It seems to be trivial, but I couldn't find anything.
I have a metric space, let us say $\mathbb{R}_{>0}^3$, and a equivalence relation $\sim$, let's say $(x_1,y_1,z_1)\sim(x_2,y_2,z_2)$ if $z_1=z_2$ and $\exists p>0$ such that $px_1=x_2$ and $py_1=y_2$. Then, the quotient space $\mathbb{R}_{>0}^3/\sim$ can be identified with $\mathbb{R}_{>0}^2$. So far, so good.
Now, I want to plot a given trajectory both in $\mathbb{R}_{>0}^3$ and $\mathbb{R}_{>0}^3/\sim$. As a representation of $\mathbb{R}_{>0}^3/\sim$ as $\mathbb{R}_{>0}^2$ I have chosen a 2D slice through, let's say, $y_2=const$. Then I projected all points of the trajectory on the 2D slice such that each point stays in its equivalence class, and then plotted that trajectory in $\mathbb{R}_{>0}^2$ as a representation of the trajectory in the quotient space.
My problem: What is a suitable labeling of the axes? One axis I would like to label something along the line of "$z$", but labeling the other "$x$" would be misleading. $[z]$ and $[x,y]$ also feels wrong.
Note: (i) Due to technical reasons I have to project on a slice with $y=const$. (ii) The axes labels should somehow tell something about their relationship with x,y, and z. (iii) The equivalence relation is in general more complex.
As you say, one coordinate can be called $z$, and this is fine.
The other coordinate can be represented best by the ratio $x/y$ (or $y/x$).