What would happen if a standard Cartesian coordinate system were intersected by a polar coordinate system in terms of the graph? What could it show?

Question

In a standard Cartesian system you have X, Y, and if you're going into the 3rd dimension Z.

In a polar coordinate system you have theta and Y.

If you intersected the two coordinate systems could you graph imaginary numbers?

Why/ why not would this be possible?

Answer 1

I'm not an expert in the field, however the coordinate systems are just measurements! So if you have a point in space, after fixing an Origin $O$ (which I guess it would be the same for both coordinate systems, in your question), the coordinate systems are just a way of describing where the point is, with respect to the standard basis vector for that coordinate system. Which are $\vec{i},\vec{j},\vec{k}$ and $\vec{e_r}, \vec{e_\theta},\vec{e_\phi}$.

Hence, a line or a plane, a sphere or whatever you have can be described in two different ways, however the given object is still the same! The shape of them is coordinate-independent!

There's not such thing, as far as I know, of intersecting two coordinate systems. We already have a way of graphing complex numbers, which is the plane with the real $x$ axis, called $Re(z)$ and the Imaginary $y$ axis, called $Im(z)$.

I guess what you want to do is to graph complex numbers in 3D? Well technically, if you have a function $f:t\in\mathbb{R}\to z(t):=x(t)+iv(t)\in\mathbb{C}$, you have a function with a graph in 3 dimensions, however we normally just graph the "path" in a 2D plane.

Indeed, you would have an axis for $t$, one for $x(t)$ and one for $y(t)$.

Answer 2

My intuition tells me you believe that the $y$ in the Cartesian system ('$x$-$y$') and the $y$ in the polar system ('$y$-$\theta$') mean the same thing. In polar coordinates we actually usually use $r$-$\theta$ notation precisely to mark the difference, $r$ means radius (or distance to the origin) and $y$ means height (or signed distance to the $x$-axis).

They are related by the usual transformation $y=r\cdot\sin(\theta)$, but there's nothing else and that certainly doesn't make them the same thing. For one, $y$ can assume all real values, while $r$ must be non-negative.