3.1 Contour Maps

A function of two variables assigns a height $z = f(x, y)$ to every point of the plane, and its graph is a landscape. Landscapes are hard to carry around, so hikers long ago invented something better: the map. A contour map of $f$ draws, on the flat plane, the level curves $f(x, y) = c$ for a list of equally spaced heights $c$ — each curve is the set of points where the landscape stands at exactly that height. The skill this section builds is fusion: seeing map and terrain as one object, so that a glance at crowded curves tells you "steep ground here" without any computation.

The figure stages the definition so you can run it in both directions. The landscape is a two-hill surface; the map is a separate object lying flat beneath it, and in the XYZ 3D editor the map's altitude is under your control.

A half-transparent surface with two rounded hills of different heights, a taller one near the point (1, 0) and a shorter one near the point (-1, 1), above a flat sheet at height zero carrying pink nested contour loops at twelve levels. The five lowest levels each draw one loop around both hills; the next three each draw a separate loop around each hill; the four highest circle only the taller hill. — The two-peak surface $z = 3e^{-((x-1)^2+y^2)} + 2e^{-((x+1)^2+(y-1)^2)}$ , half-transparent, above a flat sheet of pink nested loops at twelve levels — a contour plot built from the same expression, resting at $z = 0$ . Open in XYZ 3D (↗), the sheet has a z-offset you can type, so the whole stack of level curves can ride up into the surface, each loop seating itself when the sheet reaches that loop's own height.

Explore

Orbit until you look straight down the $z$ -axis, so the contour plot reads as a flat map and the surface is only a colored blur behind it. From the map alone: how many peaks does this landscape have, and which one is taller? What about the map told you?
Still looking down, find where the loops crowd closest together and where they spread farthest apart. Then orbit to a side view and predict: walking through the crowded region, what would the ground feel like underfoot? Through the spread-out region?
Open the figure in XYZ 3D (↗), select the contour plot, and type its z offset up from 0 one level at a time. The twelve levels are about $0.23$ apart: $0.23, 0.46, 0.70, 0.93, \ldots, 2.78$ . At each value one level's loops seat themselves exactly where the plane cuts the landscape, while the loops that landed earlier lift off above it. In what order do the loops land — and which peak runs out of loops to seat first?
Type the offset back to $1.16$ , the fifth level. Which loops now float above the surface, which are hidden inside the hills, and which one is seated? Switch between $0.93$ and $1.16$ and watch how far the seated loop shifts across the map: where it barely moves, the wall is steep. Is that the crowded region you flagged in step 2?
Still in XYZ 3D, drag the plot's Levels slider from 12 up to 20. The levels are equally spaced in height, so what does tighter spacing on the map now say, precisely, about the surface between two neighboring curves?
Sketch, on paper, the map you expect for $4 - x^2 - y^2$ — the example below does the algebra — then, in XYZ 3D, edit both objects' expressions to it and look straight down. Where were the rings more crowded than you drew?

The map is the surface, disassembled

A level curve is a slice: cut the surface with the horizontal plane $z = c$ , and the cut edge, dropped straight down to the page, is the curve $f(x, y) = c$ . Raising the z-offset runs this definition in reverse — the plane climbs, and at each level's own height it seats that level's curve where the slice happens. Because the $c$ -values are equally spaced, the horizontal gap between neighboring curves is inversely tied to how fast $f$ climbs there: crowded curves mean steep ground.

The two-hill specimen shows what else a map encodes. The taller summit, near $(1, 0)$ , tops out around $3.01$ ; the shorter one, near $(-1, 1)$ , at about $2.02$ . Levels above $2.02$ therefore cut only the taller hill — which is why more loops encircle it. Counting rings compares summits.

Example 1 The rings of a paraboloid, before you plot them

For $g(x, y) = 4 - x^2 - y^2$ , find the level curves at $c = 3, 2, 1, 0$ and decide whether the rings crowd together or spread apart as you move outward from the center.

Show solution

Setting $g = c$ gives $x^2 + y^2 = 4 - c$ : a circle of radius $\sqrt{4 - c}$ , so the four levels are circles of radii

$\sqrt{1} = 1,\quad \sqrt{2} \approx 1.414,\quad \sqrt{3} \approx 1.732,\quad \sqrt{4} = 2.$

The gaps between successive rings, moving outward, are about $0.414$ , then $0.318$ , then $0.268$ — shrinking. The rings crowd as you move out, because the bowl steepens: along the ring of radius $r$ the height $4 - r^2$ falls at rate $2r$ , so the farther out you stand, the less horizontal run a one-unit drop requires. Most people sketch the rings evenly spaced; the map knows better, and now so do you.

An original work of XYZ Homework, built around interactive XYZ 3D figures. Aligned to OpenStax Calculus Volume 3 (Strang & Herman), © OpenStax (Rice University), licensed CC BY-NC-SA 4.0; no OpenStax content is reproduced, and this work is not affiliated with or endorsed by OpenStax or Rice University. License: CC-BY-NC-SA-4.0.