Introduction to polar coordinates

We are already familiar with the Cartesian coordinate system, which is built on two orthogonal axes. The polar coordinate system is a way of describing the same points from a different perspective. Rather than $x$ and $y$, it uses the radial and angular coordinates $r$ and $\varphi$, respectively. Let's look at how these new coordinates work.

Navigating Antarctica with polar coordinates

Most maps include a compass rose – a symbol to remind you that each cardinal direction (North, South, East, and West) is represented by a certain direction on the map. Near the poles, however, this assumption breaks down. Observe:

If you're standing at the South Pole, every direction is north. You can walk 20 degrees north to get to the Ross Ice Shelf, or about 30 degrees in a different north and swim in the Weddel Sea. So the latitude of a point gives us its distance from the south pole. Our first polar coordinate $r$ does the same thing, but with the origin $(0,0)$ standing in for the pole.

Why the letter $r$? For a hint, notice that the latitudes 20°S, 40°S, and 60°S are represented by circles. For any positive number $c$, the equation $r = c$ will graph a circle with radius $c$ – every point is a distance $c$ from the origin. So we call the first coordinate $r$ for radius. (Some people prefer to use $\rho$, the Greek letter rho, for radius. The notation of polar coordinates is not as universal as the Cartesian $x$ and $y$.)

Since $r$ represents distance without direction, it makes sense for the second coordinate $\varphi$ to represent direction without distance. On the map above, that's exactly what the longitudes (East-West) are doing. Each straight line marks a direction leading away from the pole. Using both coordinates, we can say for example that South Orkney Island is close to the point (60°S, 30°W).

On the plane, of course, we don't care about East or West, so those would just be plain numbers. Also, while the 0° line on the map points straight up, on the plane it goes to the right, along the positive $x$–axis, and the angle increases as you turn counterclockwise. The letter $\varphi$ (phi) doesn't stand for anything; sometimes you'll see a $\theta$ (theta) instead.

Points on the plane

That's enough theory for now. What does it actually look like when you plot points this way?

There are two points in the figure above. The green one has coordinates $(3,60\degree)$, which means that after turning the positive $x$–axis $L$ by $60$ degrees and counting $3$ on it, we will get to that point. Another way to get there is to draw a circle of radius $3$ and measure an arc that goes $1/6$ of the way around, because $\frac{60\degree}{360\degree} = 1/6$ of a circle. The blue point has coordinates $(4,210\degree)$, so it's farther from the origin. As the angular coordinate $\varphi$ approaches $360\degree$, the point gets closer to $L$, and when it's more than that, something strange happens: the same point can be labeled with multiple pairs of coordinates!

The green point above, as we've seen, can be found at $(3,60\degree)$ – a $60\degree$ turn of the axis – but it also has the coordinates $(3, 420\degree)$, corresponding to a complete rotation plus another $60\degree$. Or $(3,-300\degree)$, if you turn clockwise instead. Some authors would even call it $(-3,240\degree)$ – turn a little past the blue point and then count backwards – but many do not allow negative values of $r$. What this means is that polar coordinates are not unique. Unlike the Cartesian world, in which we can talk about the $x$–coordinate of a particular point, here there are infinitely many $\varphi$–coordinates that work equally well.

How do we measure angles?

By now you're starting to understand the crucial role that angles play in polar coordinates. But there's something we've left out: angles in mathematics are usually measured with radians, not degrees. You may remember radians from trigonometry, but it never hurts to review.

Degree measures are useful in certain situations because the number $360$ has a lot of divisors; it's easy to recognize a certain measure as $1/6, \, 1/10,$ or even $1/72$ of a full circle. Their weakness is in measuring arc length, or pieces of a circle's circumference. These calculations come up in physics whenever an object moves in a circular path. With an interior angle of $d\degree$and a radius of $r$, the arc length $s$ is given by $s = \frac{dr\pi}{180}$. It's not impossible to remember, but it's not obvious either. Enter the radian:

When you measure an angle in radians, the constant in that formula disappears, leaving $s = r\theta$. In the figure above, it evaluates to $1\cdot1 = 1$. In words, an angle of $1$ radian intercepts an arc whose length is exactly the radius of the circle. Isn't that convenient?

In practice, it's rare to see a measure of $1,2,$ or $3$ radians; because the full circle consists of $2\pi$ radians, most angle measures you encounter will be multiples of $\pi$. Here's a cheat sheet of degree–radian conversions:

$\def\arraystretch{1.4} \begin{array}{c|c} \text{deg} & \text{rad} \\ \hline 0\degree & 0 \\ 30\degree & \frac{\pi}{6} \\ 45\degree & \frac{\pi}{4} \\ 90\degree & \frac{\pi}{2} \\ 180\degree & \pi \\ 360\degree & 2\pi \end{array}$

If you need to convert another angle, just use these formulas, where $d$ the angle measure in degrees and $\theta$ in radians:

$d = \frac{180}{\pi} \: \theta$
$\theta = \frac{\pi}{180} \: d$

Check these against the table for $360$, $180$, and $90$ degrees, making sure that they're correct both ways. From this point on we will use radians to measure our angles.

Conclusion

So, now it's obvious to us that Cartesian coordinates are not the only way to set a coordinate grid in space. For example, we can use the polar coordinate system as an alternative. The polar coordinates of a certain point $M$ are two parameters $r$ and $\varphi$. The first of them is called the radius and defines the distance between $M$ and the origin $(0, 0)$. The second one specifies the angle that we have to turn the positive $x$-axis to get the radius. Already tired of theory? Well, it's time to consolidate your knowledge and practice!