Undergrad Cartesian to Polar form.... Is it just a transformation of the plane?

[srfriggen]

TL;DR

I am trying to get a better intuition for graphing in polar form and would appreciate any insight

Hello,

Today I started to think about why graphs, of the same equation, look different on the Cartesian plane vs. the polar grid. I have this visualization where every point on the cartesian plane gets mapped to a point on the polar grid through a transformation of the grids themselves.

Imagine the line y = 2, graphed in rectangular, for example. This, of course, is a circle in polar (r = 2) and I envision that as a transformation where the x-axis gets looped in on itself and squished down to a single point (the pole). While this is happening all of the other points in the plane warp around the center, giving us the polar graph we all know. This, in turn, affects the line by wrapping it in on itself so that it forms a circle.

Would love to hear whether this intuition is valid

 

[Office_Shredder]

I think it's mostly not valid. Why does y=2 become r=2 and not $\theta=2$?

The consistent way of translating coordinates is y=2 becomes $r\sin(\theta)=2$ which has the exact same graph (a horizontal line).

 

[srfriggen]

I'll try to be more clear. If you want the equation, of a line, to look like a line in polar, then you have to use the conversion, as you said. What I'm talking about is, if on the x / y-axis you represent angles on the x-axis, and distance from zero on the y-axis (per usual), then the equation y = 2 and r = 2 will make a different shape, but both have the same inputs and outputs.

Think about the rectangular graph of a cardioid. It is a sine curve, on the x/y, with one zero between 0 and 2pi. y = 2+2sin(theta), for example. That same equation, in polar (just replace y with r) yields a shape that is different but still maintains a key feature which is it only has one zero from 0 to 2pi.

Every point on the rectangular graph has a 1-1 correspondence with a point on the polar graph.

The polar shape of the function y = 2 + 2sin(theta) can be thought of as wrapping that sinusoidal curve around the polar axis. If you do it the right way you'll get the cardioid shape and all of the other points will line up as well, while the x-axis shrinks to a single point.

I really wish I knew how to make an animation of what I'm talking about. I've attached a drawing. The black line is the x-axis which gets squished to the zero point of the pole. The blue line is a line y = a, where a is some constant.

 

[Office_Shredder]

I understand. I guess just to start

Every point on the rectangular graph has a 1-1 correspondence with a point on the polar graph.

This isn't true, since $x$ and $x+2\pi$ get mapped to the same point.

 

[srfriggen]

Ah yes, very true.

But do you get what I'm saying about how a cardioid is like a sine curve that's wrapped in a circle?

 

[DaveE]

I think this would be more clear, to us and perhaps to you, if you separated the ideas of a coordinate system and transformations. Normally people wouldn't consider a change from y=2 to r=2 as a change in the coordinate system, they would think of that as an operation (or operator) that changes a function. Of course there are lots of similarities, so you're not wrong. But you are using language that isn't standard. For example: when you say y=2 transformed to polar coordinates, I think of y=r⋅sin(Θ)=2, same function, same shpe, etc. just a different description with different coordinates.

 

[srfriggen]

Hi Dave,

Thank you for your reply. It seems you do visually get what I'm saying... another example could be how the curve y = 2+2sin(x) , if wrapped around on itself, would form a cardioid.

Can you help to polish up my thoughts? I feel like a simple animation (like a 3blue1brown style animation) would get to the heart of what I'm envisioning. I just can't get it into words (I don't have the same vocabulary as you do).

 

[srfriggen]

I found it! I found what I was envisioning! Check out the gif on the wikipedia site

https://commons.wikimedia.org/wiki/File:Cartesian_to_polar.gif#/media/File:Cartesian_to_polar.gif

 

[srfriggen]

Why the skeptical face?