Do we use the x-axis for defining polar coordinates?

Question

"To represent any point in a two dimensional space, you need two variables"

right?

For example (using our good old friend, The cartesian coordinates) we can talk about any point in two dimensional space using two variables, $x$ and $y$.

All fine and good!

Enter Polar coordinates.

Now this system uses the renowned $r$ and $\theta$ where theta is defined as the angle the vector makes with the x-axis.

AHA! "Angle made with the x-axis" so we need to define what the x-axis is? so does that mean we are using THREE things to talk about a point?

ps: My doubt can be extended for three dimensional spherical coordinates but lets keep it simple.

Answer 1

In polar coordinates, we're referencing a particle by its distance from the origin and angle relative to a reference point. There's no reason that reference must be the $x$ axis. It can be the $y$ axis or the direction of any other unit vector.

Answer 2

You need a designated origin and a designated line through that origin called the $x$-axis as a starting point to define a Cartesian coordinate grid as well. That's no different. The only difference is how you go from there to assigning a pair of numbers to each point in the plane.

The Cartesian coordinate system measures how far you have to go along the $x$-axis from the origin, then how far you have to go orthogonally to the $x$-axis to reach your point. The polar coordinate system tells you how far you have to go to get in a straight line from the origin to your point, and in what direction (relative to the designated $x$-axis) you have to go to get there.

So the moral of the story is that once you have a coordinate system in place, any reasonable coordinate systems in the plane (like Cartesian, polar or elliptic) will assign to each point a pair of numbers. But you do need a couple of parameters to fully specify any coordinate system.