Application for functions of the shape $r = f(\theta)$ 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.
 A: 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.
A: An important class of curves with equations of this form is spirals. They are transcendental curves, hence cannot be expressed by a polynomial equation relating $x$ and $y$ coordinates. Archimedean spiral was the first to be studied. Anyone who coiled a rope has encountered this shape. Logarithmic spiral has many applications to modeling shapes of objects in nature, from mollusks to cyclones and spiral galaxies.   
A: Parametrizations of curves $r=f(\theta)$ were considered even before Descartes coordinate system was introduced. In the science which at that time was the main application of mathematics, namely in astronomy. See any introductory course of astronomy or celestial mechanics.
A: I don't know if the application that I'm going to refer is well-known (an invention of the author of [1]). I've read it from a book, Wikipedia has an article dedicated to the author E. O. Wilson, that refers the application in chapter 14 of the book [1] (I know the Spanish edition of the book Editorial Planeta, Crítica 2022); the chapter is potentially interesting for mathematicians, and are included the  related articles in the bibliography of this book.
Wikipedia has an article including the section In engineering applications from the Wikipedia article dedicated to Logarithmic spiral.
If I understad well whirls don't fit with your question/post, but I include the following references as general reference in case that you or the readers want to know these notes that I consider interesting, if I remember well [2] include notes about whirls, and [3] seems interesting for me (is a note in Spanish language).
Other instances are the following, 1) this time is an historical example, is from the book [4], see Figure 2.8 and the preceding subsection of Chapter 2; and 2) the meaning of article [5] (I don't know what's the revelance of this article published from a journal of the American Chemical Society, but I believe that sure it is interesting).
Please let me to know (you or your colleagues) if my post isn't a good answer that I can to delete it in next few hours, many thanks.
References:
[1] Edward O. Wilson, Tales from the Ant World, Published by Liveright (2020).
[2] Leo Zippin, Uses of Infinity, Dover Publications (2000).
[3] Juan Luis Varona, $\zeta(2k)$ versus $\zeta(k)$: una relación geométrica en una espiral, Miniaturas matemáticas, La Gaceta de la RSME
Vol. 18 (2015), Núm. 2, Pág. 352.
[4] Michio Kaku, Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and The Tenth Dimension, Oxford University Press (1994).
[5] Audrey R. Sulkanen, Minyuan Wang, Logan A. Swartz, Jaeuk Sung, Gang Sun, Jeffrey S. Moore, Nancy R. Sottos and Gang-yu Liu, Production of Organizational Chiral Structures by Design, J. Am. Chem. Soc.  144, 2 (2022), pages 824–831.
