum [latex]|r|[/latex]. We may find additional information by calculating values of [latex]r[/latex] when [latex]\theta =0[/latex]. These points would be polar axis intercepts, which may be helpful in drawing the graph and identifying the curve of a polar equation.

Example 2: Finding Zeros and Maximum Values for a Polar Equation

Using the equation in Example 1, find the zeros and maximum [latex]|r|[/latex] and, if necessary, the polar axis intercepts of [latex]r=2\sin \theta[/latex].

Show Solution

To find the zeros, set [latex]r[/latex] equal to zero and solve for [latex]\theta[/latex].

[latex]\begin{align}&2\sin \theta =0 \\ &\sin \theta =0 \\ &\theta ={\sin }^{-1}0 \\ &\theta =n\pi&& \text{where }n\text{ is an integer} \end{align}[/latex]

Substitute any one of the [latex]\theta[/latex] values into the equation. We will use [latex]0[/latex].

[latex]\begin{align} &r=2\sin \left(0\right) \\ &r=0 \end{align}[/latex]

The points [latex]\left(0,0\right)[/latex] and [latex]\left(0,\pm n\pi \right)[/latex] are the zeros of the equation. They all coincide, so only one point is visible on the graph. This point is also the only polar axis intercept.

To find the maximum value of the equation, look at the maximum value of the trigonometric function [latex]\sin \theta[/latex], which occurs when [latex]\theta =\frac{\pi }{2}\pm 2k\pi[/latex] resulting in [latex]\sin \left(\frac{\pi }{2}\right)=1[/latex]. Substitute [latex]\frac{\pi }{2}[/latex] for [latex]\mathrm{\theta .}[/latex]

[latex]\begin{align}&r=2\sin \left(\frac{\pi }{2}\right) \\ &r=2\left(1\right) \\ &r=2 \end{align}[/latex]

Analysis of the Solution

The point [latex]\left(2,\frac{\pi }{2}\right)[/latex] will be the maximum value on the graph. Let’s plot a few more points to verify the graph of a circle.

[latex]\theta[/latex]	[latex]r=2\sin \theta[/latex]	[latex]r[/latex]
0	[latex]r=2\sin \left(0\right)=0[/latex]	[latex]0[/latex]
[latex]\frac{\pi }{6}[/latex]	[latex]r=2\sin \left(\frac{\pi }{6}\right)=1[/latex]	[latex]1[/latex]
[latex]\frac{\pi }{3}[/latex]	[latex]r=2\sin \left(\frac{\pi }{3}\right)\approx 1.73[/latex]	[latex]1.73[/latex]
[latex]\frac{\pi }{2}[/latex]	[latex]r=2\sin \left(\frac{\pi }{2}\right)=2[/latex]	[latex]2[/latex]
[latex]\frac{2\pi }{3}[/latex]	[latex]r=2\sin \left(\frac{2\pi }{3}\right)\approx 1.73[/latex]	[latex]1.73[/latex]
[latex]\frac{5\pi }{6}[/latex]	[latex]r=2\sin \left(\frac{5\pi }{6}\right)=1[/latex]	[latex]1[/latex]
[latex]\pi[/latex]	[latex]r=2\sin \left(\pi \right)=0[/latex]	[latex]0[/latex]

Figure 4

 Graphing Circles and the 5 Classic Polar Curves

Investigating Circles

Now we have seen the equation of a circle in the polar coordinate system. In the last two examples, the same equation was used to illustrate the properties of symmetry and demonstrate how to find the zeros, maximum values, and plotted points that produced the graphs. However, the circle is only one of many shapes in the set of polar curves.

There are five classic polar curves: cardioids, limaҫons, lemniscates, rose curves, and Archimedes’ spirals. We will briefly touch on the polar formulas for the circle before moving on to the classic curves and their variations.

Example 3: Sketching the Graph of a Polar Equation for a Circle

Sketch the graph of [latex]r=4\cos \theta[/latex].

