What is the difference in level curves from 2 to 3 dimensions?

Question

Trouble with intuition on level curves.

In 3 D "level" can also be vertical, is that true?

I assume they can be level to ANY of the axis and my second question is what is a level curve when using only 2D?

Answer 1

Your question is a little hard to understand, but I think I know the general concept you're after, so here goes.

When I think of level curves, I like to think of the analogy of an island where you can vary the water level. A "3D" function that maps two coordinates to a single output $z = f(x,y)$ can be thought of as defining the surface of the earth around an island. The level curves of $f$ can be thought of as the various shorelines you get as you change the water level up and down.

We can play the same game in 2D where we have a single input, single output function $y = f(x)$ which can be thought of as defining the surface of a two dimensional world, and whose water levels are horizontal lines. In this case, the level "curves" are the particular set of intersection points that a given horizontal line makes with the graph of $y = f(x)$.

Observe that the level curves of a surface embedded in 3D space are curves in 2D space, and the level "curves" of a curve embedded in 2D space are points in 1D space. Thus, if you imagine a 4D function like $w = f(x,y,z)$ which defines a 3D solid embedded in 4D space (and hence not really visualizable), the level curves will be 2D surfaces embedded in 3D space (hence they are often referred to as level surfaces).

As for your first question, can level sets be taken with respect to any axis? I think the answer is No. When we talk about the level curves of a function from $\mathbb{R}^n$ to $\mathbb{R}$ (i.e. takes possibly multiple inputs, but returns only one output), we usually mean the set of intersection points the function makes with a constant function of the same dimension. That is, sets of the form $$ \{(x_1, ..., x_n)\colon\ f(x_1, ..., x_n) = c\} $$ where $c$ represents the "water level".

This geometrically means the "water levels" are always taken to be perpendicular to the output axis (e.g. $y$ in 2D, $z$ in 3D, $w$ in 4D, etc.)

However, this is math. If you want to define level curves a different way whereby you can take level curves with respect to other axes (whatever that means), you're more than free to do so. The real question is if there is any reason to do so.