From polar to cartesian coordinates

When we’re looking at the plane, we’ve learned to love Cartesian coordinates. They’ve been so useful in so many ways. It turns out, though, that there is another alternate method to assign coordinates to points in the plane, which is really very different. It’s called the method of polar coordinates. As we’ll see, it’s a beautiful subject that leads to beautiful diagrams. How does it work exactly? Well, polar coordinates begin by specifying two things– first, a point in the plane that we call the pole, and then a ray emanating from that point that we call the polar axis. And that’s all there is for describing polar coordinates. How does it work?

If I give myself a typical point in the plane, what are its coordinates– its polar coordinates– relative to this mechanism? Well, the first coordinate is simply the distance from the pole to the point. Let’s call that r. That’s the first polar coordinate. The second coordinate is going to be the angle theta determined by that line segment and the polar axis. And so these are the polar coordinates of my point– r and theta. As you can see, they determine the position of the point. There’s something a little different, though, because they wouldn’t be the only polar coordinates that would determine the position of the point.

For example, if I were to give you the polar coordinates r and theta plus 2 pi, well, that would be exactly the same point that you would get, because adding 2 pi to theta just brings you back to the same position. Similarly, minus 2 pi would do the same thing. More generally, the polar coordinates of that point can be thought of as being r on the one hand and theta plus any integer multiple of 2 pi, k being positive or negative as an integer. And another kind of ambiguity arises at the pole itself. The pole is when r is 0. But what is the angle theta at the pole? There is no obvious unique choice.

It could be any theta, as long as r is 0, and you’ll be at the pole. So these coordinates work somewhat differently from Cartesian coordinates that were so well-defined for any point. And yet, we’ll see that they’re extremely useful in certain circumstances. Just to get a hint of why they’re going to be useful, suppose in polar coordinates I ask you, what is the level set of the set of points r, theta, for which r equals 2? What a simple equation, r equals 2. What does that describe as a locus of points? Answer, to say that your distance 2 from the pole is to say that you’re on a circle of radius 2 around that pole.

You see how easy it is to describe a circle in polar coordinates. We’ll see other examples later. Now, a circle could have been described in Cartesian coordinates, of course, which raises the issue, what is the connection between polar coordinates and Cartesian coordinates? Let’s first think about, given the polar coordinates, how do you find the Cartesian ones? Well, we have to put in the Cartesian coordinates if we’re going to compare them. And the usual way is to place the origin of the Cartesian coordinates at the pole and to place the positive x-axis for your Cartesian coordinates along the polar axis and then the Cartesian axis corresponding to y goes where it has to go.

So this is the canonical configuration when you want to speak of both polar coordinates and Cartesian coordinates. And how do you find the Cartesian coordinates of a given point with polar coordinates as are r and theta? Well, you can do the following. You can drop a vertical down from the point to the x-axis. And you see that the sine of theta, by our triangle definition of sine– you know, opposite over hypotenuse– is y over r, and the cosine of theta is x over r. Therefore, you find x and y– x is r cos theta, y is r sine theta. That was pretty easy.

And by the way, this will work even if theta is not in the first quadrant and if theta is negative, and so forth.