How can changes between coordinate systems create functions from non-functions?

Question

I have recently been reviewing polar coordinates and coordinate transformations. I never really understood the purpose of using polar coordinates...but then I came across this quote: "Frequently, polar coordinates are used where Cartesian coordinates would produce a relationship that isn’t a function".

The authors go on to demonstrate that it is possible for a shape on a graph (say a circle) to be a function using polar coordinates but not a function when using Cartesian coordinates. In the case of a circle defined by r=sin(theta) (refer to attached photo), the authors state that the polar form is a function but the only way to represent this equation as a function using Cartesian coordinates is "through the union of more than one function".

So, I guess my question is, "What exactly, at an abstract level, is going on here that allows the same shape (e.g. a circle defined by r=sin(theta)) to be equivalently described by different coordinate systems...where in one equation, there is a functional description but in another equation (using different coordinates) there is only a relational description".

Is there a nice intuitive picture of sets (input and output) that can demonstrate what is going and how the inputs from one coordinate system are related to the inputs of another coordinate system? Where exactly in this abstract procedure is one system 'becoming' a function? Said differently, how is the conversion between two coordinate systems imbuing one system with a new property of "function" whereas the previous system did not have such a property?

Answer 1

The circle intersects the $y$ axis either twice or not at all, for any value of $x$. Functions, strictly interpreted, have to be single-valued, thus the circle cannot be $y=f(x)$.

But, a circle around the origin (0,0) intersects the positive $r$ axis only once. hence, it can be described by $r(\theta)=R=$constant.

This is the key difference: does the shape intersect the "dependent axis" ($y$ or $r$) at most once? Yes: function, No: not function.

Your example of $r(\theta)=\sin(\theta)$ is a degenerate case. It seems to intersect the $r$ axis twice, once at $r=0$ and once at $r=\sin(\theta)$. It is tangential to the $x$ axis, and if you move it upwards even slightly, it is no longer a function even in polar coordinates.