Visualizing "complicated" level sets, e.g. $\sin(x)-\sin(y)=k$?
I can picture that there's wave-like motion in both directions and there's some kind of symmetry pattern, because both functions are $\sin(x)$, however, since the other is $-\sin(y)$, then particularly, it's a $\pi$ shifted $\sin(x)$. So this reads that the symmetry pattern should be "shifted" in $y$ direction, but not on $x$.
However, it becomes complicated to deduce more about this. Are there some general techniques other than merely visualizing function values, like I did above?
EDIT:
I also figured on the above function that of course I can visualize the function's plot, which should intuitively be a 2D wave surface that's asymmetric with regards to $y$-axis. Then if one considers intersecting this with a plane to get level curves, then one can imagine the pattern being "circular" concentrations that occur symmetrically, but in $y$ direction there should be a "half period" shift visible.
Another example function:
$$\frac{(x-y)}{1+x^2+y^2}$$