1.3 Polar Coordinates

Defining Polar Coordinates

To find the coordinates of a point in the polar coordinate system, consider Figure 1.27. The point $P$ has Cartesian coordinates $\left(x,y\right).$ The line segment connecting the origin to the point $P$ measures the distance from the origin to $P$ and has length $r.$ The angle between the positive $x$-axis and the line segment has measure $\theta .$ This observation suggests a natural correspondence between the coordinate pair $\left(x,y\right)$ and the values $r$ and $\theta .$ This correspondence is the basis of the polar coordinate system. Note that every point in the Cartesian plane has two values (hence the term ordered pair) associated with it. In the polar coordinate system, each point also has two values associated with it: $r$ and $\theta .$

Figure 1.27 An arbitrary point in the Cartesian plane.

Using right-triangle trigonometry, the following equations are true for the point $P\text{:}$

Furthermore,

Each point $\left(x,y\right)$ in the Cartesian coordinate system can therefore be represented as an ordered pair $\left(r,\theta \right)$ in the polar coordinate system. The first coordinate is called the radial coordinate and the second coordinate is called the angular coordinate. Every point in the plane can be represented in this form.

Note that the equation $\text{tan}\theta =y\text{/}x$ has an infinite number of solutions for any ordered pair $\left(x,y\right).$ However, if we restrict the solutions to values between $0$ and $2\pi$ then we can assign a unique solution to the quadrant in which the original point $\left(x,y\right)$ is located. Then the corresponding value of r is positive, so $r^2=x^2+y^2.$

The polar representation of a point is not unique. For example, the polar coordinates $\left(2,\frac{\pi }{3}\right)$ and $\left(2,\frac{7\pi }{3}\right)$ both represent the point $\left(1,\sqrt{3}\right)$ in the rectangular system. Also, the value of $r$ can be negative. Therefore, the point with polar coordinates $\left(\mathrm{-2},\frac{4\pi }{3}\right)$ also represents the point $\left(1,\sqrt{3}\right)$ in the rectangular system, as we can see by using Equation 1.8:

Every point in the plane has an infinite number of representations in polar coordinates. However, each point in the plane has only one representation in the rectangular coordinate system.

Note that the polar representation of a point in the plane also has a visual interpretation. In particular, $r$ is the directed distance that the point lies from the origin, and $\theta$ measures the angle that the line segment from the origin to the point makes with the positive $x$-axis. Positive angles are measured in a counterclockwise direction and negative angles are measured in a clockwise direction. The polar coordinate system appears in the following figure.

Figure 1.28 The polar coordinate system.

The line segment starting from the center of the graph going to the right (called the positive x-axis in the Cartesian system) is the polar axis. The center point is the pole, or origin, of the coordinate system, and corresponds to $r=0.$ The innermost circle shown in Figure 1.28 contains all points a distance of 1 unit from the pole, and is represented by the equation $r=1.$ Then $r=2$ is the set of points 2 units from the pole, and so on. The line segments emanating from the pole correspond to fixed angles. To plot a point in the polar coordinate system, start with the angle. If the angle is positive, then measure the angle from the polar axis in a counterclockwise direction. If it is negative, then measure it clockwise. If the value of $r$ is positive, move that distance along the terminal ray of the angle. If it is negative, move along the ray that is opposite the terminal ray of the given angle.

Example 1.11

Plotting Points in the Polar Plane

Plot each of the following points on the polar plane.

1. $\left(2,\frac{\pi }{4}\right)$
2. $\left(\mathrm{-3},\frac{2\pi }{3}\right)$
3. $\left(4,\frac{5\pi }{4}\right)$

Solution

The three points are plotted in the following figure.

Figure 1.29 Three points plotted in the polar coordinate system.

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.

Example 1.12

Graphing a Function in Polar Coordinates

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

Solution

Because the function is a multiple of a sine function, it is periodic with period $2\pi ,$ so use values for $\theta$ between 0 and $2\pi .$ The result of steps 1–3 appear in the following table. Figure 1.30 shows the graph based on this table.

$\theta$	$r=4\text{sin}\theta$		$\theta$	$r=4\text{sin}\theta$
0	0		$\pi$	0
$\frac{\pi }{6}$	$2$		$\frac{7\pi }{6}$	$\mathrm{-2}$
$\frac{\pi }{4}$	$2\sqrt{2}\approx 2.8$		$\frac{5\pi }{4}$	$\mathrm{-2}\sqrt{2}\approx \mathrm{-2.8}$
$\frac{\pi }{3}$	$2\sqrt{3}\approx 3.4$		$\frac{4\pi }{3}$	$\mathrm{-2}\sqrt{3}\approx \mathrm{-3.4}$
$\frac{\pi }{2}$	$4$		$\frac{3\pi }{2}$	$-4$
$\frac{2\pi }{3}$	$2\sqrt{3}\approx 3.4$		$\frac{5\pi }{3}$	$\mathrm{-2}\sqrt{3}\approx \mathrm{-3.4}$
$\frac{3\pi }{4}$	$2\sqrt{2}\approx 2.8$		$\frac{7\pi }{4}$	$\mathrm{-2}\sqrt{2}\approx \mathrm{-2.8}$
$\frac{5\pi }{6}$	$2$		$\frac{11\pi }{6}$	$\mathrm{-2}$
			$2\pi$	0

Figure 1.30 The graph of the function $r=4\text{sin}\theta$ is a circle.

