Why does the radius come before the angle?

Question

Based on my understanding, when delineating two variables (for a coordinate system or otherwise) convention is to label the 'independent variable' first, then the 'dependent variable'. So for a function $f(x)=y$ we write a 'point' as $$(x,f(x))$$ or $$(x,y)$$ First comes the 'input', then the output.

Yet for polar coordinates (or parametric coordinates) we label them $$(r,\theta)$$ even though the function is normally as defined as $$r(\theta)$$ Furthermore, when applying polar's to physical situations, intuitively I had always thought that $\theta$ was the independent variable.

Is there a historical reason for this convention?

I appreciate the arbitrary nature of labelling but it seems odd that polar's and cartesians differ. My question is concerned with historical provenance.

Answer 1

Plotting in $\mathbb{R}^2$ (i.e., $(x,y)$-space) does not intrinsically have an "independent variable / dependent variable" relationship. Yes, it shows up when we want to graph functions like $(x,f(x))$, but you can also plot many other things. For instance, you could parametrize a path in $\mathbb{R}^2$ (imagine tracking the movement of a car on a map) using a parameter $t$ using $(x(t), y(t))$. Here, the "independent variable" is $t$, while $x(t)$ and $y(t)$ are "dependent variables."

Likewise, polar coordinates do not intrinsically encode this relationship either: $r$ and $\theta$ are just parameters, and we do not assume any dependence on each other unless there is context. The ordering is just convention and [as far as I know] is arbitrary.

Answer 2

The coordinate $y$ in two dimensional cartesian coordinates is not a function of $x$, and neither is $\theta$ a function of $r$. You seem to be mistaking a graph of a function $f:\mathbb{R}\rightarrow\mathbb{R}$, which can be plotted in two dimensions with the independent variable on the $x$ axis and the dependent variable in the $y$ axis, with a point in $\mathbb{R}^2$, given by two independent coordinates.