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Contour maps

When drawing in three dimensions is inconvenient, a contour map is a useful alternative for representing functions with a two-dimensional input and a one-dimensional output.

The process

Contour maps are a way to depict functions with a two-dimensional input and a one-dimensional output. For example, consider this function:
With graphs, the way to associate the input (x,y) with the output f(x,y) is to combine both into a triplet (x,y,f(x,y)), and plot that triplet as a point in three-dimensional space. The graph itself consists of all possible three-dimensional points of the form (x,y,f(x,y)), which collectively form a surface of some kind.
But sometimes rendering a three-dimensional image can be clunky, or difficult to do by hand on the fly. Contour maps give a way to represent the function while only drawing on the two-dimensional input space.
Here's how it's done:
  • Step 1: Start with the graph of the function.
  • Step 2: Slice the graph with a few evenly-spaced level planes, each of which should be parallel to the xy-plane. You can think of these planes as the spots where z equals some given output, like z=2.
  • Step 3: Mark the graph where the planes cut into it.
  • Step 4: Project these lines onto the xy-plane, and label the heights they correspond to.
In other words, you choose a set of output values to represent, and for each one of these output values you draw a line which passes through all the input values (x,y) for which f(x,y) equals that value. To keep track of which lines correspond to which values, people commonly write down the appropriate number somewhere along each line.
Note: The choice of outputs you want to represent, such as {2,1,0,1,2} in this example, should almost always be evenly spaced. This makes it much easier to understand the "shape" of the function just by looking at the contour map.

Example 1: Paraboloid

Consider the function f(x,y)=x2+y2. The shape of its graph is what's known as a "paraboloid", the three-dimensional equivalent of a parabola.
Here's what its contour map looks like:
Notice that the circles are not evenly spaced. This is because the height of the graph increases more quickly as you get farther away from the origin. Therefore, increasing the height by a given amount requires a smaller step away from the origin in the input space.

Example 2: Waves

How about the function f(x,y)=cos(x)sin(y)? Its graph looks super wavy:
And here is its contour map:
One feature worth pointing out here is that peaks and valleys can easily look very similar on a contour map, and can only be distinguished by reading the labels.

Example 3: Linear function

Next, let's look at f(x,y)=x+2y. Its graph is a slanted plane.
This corresponds to a contour map with evenly spaced straight lines:

Example 4: Literal map

Contour maps are often used in actual maps to portray altitude in hilly terrains. The image on the right, for example, is a depiction of a certain crater on the moon.
Contour map of South Ray crater on the moon, from wikipedia
Imagine walking around this crater. Where the contour lines are close together, the slope is rather steep. For instance, you descend from 7700 meters to 7650 meters over a very short distance. At the bottom, where lines are sparse, things are more flat, varying between 7650 meters and 7628 meters over larger distances.


The lines on a contour map have various names:
  • Contour lines.
  • Level sets, so named because they represent values of (x,y) where the height of the graph remains unchanged, hence level.
  • Isolines, where "iso" is a greek prefix meaning "same".
Depending on what the contour map represents, this iso prefix might come attached to a number of things. Here are two common examples from weather maps.
  • An isotherm is a line on a contour map for a function representing temperature.
  • An isobar is a line on a contour map representing pressure.

Gaining intuition from a contour map

You can tell how steep a portion of your graph is by how close the contour lines are to one another. When they are far apart, it takes a lot of lateral distance to increase altitude, but when they are close, altitude increases quickly for small lateral increments.
The level sets associated with heights that approach a peak of the graph will look like smaller and smaller closed loops, each one encompassing the next. Likewise for an inverted peak of the graph. This means you can spot the maximum or minimum of a function using its contour map by looking for sets of closed loops enveloping one another, like distorted concentric circles.

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