A curiosity about the polar coordinate system

Question

I have a very simple question about polar coordinate system. According to Wikipedia, the angle is defined as:

where atan2(y,x) is defined as "a common variation on the arctangent function"

Since I'm not an advanced student in calculus, I was trying to determine these values as an exercise, just by appling the inverse relation between tangent and arctangent function, and by noticing that the ratio of the argument of the arctangent function is in fact the ratio between the sinus and the cosinus of the angle. I've found correctly the values for the angle in the last three cases of the system above, but I don't understand why I should add and subtract \pi in the second and third case respectively. Isn't the period of the tangent function equal to \pi itself? Why am I wrong? I suppose that there is some advanced additional stuff that I can't understand! As usual, thank you!

Answer 1

The point is we can't use $\arctan\frac{y}{x}$ in general because $(a,\,b)$ is antipodal to $(-a,\,-b)$, which achieves the same ratio.

Answer 2

The point of a coordinate system is that given a radius and an angle you define one and only one point in the plan.

Adding and substracting $\pi$ when $(x,y)$ lies "in the left quarters" ensures that the angle is a continuuous, increasing function to $(-\pi$ ; $\pi]$ when you rotate counter-clockwise from $(-1;0)$ along the unit circle.

Hence from a given $(r,\theta) \in (0,\infty)\times(-\pi,\pi]$ you define a single point of the plan minus its origin.

Answer 3

The problem is that $\frac{a}{b}$ and $\frac{-a}{b}$ result in the same value when you plug them into $tan(x)$, which means that if we are going the other way around (from $tan(x)$ to $x$), then there is no clear solution. Which of the two should be returned. This comes down to the function $tan(x)$ not being one-to-one.

Now in the $(x,y)$-plane when you know the sign of $x$ and of $y$ then you know in which quadrant they lie and thus (coming polar coordinates) in which "quarter" of the circle they are in. For example if $x$ is positive and $y$ is negative, then I know that the angle this point will have in polar coordinates will be between 270 and 360 degrees or $\frac{3\pi}{2}$ and $\frac{4\pi}{2} = 2 \pi$.

This allows me to choose the appropriate value when using $arctan(x)$ up there. So these cases are needed because $tan(x)$ is not one-to-one and they are defined using this additional knowledge about the position of a point given the signs of its components.