Polar Coordinates

On the coordinate plane we pin a point down with two numbers: how far across and how far up, the ordered pair (x, y). But that is not the only way to say where something is. Stand at the centre of a radar screen and a blip is described far more naturally by two different numbers:

how far away it is — a distance we call r;
which direction it lies in — an angle we call \theta (the Greek letter "theta"), measured counter-clockwise from the positive x-axis.

Together (r, \theta) are the polar coordinates of the point. The centre we measure from is called the pole (it plays the role of the origin), and the positive x-axis is the polar axis — the direction we call \theta = 0. "How far, and which way" — that is the whole idea.

Drag r and the point slides straight out from the pole along a spoke; drag \theta and it swings around the circle. Every point on the plane is reached by one distance and one direction.

A radar dish spins and, whenever it catches an echo, it knows exactly two things: how long the pulse took to return (that gives the range, our r) and which way the dish was pointing (the bearing, our \theta). That is literally a polar coordinate. Sonar on a submarine works the same way, an air-traffic controller reads "range and bearing" off the scope, and a ship reports its position to a lighthouse as "three miles out, bearing north-east." Nobody says "go 2.1 km east and 2.1 km north" — polar is simply the honest language of anything that sweeps a circle. Even a spiral galaxy is easiest to describe as "how far from the core, at what angle."

From polar to Cartesian: x = r\cos\theta,\ y = r\sin\theta

A polar address and a Cartesian address describe the same point, so we can always translate between them. Drop a right triangle from the point to the x-axis: the hypotenuse has length r and sits at angle \theta, so plain trigonometry reads off the two legs:

x = r\cos\theta \qquad y = r\sin\theta

That is the polar-to-Cartesian conversion, and it is the one you will use most, because it never needs any quadrant care — you just feed in r and \theta and out come x and y.

Polar → Cartesian: x = r\cos\theta and y = r\sin\theta.
Cartesian → polar: r = \sqrt{x^2 + y^2} and \tan\theta = \dfrac{y}{x} — then check the point's quadrant to choose the right \theta.

Worked example 1 — polar to Cartesian

Convert (r, \theta) = \left(4, \tfrac{2\pi}{3}\right) to (x, y). The angle \tfrac{2\pi}{3} is 120^\circ, which sits in the second quadrant, so we expect a negative x and a positive y:

x = 4\cos 120^\circ = 4\left(-\tfrac{1}{2}\right) = -2, \qquad y = 4\sin 120^\circ = 4\left(\tfrac{\sqrt{3}}{2}\right) = 2\sqrt{3}.

So the point is (-2,\ 2\sqrt{3}) \approx (-2,\ 3.46) — and, sure enough, it lands up and to the left, exactly where a 120^\circ direction should put it.

From Cartesian to polar — and minding the quadrant

Going the other way, the distance is just Pythagoras from the origin, r = \sqrt{x^2 + y^2}, and the direction comes from \tan\theta = \tfrac{y}{x}. The catch is that \tan can't tell opposite directions apart — (1, 1) and (-1, -1) give the same ratio \tfrac{y}{x} = 1 even though they point opposite ways. So you must look at which quadrant the point is actually in and pick the matching angle.

Worked example 2 — Cartesian to polar

Convert (x, y) = (-1, 1) to polar. First the distance:

r = \sqrt{(-1)^2 + 1^2} = \sqrt{2}.

Now the angle. A calculator's \arctan\!\left(\tfrac{1}{-1}\right) = \arctan(-1) = -45^\circ — but that answer points into the fourth quadrant, whereas (-1, 1) (negative x, positive y) clearly lives in the second. Add 180^\circ to swing it around to the correct side:

\theta = -45^\circ + 180^\circ = 135^\circ = \tfrac{3\pi}{4}.

So (-1, 1) = \left(\sqrt{2},\ \tfrac{3\pi}{4}\right) in polar form. Always sketch the point — the picture tells you the quadrant, and the quadrant fixes the angle.

Polar coordinates are not unique — one point has infinitely many names. These all point at the very same place:

(r, \theta) and (r, \theta + 2\pi) — spinning a full turn lands you back where you started, so you can add 360^\circ to \theta as often as you like.
(r, \theta) and (-r, \theta + \pi) — a negative r means "measure the distance backwards," so facing the opposite way and stepping backwards reaches the same spot.
the pole itself is r = 0 for every angle \theta — at the centre, direction stops mattering.

And the second trap: when converting (x, y) \to (r, \theta), the calculator's \arctan\!\left(\tfrac{y}{x}\right) only ever returns an angle between -90^\circ and 90^\circ — the right-hand half of the plane. If your point is really on the left (quadrant II or III), add 180^\circ. Check the quadrant, always.

Curves that polar coordinates make easy

The real payoff is that some shapes which are messy in (x, y) become almost trivial in (r, \theta). Watch what happens when you hold one coordinate fixed and let the other roam:

r = a — every point the same distance a from the pole, any direction. That is a circle of radius a centred at the pole — one tiny equation for what Cartesian writes as x^2 + y^2 = a^2.
\theta = \alpha — every point in a single fixed direction, any distance. That is a ray (a straight line through the pole) at angle \alpha.
r = 2a\cos\theta — here the distance changes with the angle, tracing a circle of radius a that passes through the pole, centred at (a, 0). (With a = 2 that is r = 4\cos\theta.)

Step through the diagram to see all three:

Multiply both sides by r: r^2 = 2a\,r\cos\theta. Now use the conversions — r^2 = x^2 + y^2 and r\cos\theta = x — to get x^2 + y^2 = 2ax, i.e. (x - a)^2 + y^2 = a^2. That is a plain circle of radius a centred at (a, 0), and since (0,0) satisfies it, the circle runs right through the pole. Polar made the equation short; Cartesian revealed the shape.

Polar coordinates are one way to describe a curve by something other than a bare y = f(x); another is to let both x and y follow a shared parameter, the idea of parametric equations.

See it: read the radar blip

Here is a fresh blip each time. Read its distance r off the concentric circles and its angle \theta off the spokes, then check yourself against the label. Press Refresh for a new one.