Last week, I started studying implicit functions during an Analysis II lecture. The thing is, I quickly struggled to visualize subsets of $ \mathbb{R^3} $ without computer aid, given their cartesian equations. Take for example the subset: $ D = \{ (x, y, z) \in \mathbb{R^{3}} | x + y + z - 2 = 0 \} $. If we consider this as the level set $ 0 $ of a function from $ \mathbb{R^3} $ to $ \mathbb{R} $, it's not really intuitive to visualize its shape. (In this case it's trivial because I chose a simple example for the sake of the question. Some equations are easily recognizable, such as the equation of a plane, used in this example, or the equation of a shifted sphere of radius $ r $ given by $ (x- a)^2 + (y - b)^2 + (z -c)^2 = r^2 $, but I believe that's about it. For my part, I don't know many more).
Now, if we consider $ D $ as the graph of a function from $ \mathbb{R^2} $ to $ \mathbb{R} $ given by $ z \equiv f(x, y) = 2 - x - y $, then it becomes easier to see that it's a plane (which intersects the $ xy $ plane at the line joining the points $ (1, 0)$ and $ (0, 1)$). My question then is if there is a way, or if someone has some trick to somehow guess, even if approximately, what the shape of the subset $ D \subset \mathbb{R^{3}}$ looks like given its level set form, without solving for $ z $. (Potentially for subspaces of $ \mathbb{R^n} $ as well, huh). I know my question is quite a bit general, but still, it had me wondering...
PS: If the question isn't clear, is redundant, you don't like the formulation, there's an inappropriate tag, you believe it doesn't contribute to other people, or anything else please let me know. I really don't want to lose my right to ask questions, which might happen if I get downvoted or someone closes my question . It really helps me out and makes learning math more fun by sharing and receiving other people's pov and insights. Have a great one, thanks.