Polar Coordinates functional notation.

1. Jan 14, 2013

That Neuron

I've always been curious why points in polar coordinates are defined as (r,θ) when all equations (including parametric equations formed from them) are defined as r=f(θ).

Considering that point in cartesian coordinates are defined as (x,y) where y=f(x).

Also a,b=(r,θ) ∫1/2[f(θ)]2 further implies that θ is the domain.

I just find this odd notation wise, and am wondering if anyone can provide me with a reason for this seeming discrepancy.

:) Thanks!

2. Jan 15, 2013

Stephen Tashi

That's a good question. I conjecture it's just tradition and a matter of convenience.

Polar coordinates for a point are not unique (even though it's common to hear math people talk about "the" polar coordinates of a point). The point $(r,\theta)$ is the same as the point $(r,\theta + 2 \pi )$.

if you want to define points in the cartesian plane that have the polar form $(f(\theta), \theta)$ you have to be careful to make $f(\theta) = f(\theta + 2 \pi )$. This happens "naturally" with trigonmetric functions such as $f(\theta) = \sin{\theta}$.

If you want to define a function by points in the cartesian plane that have the polar form $(r, g(r) )$ then you have less to worry about. However you can't describe a curve with a set of points having the same radius and several different principal angles.

I suspect polar coordinates for curves are most often used when we need a shape where two points with the same radii can have different principal angles. These are often written using trig functions so it isn't a problem to insure that $f(\theta) = f(\theta + 2\pi)$.

3. Jan 15, 2013

chiro

It is subtle, but a parameterization can map say a value on the real line to that of say [0,2pi) with the simple example being a circle with x = rcos(t), y = rsin(t) for t = [0,infinity).

Its subtle, but I think its worth noting.

4. Jan 19, 2013

That Neuron

But wouldn't a function defined as $(r, g(r) )$ not be an actual function since $f(\theta) = f(\theta + 2 \pi )$, so for every r there would be a myriad of possible values of $\theta$ that would be solutions. Just as we can only define inverse trigonometric (sine for example) functions on a limited ($[0,2pi)$ domain.

Perhaps this is why we can only define polar functions with $\theta$ as the domain?

Sorry if I seem a little distracted, but I've just been digesting a bunch of Mathematical Grammar and set logic, so my mind is completely scrambled :) haha.

Last edited: Jan 19, 2013
5. Jan 19, 2013

Stephen Tashi

My notation (r, g(r)) assumes g(r) is a function from the non-negative real numbers to the real numbers. So each r is mapped to only a single g(r).

Your question about solving for theta is relevant to the case of ( f(theta), theta). It is correct that f(theta) must be a function with the property that f(theta) = f(theta + 2 pi).

If we need to write a function whose graph is a spiral, we have to introduce a parameter and make both radius and angle depend on the parameter in the form ( r(t), theta(t)). So it isn't correct to say that there is only one form for a graph in polar coordinates. It's just that (f(theta), theta) is a very commonly encountered form.