# Polar Coordinates functional notation.

• That Neuron
In summary, the use of (r,θ) for points in polar coordinates and r=f(θ) for equations is likely due to tradition and convenience. This notation is also used because polar coordinates for a point are not unique and can have different principal angles. There is also a difference in defining a curve with points having the same radius and different principal angles. Additionally, polar functions with θ as the domain are more commonly used as they can describe shapes where points with the same radius can have different principal angles. However, for functions defined as (r, g(r)), each r is mapped to only a single g(r) and the notation (f(θ), θ) must have the property that f(θ) = f
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!

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

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.

Stephen Tashi said:
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.

Last edited:
That Neuron said:
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.

## What are polar coordinates and how are they different from Cartesian coordinates?

Polar coordinates are a type of coordinate system used to locate points in a two-dimensional plane. They are different from Cartesian coordinates in that they use a distance and an angle to represent a point, rather than using two perpendicular lines.

## What is the functional notation for polar coordinates?

The functional notation for polar coordinates is (r, θ), where r represents the distance from the origin and θ represents the angle from the positive x-axis to the point.

## How do I convert polar coordinates to Cartesian coordinates?

To convert from polar coordinates to Cartesian coordinates, you can use the following formulas:
x = r * cos(θ)
y = r * sin(θ)
Where r is the distance and θ is the angle.

## What are some real-world applications of polar coordinates?

Polar coordinates are commonly used in navigation, astronomy, and engineering. They are also used in mapping and geographic information systems to represent points on a map.

## Is functional notation the only way to represent polar coordinates?

No, there are other ways to represent polar coordinates, such as using the magnitude and direction form (r, θ). However, functional notation is the most commonly used and preferred method of representing polar coordinates in mathematics and science.

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