A fairly ubiquitous object in elementary calculus is a function of the shape $r = f(\theta)$, where $r$ is the radius and $\theta$ the argument. Common examples include the cardiod and limacon, and of course the circle can also be expressed this way. 


I understand the value of the substitution $x = r\cos(\theta), y = r\sin(\theta)$ since this helps with understanding complex variables, and when doing contour integration helps immensely. But I don't recall in my entire undergraduate career, did I ever encounter a function of the shape $r = f(\theta)$ past the examples shown in first year calculus.


Are these objects of any importance to any mathematically related field? If so, what are they? If not, then why are these objects still so dominant in the curriculum?


I understand one 'application' is arc length computation, which in Cartesian coordinates is notoriously difficult (to do exactly). For instance, the arc length of $y= \sin(x)$ say is 'difficult' since $\sqrt{1 + \cos^2(x)}$ has no anti-derivative in elementary functions. If we have $r = f(\theta)$ then the integral to be computed looks something like $\displaystyle \int\_{\alpha}^\beta f(\theta)d\theta$, which is fairly straightforward. But this 'application' doesn't really simplify things in practice, since a relationship of the form $r = f(\theta)$ itself is rare and difficult to obtain, and numerical methods can give us practical ways to compute arc length anyway.


Any insight would be appreciated.



Application for functions of the shape $r = f(\theta)$
The most spectacular application is the theory of orbits under a central force field. This is basically what Kepler and Newton did (not, of course, using
this notation). One of Kepler's key observations on the orbit of Mars was the constancy of a certain quantity associated with points on the orbit. In modern terms he had noticed that $r(1+ e\cos \theta)$ was constant along the orbit. We now recognise this instantly as the equation of an ellipse with the origin (the sun) at a focus. I suggest that you look up the theorem of Binet for more information on this theme. There are other applications of the use of forms $r=f(\theta)$ (even better $rf(\theta)=1$). Space is too short to give a detailed list but there are many such, e.g., unifying the basic explicit solutions of the motivating problems of the elementary calculus of variations.


