Graphs and Symmetry of Polar Curves

Learning Outcomes

• Sketch polar curves from given equations
• Convert equations between rectangular and polar coordinates
• Identify symmetry in polar curves and equations

Polar Curves

Now that we know how to plot points in the polar coordinate system, we can discuss how to plot curves. In the rectangular coordinate system, we can graph a function $y=f\left(x\right)$ and create a curve in the Cartesian plane. In a similar fashion, we can graph a curve that is generated by a function $r=f\left(\theta \right)$.

The general idea behind graphing a function in polar coordinates is the same as graphing a function in rectangular coordinates. Start with a list of values for the independent variable ($\theta$ in this case) and calculate the corresponding values of the dependent variable $r$. This process generates a list of ordered pairs, which can be plotted in the polar coordinate system. Finally, connect the points, and take advantage of any patterns that may appear. The function may be periodic, for example, which indicates that only a limited number of values for the independent variable are needed.

Problem-Solving Strategy: Plotting a Curve in Polar Coordinates

1. Create a table with two columns. The first column is for $\theta$, and the second column is for $r$.
2. Create a list of values for $\theta$.
3. Calculate the corresponding $r$ values for each $\theta$.
4. Plot each ordered pair $\left(r,\theta \right)$ on the coordinate axes.
5. Connect the points and look for a pattern.

Example: Graphing a Function in Polar Coordinates

Graph the curve defined by the function $r=4\sin\theta$. Identify the curve and rewrite the equation in rectangular coordinates.

Watch the following video to see the worked solution to Example: Graphing a Function in Polar Coordinates.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

try it

Create a graph of the curve defined by the function $r=4+4\cos\theta$.

The graph in the previous example was that of a circle. The equation of the circle can be transformed into rectangular coordinates using the coordinate transformation formulas in the theorem. The example after the next gives some more examples of functions for transforming from polar to rectangular coordinates.

Example: Transforming Polar Equations to Rectangular Coordinates

Rewrite each of the following equations in rectangular coordinates and identify the graph.

1. $\theta =\frac{\pi }{3}$
2. $r=3$
3. $r=6\cos\theta -8\sin\theta$

Watch the following video to see the worked solution to Example: Transforming Polar Equations to Rectangular Coordinates.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

try it

Rewrite the equation $r=\sec\theta \tan\theta$ in rectangular coordinates and identify its graph.

We have now seen several examples of drawing graphs of curves defined by polar equations. A summary of some common curves is given in the tables below. In each equation, a and b are arbitrary constants.

A cardioid is a special case of a limaçon (pronounced “lee-mah-son”), in which $a=b$ or $a=-b$. The rose is a very interesting curve. Notice that the graph of $r=3\sin2\theta$ has four petals. However, the graph of $r=3\sin3\theta$ has three petals as shown.

If the coefficient of $\theta$ is even, the graph has twice as many petals as the coefficient. If the coefficient of $\theta$ is odd, then the number of petals equals the coefficient. You are encouraged to explore why this happens. Even more interesting graphs emerge when the coefficient of $\theta$ is not an integer. For example, if it is rational, then the curve is closed; that is, it eventually ends where it started (Figure 10 (a)). However, if the coefficient is irrational, then the curve never closes (Figure 10 (b)). Although it may appear that the curve is closed, a closer examination reveals that the petals just above the positive x axis are slightly thicker. This is because the petal does not quite match up with the starting point.

Since the curve defined by the graph of $r=3\sin\left(\pi \theta \right)$ never closes, the curve depicted in Figure 10 (b) is only a partial depiction. In fact, this is an example of a space-filling curve. A space-filling curve is one that in fact occupies a two-dimensional subset of the real plane. In this case the curve occupies the circle of radius 3 centered at the origin.

Suppose a curve is described in the polar coordinate system via the function $r=f\left(\theta \right)$. Since we have conversion formulas from polar to rectangular coordinates given by

$\begin{array}{c}x=r\cos\theta \hfill \\ y=r\sin\theta ,\hfill \end{array}$

it is possible to rewrite these formulas using the function

$\begin{array}{c}x=f\left(\theta \right)\cos\theta \hfill \\ y=f\left(\theta \right)\sin\theta .\hfill \end{array}$

This step gives a parameterization of the curve in rectangular coordinates using $\theta$ as the parameter. For example, the spiral formula $r=a+b\theta$ from Figure 7 becomes

$\begin{array}{c}x=\left(a+b\theta \right)\cos\theta \hfill \\ y=\left(a+b\theta \right)\sin\theta .\hfill \end{array}$

Letting $\theta$ range from $-\infty$ to $\infty$ generates the entire spiral.

Symmetry in Polar Coordinates

When studying symmetry of functions in rectangular coordinates (i.e., in the form $y=f\left(x\right)$), we talk about symmetry with respect to the y-axis and symmetry with respect to the origin. In particular, if $f\left(-x\right)=f\left(x\right)$ for all $x$ in the domain of $f$, then $f$ is an even function and its graph is symmetric with respect to the $y$-axis. If $f\left(-x\right)=-f\left(x\right)$ for all $x$ in the domain of $f$, then $f$ is an odd function and its graph is symmetric with respect to the origin. By determining which types of symmetry a graph exhibits, we can learn more about the shape and appearance of the graph. Symmetry can also reveal other properties of the function that generates the graph. Symmetry in polar curves works in a similar fashion.

theorem: Symmetry in Polar Curves and Equations

Consider a curve generated by the function $r=f\left(\theta \right)$ in polar coordinates.

1. The curve is symmetric about the polar axis if for every point $\left(r,\theta \right)$ on the graph, the point $\left(r,-\theta \right)$ is also on the graph. Similarly, the equation $r=f\left(\theta \right)$ is unchanged by replacing $\theta$ with $-\theta$.
2. The curve is symmetric about the pole if for every point $\left(r,\theta \right)$ on the graph, the point $\left(r,\pi +\theta \right)$ is also on the graph. Similarly, the equation $r=f\left(\theta \right)$ is unchanged when replacing $r$ with $-r$, or $\theta$ with $\pi +\theta$.
3. The curve is symmetric about the vertical line $\theta =\frac{\pi }{2}$ if for every point $\left(r,\theta \right)$ on the graph, the point $\left(r,\pi -\theta \right)$ is also on the graph. Similarly, the equation $r=f\left(\theta \right)$ is unchanged when $\theta$ is replaced by $\pi -\theta$.

The following table shows examples of each type of symmetry.

Figure 11.

Example: using Symmetry to Graph a Polar Equation

Find the symmetry of the rose defined by the equation $r=3\sin\left(2\theta \right)$ and create a graph.

Watch the following video to see the worked solution to Example: using Symmetry to Graph a Polar Equation.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

try it

Determine the symmetry of the graph determined by the equation $r=2\cos\left(3\theta \right)$ and create a graph.