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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.

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    $\begingroup$ Curves show up all over the place, and having many ways to describe them is undoubtedly good, isn't it? Beyond that, I really cannot see what you expect as an answer! I know a proof of the isoperimetric inequality using polar coordinates and Fourier series: does that count as an application? A change of coordinates cannot have «importance» in a mathematical field in any sensible sense, I think. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Nov 28, 2013 at 14:39
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    $\begingroup$ If you want a specific example, the central piece of the Mandelbrot set is a cardiod. So that's an important place where a cardiod appears, and as you noted, probably the easiest way to describe a cardiod is via an $r=f(\theta)$ equation. $\endgroup$
    – Joe Silverman
    Commented Dec 21, 2013 at 19:55
  • $\begingroup$ With respect, I have to disagree strongly with the first comment, especially the final sentence thereof. Surely one of the BIG themes of mathematics is the use of special frames for specific problems---think canonical forms. Examples: the Jordan canonical form, the spectral theorem for normal matrices, bounded and $\endgroup$
    – user6891
    Commented Dec 22, 2013 at 12:17
  • $\begingroup$ unbounded normal operators. Where did Fourier series and spherical functions come from if not from here and where would mathematics be without them? The potential usefulness of a parametrisation as in the OP in the presence of central symmetry seems pretty obvious to me and, of course, it is the exploitation of this and more sophisticated symmetries which lie at the centre of the above and many more key mathematical theories and concepts. $\endgroup$
    – user6891
    Commented Dec 22, 2013 at 12:29
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    $\begingroup$ You should consider re-asking this question at matheducators.stackexchange.com $\endgroup$
    – Brian Rushton
    Commented May 1, 2014 at 17:56

4 Answers 4

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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.

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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.

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  • $\begingroup$ Since you mention spirals, a really remarkable family of these are the so-called McLaurin spirals which were discovered by this Scottish mathematcian in the 18 th century. They are those with $f$ of the form $(\cos(n \theta))^{\frac 1n}$ and have a plethora of remarkable properties, e.g., they are all orbits for power laws. The logarithmic spiral is a (degenerate) special case. They are also catenaries for such laws. $\endgroup$
    – user6891
    Commented Dec 21, 2013 at 22:23
  • $\begingroup$ @user6891, is that Maclaurin, or a different Scottish mathematician? I cannot easily find a description of McLaurin or Maclaurin spirals. $\endgroup$
    – LSpice
    Commented May 12, 2021 at 15:19
  • $\begingroup$ @LSpice The mathematician in question was from the Gaeltacht so that his name was gaelic. At that time the system of transliterating names into english was in flux, in particular with respect to the patronymic prefix and during his lifetime more than one version was used for his surname. By the way, if you are interested in his spirals and the related catenaries, you can find material in the arxiv preprint 1102.1579. The decisive fact is that the $f$ which arises has the property that $f+f''$ is proportional to a power of $f$. $\endgroup$
    – terceira
    Commented Dec 29, 2022 at 6:04
  • $\begingroup$ The arXiv preprint referenced by @terceira: Cooper - On the relevance of the differential expressions $f^2 + f'^2$, $f + f''$ and $f f'' - f'^2$ for the geometrical and mechanical properties of curves. Thanks! $\endgroup$
    – LSpice
    Commented Dec 29, 2022 at 20:04
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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.

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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.

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  • $\begingroup$ Online bookstores as Barnes & Noble include the table of content of the books that they offer, you can to read the title of this chapter 14 from their homepage. Google Books also show titles of chapters. $\endgroup$
    – user142929
    Commented Nov 8, 2022 at 15:34
  • $\begingroup$ I've designed a flag (really a cylinder since is the quotient of two sides of a rectangle that I evoke made of textile patches: patches of different colors, and reflective, fluorescent or phosphorescent properties). I can to edit in next few days (if some user is interested) a comment in memory of the idea of Professor adding this design as a file from Imgur, together simple notes/questions about if it is possible to expand his idea). $\endgroup$
    – user142929
    Commented Nov 8, 2022 at 15:41
  • $\begingroup$ You give a preface and an afterword to the answer, but I can't tell what the actual answer is. Is it the logarithmic spiral (also discussed by @VictorProtsak), or some application of it, or ...? If your example is the content of the articles, could you say something about that content beyond just the references? $\endgroup$
    – LSpice
    Commented Dec 28, 2022 at 22:11
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    $\begingroup$ I'm not a professional @LSpice but, for example, any person can to read the content of my last edit, the excerpts that I refer for the book of Michio Kaku from Google Books, and the abstract for the article [5] from the web of the ACS. Thus my last contribution (the paragraphs that I've added) is an historical remark and an application in Chemistry. I've edited both references in the same paragraph since I think that it's interesting to contrast both ideas. I always try to be extremely respectful when commenting on copyrighted documents, because I don't know how much material I can quote. $\endgroup$
    – user142929
    Commented Dec 29, 2022 at 17:12
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    $\begingroup$ Any case I understand your words, you're right. So I will try to be more concise when I edit a post. Many thanks @LSpice $\endgroup$
    – user142929
    Commented Dec 29, 2022 at 17:18

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