Plotting polar equations and scale invariance

[fog37]

TL;DR Summary

Polar coordinates, plotting polar equations, scale invariance

Hello,
In the plane, using Cartesian coordinates $x$ and $y$, an equation (or a function) is a relationship between the $x$ and $y$ variables expressed as $y=f(x)$. The variable $y$ is usually the dependent variable while $x$ is the independent variable.

The polar coordinates $r$ and $\theta$ can be used instead of $x$ and $y$ to describe the same mathematical object. The variable $r$ is the dependent variable and a general polar equation has the form $r=g(\theta)$. In general, when plotting a polar function like $r=g(\theta)$, we plot it over the Cartesian axes instead of labeling the axes $r$ and $\theta$. Why? For example, the polar equation of a circle of radius 4 centered at the origin is $r=4$ and the graph looks like a circle if plotted in the x-y plane. However, if plotted with the axes labeled $r$ and $\theta$, the graph of $r=4$ would just be a horizontal straight line. Same goes for other coordinate systems: we always plot using the corresponding $x=rcos\theta$ and $y=rsin\theta$ values that derive from the specific $r$ and $\theta$.

Scale Invariance: I read about polar equations describing shapes that are scale invariant: if scaled up by a constant, the enlarged shape is represented by the same parent equation, just rotated. I don't think all polar equations behave like this. What makes certain polar functions be scale invariant while others are not? Example: $r=e^{\theta}$, if scaled up by a constant$A$, $Ae^{(\theta)}=e^{(\theta+\theta_0)}$ with the scale parameter $A=e^{\theta_0}$. I guess only polar functions involving exponential functions have this property, correct? These shapes are said to be specified only in terms of angles. Isn't that true for any polar equation since $r=g(\theta)$ and we can always express $r$ in terms of $\theta$?

Thanks!

 

[mfb]

The polar coordinates $r$ and $\theta$ can be used instead of $x$ and $y$ to describe the same mathematical object.

If you want to do this you should keep the relation between the coordinate frames, i.e. draw theta as angle and r as radius, as you described. If you make a Cartesian r,theta plane you lose that relation.

if scaled up by a constant, the enlarged shape is represented by the same parent equation, just rotated.

That is only true for exponential functions, right.

Isn't that true for any polar equation since $r=g(\theta)$ and we can always express $r$ in terms of $\theta$?

You already did so.

 

[fog37]

If you want to do this you should keep the relation between the coordinate frames, i.e. draw theta as angle and r as radius, as you described. If you make a Cartesian r,theta plane you lose that relation. That is only true for exponential functions, right.You already did so.

Thanks mfb. I am interested in better understanding what you mean by "lose the relation". From an origin $O$, an arbitrary point $P$ in space can be identified either by the ordered pair $(x,y)$ or by the coordinate pair $(r,\theta)$. A graph is simply the collection of points that satisfy an equation. Now, if we draw two orthogonal axes meeting at the origin, then the graph looks a certain way if we label the axes $x$ and $y$. But if we label them $r$ and $\theta$ the graph is very different. When plotting polar equations, the axes are assumed to be $x$ and $y$...,correct?

 

[mfb]

But if we label them $r$ and $\theta$ the graph is very different.

Well of course they are, you changed the axes. This is nothing special about polar coordinates. If you label them x and z you can't even draw your relation y(x) any more because there is no y.

When plotting polar equations, the axes are assumed to be $x$ and $y$

You don't need to assume this, you can show it.

 

[Stephen Tashi]

A graph is simply the collection of points that satisfy an equation.

Yes, but if you want the points to represent points in Euclidean space, there is a distance function defined between two points. If you plot polar coordinates $(r,\theta)$ by interpreting $r$ as the cartesian $y$-coordinate and $\theta$ as the cartesian $x$-coordinate then the distance between 2 points on the graph does not correspond to distance between the 2 points in Euclidean space that the points represent. That's not a mathematical contradiction, but it's a practical disadvantage when the graph is supposed to represent points in Euclidean space.

There can applications where cartesian points do not represent points where the Euclidean distance function is fundamental. For example if point $(x,y)$ represents a person of height $x$ and weight $y$, the interpretation of the Euclidean distance between two points is unclear.

 

[fog37]

Thank you Tashi.

Your reply is a step forward for me. So the space we live it is approximately and locally Euclidean in the sense that two points $P_1$ and $P_2$ have a distance which is the straight line connecting them.
In the $r$ versus $\theta$ graph, two points $(r_1, \theta_1)$ and $(r_2, \theta_2)$ surely identify the same two actual physical points $P_1$ and $P_2$ but with different coordinates and labels. But the straight line connecting them in the $r$ versus $\theta$ graph does not correspond to the Euclidean or physical distance between the points...

So, all coordinate systems (Cartesian, spherical, cylindrical, elliptical, prolate, etc.) are a valid options and just a mathematical convenience. However, graphically, is it fair and correct to state that the Cartesian system is the "closest" and more "accurately?" describes the physical space we live in?

thanks!

 

[Stephen Tashi]

So, all coordinate systems (Cartesian, spherical, cylindrical, elliptical, prolate, etc.) are a valid options and just a mathematical convenience. However, graphically, is it fair and correct to state that the Cartesian system is the "closest" and more "accurately?" describes the physical space we live in?

I don't know what you mean by "graphically". Do you mean that we agree to plot points in various coordinate systems by first finding their cartesian coordinates and then plotting the points in the usual way in the cartesian (Euclidean) system? If so, then as far as distances, angles, etc. go, our everyday experience of measuring them agrees with the corresponding predictions made with Euclidean space. If we consider space at the cosmological scale or sub-atomic scale, it isn't clear what is going on.

 

[fog37]

Yes,

By graphically, I just mean that we still plot an equation in polar, spherical, prolate, etc. coordinates using axes labelled $x,y,z$ instead of the coordinates the equations are expressed in. You mention that the reason is that plotting in Cartesian, the distances and angles as the same that we measure in our everyday space...So that choosing the axex $x,y,z$ is more useful apparently...

When I think of a circle, I think of the shape of a circle. So plotting the polar equation $r=2$ with axes $r$ and $\theta$ does not convey the same circle image...