Are polar coordinates backward?

I’d never given any thought to the order of polar coordinates until yesterday. They’re written (r, θ). That’s just how it is. But a friend pointed out two reasons why this bothers him.

First, r is typically a function of θ, just as y is typically a function of x. But in rectangular coordinates, the independent variable is the first element of a pair while in polar coordinates it is the second element.

Second, if you’re going to walk a mile northwest, how do you proceed? You first face northwest, then you walk for a mile. You don’t walk a mile to the east and then walk 135° counterclockwise along an arc centered where you started. That is, you set your θ first, then increase your r.

The (r, θ) convention isn’t going to change. Maybe the only take-away from this discussion is to appreciate someone’s confusion who sees polar coordinates for the first time and wants to reverse their order.

Related post: Why use j for imaginary unit

(I don’t use j for imaginary unit, except when writing Python code. The i notation is too firmly engraved in my brain. But I understand why j has advantages.)

8 thoughts on “Are polar coordinates backward?

  1. You write r*e^it, don’t you? Might be the reason for this “convention”. Or maybe it is because some pocket computer has set it set way. Don’t remember the good old days of TI.

  2. This is cool, I never thought of the dependent/independent difference. I still like the way it is though, here is my thought: While when you are walking you want your direction first, it does not really work when plotting a point on a graph. It is difficult to correctly plot r units in a certain direction if you are off your axis (without a computer of course:-). When teaching it, I compare it to Cartesian coordinates in that you start by finding a point on the x-axis, and then go up or down from there to find your point. In this way it is similar in polar, because you still start by finding a point on the x axis, but instead of going up or down, you rotate around the circle.

  3. If we could build mathematical notation from the ground up, there’s a lot we should change. For one, eigenvalue problems should be written as $Ax = x\lambda$ so that (NxM)(MxL) = NxL matrix size rule works for L=1.

    This reminds me of the ax I periodically grind: The UTF-16 code points are being filled full of emoji, and mathematicians still don’t have a globally unique symbol for $\pi$! Maybe a UTF-16 code point which uniquely identifies $2\pi$ could get traction . . .

  4. Another reason is when you take the log of r e^(it): log(r e^(it)) = log r + it. Apparently the (log of) the magnitude comes first.

  5. When extending to more dimensions, it’s natural to append (r, θ) to say (r, θ, φ), whereas (θ, r) to (θ, φ, r) would not.

  6. (r, \theta) satisfies the same orientation as (x,y) does, that is, you turn left from the infinitesimal dr to go in the d\theta direction. The coordinates (\theta, r) would be negatively oriented, so your areas would start to look negative instead of positive.

    Physicists do this in spherical coordinates: they usually write (\rho, \phi, \theta) instead of (\rho, \theta, \phi). The latter coordinates are still positively oriented (satisfy the right-hand rule), but the former do not.

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