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

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Discussion Overview

The discussion revolves around the transformation of equations from Cartesian to polar coordinates, specifically examining how graphs of the same equation appear differently in each coordinate system. Participants explore the implications of these transformations on the shapes of graphs and the relationships between points in both systems.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant proposes that the transformation from Cartesian to polar coordinates can be visualized as a mapping where points on the Cartesian plane warp around the center, affecting the shape of the graph.
  • Another participant challenges this view by questioning why the equation y=2 translates to r=2 instead of θ=2, suggesting that the consistent translation should be y=2 becomes r sin(θ)=2.
  • A participant clarifies that while the equations can be transformed, the shapes they represent in polar coordinates differ, particularly noting that a cardioid can be viewed as a sine curve wrapped around the polar axis.
  • One participant points out that the claim of a one-to-one correspondence between points on the rectangular and polar graphs is incorrect due to periodicity in polar coordinates.
  • Another participant emphasizes the importance of distinguishing between coordinate systems and transformations, suggesting that the language used may not align with standard mathematical terminology.
  • A participant expresses a desire for a visual representation to better convey their ideas, referencing an animation they found that illustrates the transformation concept.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the nature of the transformation and the correspondence between points in Cartesian and polar coordinates. There is no consensus on the validity of the initial intuition presented.

Contextual Notes

Participants acknowledge the complexity of transforming equations and the potential for misinterpretation of the relationship between different coordinate systems. The discussion highlights the need for precise language when discussing mathematical transformations.

srfriggen
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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
 
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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).
 
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.
 

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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.
 
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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?
 
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.
 
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).
 
Why the skeptical face?
 

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