Polar Coordinates functional notation.

[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!

 

[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 $f(\theta) = f(\theta + 2\pi)$.

 

[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.

 

[That Neuron]

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.

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.

 

[Stephen Tashi]

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.

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.