Distances and functions in polar coordinate system

Previously we looked through two different coordinate systems: Cartesian and polar. Maybe, you have already had this question: how are they connected with each other? The need to convert from one coordinate system to another constantly arises in many mathematical problems in order to simplify calculations. For example, look at the equation of a circle with radius $5$ in Cartesian coordinates: $x^2 + y^2 = 25$. In polar coordinates we can write it down like this: $r = 5$. It's much better and easier to use, isn't it?

In addition, note that in Cartesian coordinates we already know how to calculate the distance between points and build graphs of functions. There is a natural desire to learn how to do the same in polar coordinates. Let's figure out how it all works.

Conversions between Cartesian and polar coordinates

If given the polar coordinates of a point, how can you find its Cartesian coordinates? What about vice versa? That's what we'll look at in this section. To convert from polar to Cartesian coordinates, we can draw a right triangle:

Basic trig functions give us the matching equations $\cos\varphi = \frac{x}{r}$ and $\sin\varphi = \frac{y}{r}$, which reduce respectively to

$x=r \cos{\varphi}, \quad y=r \sin{\varphi}.$

Now let's go the other direction.

Getting a radius from Cartesian coordinates is easy: $r^2=x^2+y^2$, so $r = \sqrt{x^2 + y^2}$. If you only take the positive square root, each point has a single possible radius. But as you remember, each point has infinitely many polar angles, so determining the $\varphi$–coordinate requires making a choice.

First, note from the triangle above that $\tan \varphi = \frac{y}{x}$. To solve for $\varphi$, we'll need to "undo" the $\tan$ function, whose graph (in Cartesian coordinates) looks like this:

This is an unfriendly function for several reasons. Firstly, it's not injective: if we drew a horizontal line corresponding to $\frac{y}{x} = 1$, that line would intersect the graph at more than one point. Secondly, the intervals are too short: if we trace an arc from $\varphi = -\frac{\pi}{2}$ to $\varphi = \frac{\pi}{2}$, it will cover only the right half of a circle. Thirdly, it has vertical asymptotes (dotted lines) where the angles $\varphi = -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2},\dots$ don't have defined tangents, despite mapping to perfectly good points on the $y$–axis. To solve these problems, early programmers developed the $\text{atan2}$ function, which is a bit of a Frankenstein:

Armed with $\text{atan2}$, we can now send any pair $(x,y)$ to an angular coordinate in the interval $-\pi < \varphi \le \pi$. So our conversions are:

$$\begin{align*} (x,y) \to& \left(\sqrt{x^2 + y^2}, \; \text{atan2}(y,x)\right) \quad \text{and} \\ (r,\varphi) \to& \; (r\cos\varphi, \; r\sin\varphi). \end{align*}$$

Distance between two points

In the Cartesian coordinate system, the distance formula is a simple application of the Pythagorean formula. In polar coordinates, we need to bring in an extension called the law of cosines. If $a,b,$ and $c$ are the side lengths of any triangle, and the angle $\gamma$ is opposite side $c$, we have:

$$c^2 = a^2 + b^2 - 2ab\cos \gamma$$In the case that $\gamma$ is a right angle, the cosine will be $0$ and the equation reduces to the familiar $a^2 + b^2 = c^2$. To see the applicability to polar coordinates, study this figure:

We can use $r_1$ and $r_2$ as side lengths $a$ and $b$, then calculate $\gamma = \varphi_2-\varphi_1$, so that the unknown length $c$ is the distance between points $M_1$ and $M_2$. After taking the square root of both sides we're left with

$$c=\sqrt{r_1^2+r_2^2-2r_1r_2\cos{(\varphi_2-\varphi_1)}}.$$

Like most problems involving straight lines, this is easier in Cartesian coordinates than in polar.

Graphing functions in polar coordinates

In Cartesian coordinates, we typically write functions as $y = f(x)$. In polar coordinates, that becomes $r = f(\varphi)$, meaning that the dependent and independent variables are radius and polar angle, respectively.

As we mentioned, a circle centered on the origin with radius $a$ is described with the equation $r=a$. The Cartesian equivalent would be the horizontal line $y = a$.

No matter what angle you input, the radius will be $a$, so the circle includes the points $(a, 0),\, \left(a,\frac{\pi}{2}\right),\, (a,\pi),$ etc. It's trivial to show that, for any angle $\varphi$, $f(\varphi) = f(\varphi + 2\pi)$, so the whole graph can be drawn in a single rotation.

The sine and cosine functions can draw polar roses with beautifully symmetrical petals. For example, the function $r=\sin(4\varphi)$ defines this rose:

Just like the circle, this function wraps around and starts tracing over itself when $\varphi = 2\pi$. Does every function have that property?

In Cartesian coordinates, a line has the equation $y = mx + b$. In polar coordinates, the corresponding equation $r=a+b\varphi$ graphs an Archimedean spiral. Letting $a=0$ and $b = 1$, we get the following graph:

Feeling sleepy? Don't worry, we're almost done.

Conclusion

The polar coordinate system is not as prevalent as the Cartesian, but it still finds use in physics, engineering, navigation, and more. When embarking on a project that involves points on a plane, stop and consider: is there a central point around which everything else revolves? Will I need to simulate circular or spiraling trajectories? If so, you may want to use polar coordinates. As we've seen, they do have drawbacks, but in many situations they're the best option.