This is the graph of a circle. The equation $r=4\text{sin}\theta$ can be converted into rectangular coordinates by first multiplying both sides by $r.$ This gives the equation $r^2=4r\text{sin}\theta .$ Next use the facts that $r^2=x^2+y^2$ and $y=r\text{sin}\theta .$ This gives $x^2+y^2=4y.$ To put this equation into standard form, subtract $4y$ from both sides of the equation and complete the square:

$$\begin{array}{ccc}\hfill x^2+y^2-4y& =\hfill & 0\hfill \\ \hfill x^2+\left(y^2-4y\right)& =\hfill & 0\hfill \\ \hfill x^2+\left(y^2-4y+4\right)& =\hfill & 0+4\hfill \\ \hfill x^2+{\left(y-2\right)}^2& =\hfill & 4.\hfill \end{array}$$

This is the equation of a circle with radius 2 and center $\left(0,2\right)$ in the rectangular coordinate system.

The graph in Example 1.12 was that of a circle. The equation of the circle can be transformed into rectangular coordinates using the coordinate transformation formulas in Equation 1.8. Example 1.14 gives some more examples of functions for transforming from polar to rectangular coordinates.

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.

Figure 1.31

Figure 1.32

A cardioid is a special case of a limaçon (pronounced “lee-mah-son”), in which $a=b$ or $a=\text{−}b.$ The rose is a very interesting curve. Notice that the graph of $r=3\text{sin}2\theta$ has four petals. However, the graph of $r=3\text{sin}3\theta$ has three petals as shown.

Figure 1.33 Graph of $r=3\text{sin}3\theta .$

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 1.34(a)). However, if the coefficient is irrational, then the curve never closes (Figure 1.34(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.

Figure 1.34 Polar rose graphs of functions with (a) rational coefficient and (b) irrational coefficient. Note that the rose in part (b) does not fill the disk of radius 3 centered at the origin, even though it appears to. To see this, just note that for any fixed ray from the origin, there are only a countable number of values of theta for which we get a point on the graph along that ray. What we can say, though, is that for any point in the disk, there are points as close as you please to that point on the given graph.

Since the curve defined by the graph of $r=3\text{sin}\left(\pi \theta \right)$ never closes, the curve depicted in Figure 1.34(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.

Example 1.14

Chapter Opener: Describing a Spiral

Recall the chambered nautilus introduced in the chapter opener. This creature displays a spiral when half the outer shell is cut away. It is possible to describe a spiral using rectangular coordinates. Figure 1.35 shows a spiral in rectangular coordinates. How can we describe this curve mathematically?

Figure 1.35 How can we describe a spiral graph mathematically?

Solution

As the point P travels around the spiral in a counterclockwise direction, its distance d from the origin increases. Assume that the distance d is a constant multiple k of the angle $\theta$ that the line segment OP makes with the positive x-axis. Therefore $d\left(P,O\right)=k\theta ,$ where $O$ is the origin. Now use the distance formula and some trigonometry:

$$\begin{array}{ccc}\hfill d\left(P,O\right)& =\hfill & k\theta \hfill \\ \hfill \sqrt{{\left(x-0\right)}^2+{\left(y-0\right)}^2}& =\hfill & k\text{arctan}\left(\frac{y}{x}\right)\hfill \\ \hfill \sqrt{x^2+y^2}& =\hfill & k\text{arctan}\left(\frac{y}{x}\right)\hfill \\ \hfill \text{arctan}\left(\frac{y}{x}\right)& =\hfill & \frac{\sqrt{x^2+y^2}}{k}\hfill \\ \hfill y& =\hfill & x\text{tan}\left(\frac{\sqrt{x^2+y^2}}{k}\right).\hfill \end{array}$$

Although this equation describes the spiral, it is not possible to solve it directly for either x or y. However, if we use polar coordinates, the equation becomes much simpler. In particular, $d\left(P,O\right)=r,$ and $\theta$ is the second coordinate. Therefore the equation for the spiral becomes $r=k\theta .$ Note that when $\theta =0$ we also have $r=0,$ so the spiral emanates from the origin. We can remove this restriction by adding a constant to the equation. Then the equation for the spiral becomes $r=a+k\theta$ for arbitrary constants $a$ and $k.$ This is referred to as an Archimedean spiral, after the Greek mathematician Archimedes.

Another type of spiral is the logarithmic spiral, described by the function $r=a·b^{\theta }.$ A graph of the function $r=1.2\left({1.25}^{\theta }\right)$ is given in Figure 1.36. This spiral describes the shell shape of the chambered nautilus.

Figure 1.36 A logarithmic spiral is similar to the shape of the chambered nautilus shell. (credit: modification of work by Jitze Couperus, Flickr)

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

it is possible to rewrite these formulas using the function

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 1.31 becomes

Letting $\theta$ range from $\text{−}\infty$ 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 $f\left(\text{−}x\right)=f\left(x\right)$ 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 $f\left(\text{−}x\right)=\text{−}f\left(x\right)$ 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.

The following table shows examples of each type of symmetry.