Writing is thinking. To write well is to think clearly. That's why it's so hard.

~ David McCullough
 

Polar Coordinates

2018年07月31日

In high school, we learn about cartesean and polar coordinates for describing points on the plane. The formulas we are asked to learn are:

$$ \begin{array}{rlcrl} r & = \sqrt{x^2 + y^2} & & x &= r \cos \theta \\ \theta & = \tan^{-1}\left(\frac{y}{x}\right) & & y &= r \sin \theta \end{array} $$

While these four formulas are enough to get you an "A" in the class, there's actually a lot packed into these lines which unfrotunately goes left unsaid, particularly in the two equations on the left. Let's pick up a loose string, pull at it, and see what happens. The place we're going to start is the most intense of the four equations: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$.

We know that angles range from $0^\circ$ to $360^\circ$. Actually, any span ranging a total of $360^\circ$ would work, so for today, let's say the angle must be from $-180^\circ$ to $+180^\circ$. Plot the graph of $\tan^{-1}$ (in degrees) and see how the function behaves.

No matter the input value $\frac{y}{x}$ we use, the value of $\tan^{-1}(\frac{y}{x})$ must lie between $+90^\circ$ and $-90^\circ$. This is only half of the range we need!

For instance, suppose we have a point written in cartesean coordinates $(x, y) = (-1, -1)$, and we want to represent this same point in polar coordinates. Sketching a plot of the point, we see that the $\theta$ coordinate is $225^\circ$. But since we said we want our angles to fall between $-180^\circ$ and $+180^\circ$, let's go around clockwise instead and call this angle $-135^\circ$.

But we have a problem: The graph of $\tan^{-1}$ never goes above $90^\circ$ or below $-90^\circ$, so $\theta$ can't be either of these values. And if we follow our original formula to calculate $\theta$, we find that we should have $\tan^{-1}(\frac{y}{x}) = \tan^{-1}\left(\frac{-1}{-1}\right) = \tan^{-1}(1) = 45^\circ$. This is totally wrong!

Another set of equations

Some textbooks conveniently avoid this issue by writing the equations slightly differently:

$$ \begin{array}{rlcrl} r^2 & = x^2 + y^2 & & x &= r \cos \theta \\ \tan(\theta) & = \frac{y}{x} & & y &= r \sin \theta \end{array} $$

We can verify our example of $(x, y) = (-1, -1)$, using $-135^\circ$ for the value of $\theta$, just as our eyes told us it should be:

$$ \tan(\theta) = \tan(-135^\circ) = 1 = \frac{-1}{-1} = \frac{y}{x} $$

And so the equation is seen to hold.

But this second set of formulas is guilty of the crime of being technically correct. It fails to disclose an important set of ideas to the student. Why does one version of the equation work and not the other? A good student would just accept this as a fact. A great student would instead be bothered by this fact.

Polar coordinates are not unique

There are number of subtlties that polar coordinates present. In cartesean coordinates every point has a unique set of coordinates, and conversely, each set of coordinates identifies a unique point. Polar coordinates do not have this same niceness! They have some fairly severe issues with uniqueness. The point $(x, y) = (0, 0)$, for instance, can be represented as $(r, \theta) = (0, 0)$ or as $(r, \theta) = (0, 10)$ or as $(r, \theta) = (0, ?)$ where the $?$ can be replaced by your favorite number. When $r=0$, the value for $\theta$ is geometrically meaningless. And so the origin, $(x, y) = (0, 0)$, has infinitely many disinct polar representations.

Even for the points away from the origin, the $\theta$ value isn't uniquely determined. You can always add $360^\circ$ to the angle and the point is still the same point. So $(r, \theta) = (1, 0^\circ)$ can just as well be given coordinates $(1, 360^\circ)$. This is occasionally even useful: Our standing assumption for today is that all angles can be represented between $-180^\circ$ and $+180^\circ$, and this is the same principle at play.

The last place we find a failure of uniqueness actually has some explaining power. We've been focusing on $\theta$, but it turns out that $r$ isn't unique either. If $(r, \theta)$ are polar coordinates for a point, then so is $(-r, \theta + 180^\circ)$. That is, we can always negate the radius $r$ and compensate by adding $180^\circ$ to $\theta$.

You can see how this manifests in the equations. In the second version, the squaring part of $r^2$ effectively hides the original sign of $r$. And, ah ha!: $\tan(\theta)$ is periodic with a period of $180^\circ$, and so it hides the $180^\circ$ offset we might have added to compensate for the sign of $r$!

If I am lucky (and if you're lucky too), I might have conveyed a particular feeling of amazement that mathematics sometimes brings: You are working hard on a problem, trying to understand a bizarre set of phenomenon, only to find that the situation works itself out perfectly. Mathematics is full of conspiracies of logic, where things look inconsistent on the surface, but upon deeper inspection, you find out it couldn't be changed even the slightest without breaking consistency. Mathematics is a bit weird sometimes, and there's no other way it could be.

Multiple-valued mayhem

To close this out, I can't help but touch on one last aspect in the original set of questions. This time, let's start by thinking about the first equation: $r = \sqrt{x^2 + y^2}$.

Perhaps you are the kind of person who thinks to themselves that the expression $\sqrt{4}$ is equal to $+2$ or $-2$. This idea -- that one expression (involving no variables) could take on two different values in the same equation -- bothered me throughout the latter half of high school and throughout college. And even today, whenever the subject comes up, I am quick to assert my worldview that $\sqrt{x}$ should always be used to denote the principal (ie, positive) sqaure root, and never otherwise. Ocassionally, on the internet or elsewhere, I hear people mention square root is a "multiple-valued function". This is absolute blasphemy, and I will not tolerate it. Functions take on single values.

However, I will leave this open for you to think about. If we were to sin and for the night call $\sqrt{x}$ a multiple-valued function, then we could perhaps get by with saying $r = \sqrt{x^2 + y^2}$. If we did, then our other formula, $\theta = \tan^{-1}(\frac{y}{x})$ would work unimpeded, since we could always take $r$ to be negative when it was needed. On the other hand, if we require $r = \sqrt{x^2 + y^2}$ be understood under my dogmatic view that the square root is always the positive value, then we would be forced to adopt the equally sinful view that $\tan^{-1}$ is multiple-valued, taking its principle value between $-90^\circ$ and $+90^\circ$, but also all values offset an integer multiple of $180^\circ$ from that.

But if we wanted to remain wholly pious in the eyes of the logicians, we might instead look elsewhere for guidance. As we found earlier, points can have multiple polar coordinates. But if we restrict our attention to a small neighborhood of points, letting the polar coordinates vary continuously, but never letting points reach the origin, nor letting them wrap too far around the origin to meet back up with themselves, then each point in our neighborhood of interest will have a unique polar coordinate.

If, for instance, we could restrain ourselves and only ever look at points on the right half-plane (meaning, points $(x, y)$ where $x > 0$), then we could firmly stipulate that $r$ is always taken to be positive and that $\theta$ will always fall between $-90^\circ$ and $+90^\circ$. And in doing so, both sets of equations would work perfectly to tranlate points between cartesean to polar coordinates and back. We would never run into contradictions. And we would never have to invoke arguments about multiple-valued functions.

qed