Functions of two variables
The bowl $f(x, y) = x^2 + y^2$ takes a point of the plane and returns one height. Its level curves — the point sets where $f = c$ — are what shape?
With the bowl at $c = 1$, the ink curve is a circle and the probe reads $f = 1.000$ anywhere on it — one output, many inputs. Raise $c$ and the circles grow, but unevenly: equal steps of $c$ pack them tighter near the rim. On the saddle, $c = 1$ draws a hyperbola opening east–west, $c = -1$ north–south, and $c = 0$ two lines crossing at the center.
A function of two variables, $z = f(x, y)$, returns one number per point of the plane. Its graph is a surface — awkward to draw, worse to measure. A level set fixes an output instead: $\{(x, y) : f(x, y) = c\}$ collects every input that maps to $c$. Draw many level sets at equal spacing and you get this instrument: a surveyor's contour map, each curve a line of constant height.
For the bowl, $x^2 + y^2 = c$ is a circle of radius $\sqrt{c}$ — radii $1, 1.41, 1.73, 2$ — which is why the rings crowd outward. Crowding is the map's way of drawing steepness: many levels crossed in a short walk means a fast-changing $f$. A level can also be empty — slide $c$ below $0$ on the bowl and the ink curve vanishes, because no input reaches a negative height.
About this instrument
What it represents: the exact level sets of $f = x^2 + y^2$ and $f = x^2 - y^2$ on $[-2, 2]^2$, drawn with the survey-map convention: thin hairlines at every $0.25$ of height, every fifth level heavier.
What it leaves out: the curves are traced by sampling $f$ on a $64 \times 64$ grid and interpolating linearly inside each cell — marching squares — so each drawn curve is a chain of short straight segments, not the true smooth curve. Between the drawn levels the function still takes every intermediate value; the map shows a sample of levels, never all of them.
Source: W. E. Lorensen and H. E. Cline, Marching Cubes: A High Resolution 3D Surface Construction Algorithm (SIGGRAPH 1987) — the 3D ancestor of the 2D marching-squares pass this instrument runs.
The probe sits on the bowl's ink curve at $c = 1$. What is true at every point of that curve?
On the bowl's map the circles crowd together toward the rim. What does that crowding mean?
Transfer: on the saddle $f = x^2 - y^2$, what is the level set for $c = 0$?
A function of two variables assigns one output to each point of the plane. Its level curves flatten the surface into a map: each curve is one output shared by many inputs, and the spacing between curves is the slope you cannot otherwise see.