Contour map

A level curve, or contour, of a function $f(x, y)$ is the set of points where $f$ takes one value $c$. On the surface $z = f(x, y)$ these are the points at height $c$. A contour map draws level curves for equally spaced heights on the ground, the way a topographic map shows hills.

The surface is drawn as glass over a square map, with the same scale on all three axes, and its contour map lies on the floor below. Raise the water to height $c$ and every point lower than $c$ is under water. The shoreline, drawn in blue on the surface and on the map, is the level curve $f = c$, so a contour map is a stack of shorelines. The water is a threshold, not a flood: a hollow far from any other water is already a lake as soon as its bottom is below $c$.

As $c$ rises, the shoreline mostly just moves and changes shape. Inside the map it changes how it connects only at the heights of critical points, where $\nabla f = 0$, and at the tip of the cone, where there is no gradient at all. Just above the bottom of a pit a lake appears. At a pass, or saddle point, two shorelines meet or one island splits into two. At a summit an island shrinks to a point and vanishes. The ruler beside the view marks these heights.

The map is a window on a larger landscape, so the page counts only lakes and islands that stay inside it. Water that reaches the edge is open sea, and land that reaches it is mainland. The counts can therefore also change where the shoreline meets the edge. The ruler marks those heights with a dashed tick and no symbol.

A level curve through a saddle point crosses itself there. On the saddle $f = \tfrac14(x^2 - y^2)$, the level curve $f = 0$ is the pair of lines $y = \pm x$. Below 0 the water lies in two valleys, at 0 their shorelines touch at the saddle, and above 0 it is one sheet. Contours of different heights never meet, because a point has only one height.

Neighbouring contours differ in height by $\Delta c$, so where they are close the ground climbs $\Delta c$ in a short distance. Where $\nabla f \ne 0$ and neighbouring contours are close, the gap measured at right angles to the contours is about

The Profile window draws the height along a route from A to B at the same scale on both axes, with a tick at each contour the route crosses and the steepest point in yellow. Along a stretch where the route keeps climbing or falling, closer ticks mean a steeper profile.

The lines alone do not say which way is up. On $f = 2x\,e^{-(x^2 + y^2)/2}$, the rings around $(1, 0)$ and $(-1, 0)$ have the same shape, but one is a hill and the other a pit. A map needs the heights written on it, or another clue such as the water.