Which curves are representible in polar coordinates and which curves are not?

Question

I stumbled over this rather small question:

Which curves are representible in polar coordinates?

What would be a scetch of a graph of a curve that is not representible in polar coordinates?

By browsing the internet I could find lots on polar coordinates, but nothing specific about what is possible to do with them and what is not.

I guess the answer to this might be rather trivial to the people here but can someone explain quickly, what are the characteristica of curves that can be represented in polar coordinates and of the ones that are can not?

Answer 1

Polar coordinates are just another way of describing points in an $n$-dimensional space -- instead of using the displacements from zero on the various axes, you use the angles you make to the various axes, plus the distance of your point from 0. What that means is that any curve (here I assume you mean 'function') that can be described as an equation $f(x_1,x_2,...)$ can also be described by an equation $f(r,\theta_1, \theta_2,...)$. Sometimes it's convenient to convert, sometimes it is not

Answer 2

If by "curve" you mean "the image of a continuous function on an interval in the real line (including possibly the whole real line) and by "representable by polar coordinates" you mean "there is a function $h$ with the property that (almost) every point $(x, y)$ of the curve $C$ is of the form $r = h(\theta)$, i.e., $$ x = h(\theta) \cos \theta\\ y = h(\theta) \sin \theta $$ for exactly one value of $\theta$, then there are in fact limits on what you can produce. If you limit to $0 \le \theta \le 2\pi$ and $r \ge 0$ (as folks often do), then if the intersection of your curve $C$ with any ray from the origin contains more than one point, it cannot be done. If you ignore the $r \ge 0$ constraint, then the intersection of $C$ with a ray through the origin may contain two points. And if you allow $\theta$ to range over a larger interval, you can hit a ray infinitely many times.

But here's a curve you cannot generate in this "polar parametric" form:

$$ x(t) = t\\ y(t) = 0 $$

Slightly more interesting is a curve like the graph of $y = \cos x$. Proving that this cannot be parameterized by a (continuous) polar parametric form (with the "for at most one value of $\theta$" requirement) seems to require careful use of the intermediate value theorem.