Show Solution

First, testing the equation for symmetry, we find that the graph is symmetric about the polar axis. Next, we find the zeros and maximum [latex]|r|[/latex] for [latex]r=4\cos \theta[/latex]. First, set [latex]r=0[/latex], and solve for [latex]\theta[/latex] . Thus, a zero occurs at [latex]\theta =\frac{\pi }{2}\pm k\pi[/latex]. A key point to plot is [latex]\left(0,\text{ }\text{ }\frac{\pi }{2}\right)[/latex].

To find the maximum value of [latex]r[/latex], note that the maximum value of the cosine function is 1 when [latex]\theta =0\pm 2k\pi[/latex]. Substitute [latex]\theta =0[/latex] into the equation:

[latex]\begin{gathered}r=4\cos \theta \\ r=4\cos \left(0\right)\\ r=4\left(1\right)=4\end{gathered}[/latex]

The maximum value of the equation is 4. A key point to plot is [latex]\left(4,0\right)[/latex].

As [latex]r=4\cos \theta[/latex] is symmetric with respect to the polar axis, we only need to calculate r-values for [latex]\theta[/latex] over the interval [latex]\left[0,\pi \right][/latex]. Points in the upper quadrant can then be reflected to the lower quadrant. Make a table of values similar to the table below. The graph is shown in Figure 6.

[latex]\theta[/latex]	0	[latex]\frac{\pi }{6}[/latex]	[latex]\frac{\pi }{4}[/latex]	[latex]\frac{\pi }{3}[/latex]	[latex]\frac{\pi }{2}[/latex]	[latex]\frac{2\pi }{3}[/latex]	[latex]\frac{3\pi }{4}[/latex]	[latex]\frac{5\pi }{6}[/latex]	[latex]\pi[/latex]
[latex]r[/latex]	4	3.46	2.83	2	0	−2	−2.83	−3.46	4

Figure 6

Investigating Cardioids

While translating from polar coordinates to Cartesian coordinates may seem simpler in some instances, graphing the classic curves is actually less complicated in the polar system. The next curve is called a cardioid, as it resembles a heart. This shape is often included with the family of curves called limaçons, but here we will discuss the cardioid on its own.

A General Note: Formulas for a Cardioid

The formulas that produce the graphs of a cardioid are given by [latex]r=a\pm b\cos \theta[/latex] and [latex]r=a\pm b\sin \theta[/latex] where [latex]a>0,b>0[/latex], and [latex]\frac{a}{b}=1[/latex]. The cardioid graph passes through the pole, as we can see in Figure 7.

Figure 7

Example 4: Sketching the Graph of a Cardioid

Sketch the graph of [latex]r=2+2\cos \theta[/latex].

Show Solution

First, testing the equation for symmetry, we find that the graph of this equation will be symmetric about the polar axis. Next, we find the zeros and maximums. Setting [latex]r=0[/latex], we have [latex]\theta =\pi +2k\pi[/latex]. The zero of the equation is located at [latex]\left(0,\pi \right)[/latex]. The graph passes through this point.

The maximum value of [latex]r=2+2\cos \theta[/latex] occurs when [latex]\cos \theta[/latex] is a maximum, which is when [latex]\cos \theta =1[/latex] or when [latex]\theta =0[/latex]. Substitute [latex]\theta =0[/latex] into the equation, and solve for [latex]r[/latex].

[latex]\begin{gathered} r=2+2\cos \left(0\right) \\ r=2+2\left(1\right)=4 \end{gathered}[/latex]

The point [latex]\left(4,0\right)[/latex] is the maximum value on the graph.

We found that the polar equation is symmetric with respect to the polar axis, but as it extends to all four quadrants, we need to plot values over the interval [latex]\left[0,\pi \right][/latex]. The upper portion of the graph is then reflected over the polar axis. Next, we make a table of values, as in the table below, and then we plot the points and draw the graph. See Figure 8.

[latex]\theta[/latex]	[latex]0[/latex]	[latex]\frac{\pi }{4}[/latex]	[latex]\frac{\pi }{2}[/latex]	[latex]\frac{2\pi }{3}[/latex]	[latex]\pi[/latex]
[latex]r[/latex]	4	3.41	2	1	0

Figure 8

Investigating Limaçons

The word limaçon is Old French for “snail,” a name that describes the shape of the graph. As mentioned earlier, the cardioid is a member of the limaçon family, and we can see the similarities in the graphs. The other images in this category include the one-loop limaçon and the two-loop (or inner-loop) limaçon. One-loop limaçons are sometimes referred to as dimpled limaçons when [latex]1<\frac{a}{b}<2[/latex] and convex limaçons when [latex]\frac{a}{b}\ge 2[/latex].

