I don't know of where this would come up in a practical sense, but I had this crazy idea this morning that maybe if given various pairs $(r, \theta)$ in polar coordinates that one could attempt to fit a curve through them, using $\theta$ as the response variable and $r$ as the explanatory variable.
For simplicity, we could start with a "linear regression" of sorts through a set of points $\{(r_i, \theta_i)\}$.
I suppose one could naively assume a model of the form
$$r_i\cos(\theta_i) = \beta_0+\beta_1 r_i\sin(\theta_i) + \epsilon_i$$
(recalling that $x_i = r_i \sin(\theta_i)$ and $y_i = r_i \cos(\theta_i)$ in rectangular coordinates), but this ruins the point of making $\theta_i$ the "response" variable of $r_i$, given that functions of $\theta_i$ appear on both sides of the equation.
Has there been any literature developed on any sort of regression in polar coordinates?
Of course, since various values of $\theta$ are coterminal, the domain of $\theta$ will probably need to be truncated. What may also complicate things slightly is that $r > 0$.
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There is an enormous amount of statistics related to this under the name circular or directional statistics.
Your case is not so well described though.
What is the 'switch' that you were thinking of by introducing polar coordinates?
Do you mean to solve a situation of a non-negative variable, e.g. some count, as function of a cyclical variable, e.g. hour of the day, by picturing it as a curve and solve it with the mechanisms for curve fitting in euclidean coordinates?