Consider the map
$$ f: \mathbb R^2 \to \mathbb R, (x,y) \mapsto x^3 + y^3 + xy$$
This defines a surface in $\mathbb R^3$. Let's consider some level set $f(x,y) = c$:
(see here page 67)
I think of these pictures as viewed from above looking down on the x-y-plane. Using these 3 pictures to imagine what the surface should look like lead me to believe that the surface should look like this:
(apologies, this is the best I could do with the online drawing tool)
To verify this I then used an online plotter which yielded this:
Now my question is:
How is it possible that the level lines of the last graph yield the
level lines in the first three pictures? I do not see how this is
possible.
(the range of the plot I used was $-1 \le x,y \le 1$ and $-1.3 \le z \le 3$, varying the range does not seem to change the graph)
Best Answer
Your sketch doesn't even look like the graph of a smooth function - remember that it should just be a deformed plane with nowhere vertical tangents. Here's an animation I whipped up that might help your intuition for this example:
In general it's helpful to remember how changes in the topology of the level curves correspond to local features of the function: a loop appearing/disappearing corresponds to a local extremum, while a transition across a self-intersection like you see in this example at $c=0$ corresponds to a saddle point.