Polar coordinates doubt (Graph of $r \le 1$)

Question

I have a doubt. I have to plot the graph of $r \le 1$. Now, according to me, it should be a circular disc with center origin and radius 1 unit. But, some of my friends say that it should be the whole plane. According to them, $r \le 1$ means negative values of r are also allowed, but, I am not so sure about this logic. If I convert this to rectangular coordinates:

$\sqrt{x^2 + y^2} \le 1$ $\implies x^2 + y^2 \le 1$ which is the circular region I am talking about.

Also, both "|r|" and "-|r|" have the same plot in the polar coordinate system. Thus, negative values of r will have the same plot as positive values of r. This would make the question inconclusive (acc. to me).

Any hint ?

Vishwesh

Answer 1

It depends on the definition of polar coordinates you are using. For example this article claims that

What's more, one often allows negative values of r under the assumption that $(-r,\theta)$ is plotted identically to $(r,\theta\pm\pi)$.

However, that makes the question rather nonsensical since obviously you cannot plot the entire plane. It's better to check what definition you are using but most people would guess that it's restricted to $r\geq0$.