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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 Nov 28 '13 at 14:39
  • $\begingroup$ 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 Dec 21 '13 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 Dec 22 '13 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 Dec 22 '13 at 12:29
  • $\begingroup$ You should consider re-asking this question at matheducators.stackexchange.com $\endgroup$ – Brian Rushton May 1 '14 at 17:56
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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 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 Dec 21 '13 at 22:23
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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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