A General Note: Formulas for One-Loop Limaçons

The formulas that produce the graph of a dimpled one-loop limaçon are given by [latex]r=a\pm b\cos \theta[/latex] and [latex]r=a\pm b\sin \theta[/latex] where [latex]a>0,b>0,\text{and 1<}\frac{a}{b}<2[/latex]. All four graphs are shown in Figure 9.

Figure 9. Dimpled limaçons

Example 5: Sketching the Graph of a One-Loop Limaçon

Graph the equation [latex]r=4 - 3\sin \theta[/latex].

Show Solution

First, testing the equation for symmetry, we find that it fails all three symmetry tests, meaning that the graph may or may not exhibit symmetry, so we cannot use the symmetry to help us graph it. However, this equation has a graph that clearly displays symmetry with respect to the line [latex]\theta =\frac{\pi }{2}[/latex], yet it fails all the three symmetry tests. A graphing calculator will immediately illustrate the graph’s reflective quality.

Next, we find the zeros and maximum, and plot the reflecting points to verify any symmetry. Setting [latex]r=0[/latex] results in [latex]\theta[/latex] being undefined. What does this mean? How could [latex]\theta[/latex] be undefined? The angle [latex]\theta[/latex] is undefined for any value of [latex]\sin \theta >1[/latex]. Therefore, [latex]\theta[/latex] is undefined because there is no value of [latex]\theta[/latex] for which [latex]\sin \theta >1[/latex]. Consequently, the graph does not pass through the pole. Perhaps the graph does cross the polar axis, but not at the pole. We can investigate other intercepts by calculating [latex]r[/latex] when [latex]\theta =0[/latex].

[latex]\begin{gathered}r\left(0\right)=4 - 3\sin \left(0\right) \\ r=4 - 3\cdot 0=4 \end{gathered}[/latex]

So, there is at least one polar axis intercept at [latex]\left(4,0\right)[/latex].

Next, as the maximum value of the sine function is 1 when [latex]\theta =\frac{\pi }{2}[/latex], we will substitute [latex]\theta =\frac{\pi }{2}[/latex] into the equation and solve for [latex]r[/latex]. Thus, [latex]r=1[/latex].

Make a table of the coordinates similar to the table below.

[latex]\theta[/latex]	[latex]0[/latex]	[latex]\frac{\pi }{6}[/latex]	[latex]\frac{\pi }{3}[/latex]	[latex]\frac{\pi }{2}[/latex]	[latex]\frac{2\pi }{3}[/latex]	[latex]\frac{5\pi }{6}[/latex]	[latex]\pi[/latex]	[latex]\frac{7\pi }{6}[/latex]	[latex]\frac{4\pi }{3}[/latex]	[latex]\frac{3\pi }{2}[/latex]	[latex]\frac{5\pi }{3}[/latex]	[latex]\frac{11\pi }{6}[/latex]	[latex]2\pi[/latex]
[latex]r[/latex]	4	2.5	1.4	1	1.4	2.5	4	5.5	6.6	7	6.6	5.5	4

The graph is shown in Figure 10.

Figure 10. One-loop limaçon

Analysis of the Solution

This is an example of a curve for which making a table of values is critical to producing an accurate graph. The symmetry tests fail; the zero is undefined. While it may be apparent that an equation involving [latex]\sin \theta[/latex] is likely symmetric with respect to the line [latex]\theta =\frac{\pi }{2}[/latex], evaluating more points helps to verify that the graph is correct.

Another type of limaçon, the inner-loop limaçon, is named for the loop formed inside the general limaçon shape. It was discovered by the German artist Albrecht Dürer(1471-1528), who revealed a method for drawing the inner-loop limaçon in his 1525 book Underweysung der Messing. A century later, the father of mathematician Blaise Pascal, Étienne Pascal(1588-1651), rediscovered it.