From Terrain to Contour Map
A topographic map represents the height of the land above a chosen reference level. Imagine that the terrain is described by a height function \(z=f(x,y)\), where \((x,y)\) gives a horizontal location and \(f(x,y)\) gives the elevation there. A contour line joins map locations that have the same elevation.
This is the same mathematical idea as a level curve: the points on a contour at elevation \(c\) satisfy \(f(x,y)=c\). The map is a two-dimensional drawing of these curves, with elevation labels identifying their heights. Reading the map means using both the shapes of the curves and the values attached to them.
Contours do not show the land as a three-dimensional picture. Instead, they provide height information at particular levels. The pattern between contours lets us infer terrain, but a map usually does not specify the exact height at every point unless additional information is given.
Reading Elevation Labels
First establish which contours are higher and which are lower. A contour interval is the elevation difference between successive regularly spaced contours. For example, if adjacent contours are labeled \(100\) m and \(120\) m, the interval is \(20\) m. The labels, not the visual direction in which a curve bends, determine which side is higher.
When a map includes unlabeled contours between labeled ones, count them using the stated contour interval. Do not assume that the interval is the same on every map. Nor should you assume that every drawn contour is labeled: intermediate values may have to be inferred from the map’s convention.
Worked Example: Identifying a Closed Hill
A map has nested closed contours labeled, from the outside inward, \(200\) m, \(220\) m, \(240\) m, and \(260\) m. What does the pattern indicate?
The contour interval is the difference between adjacent labels:
Each contour farther inward is higher. Therefore, moving inward through the rings corresponds to increasing elevation. The enclosed area is a hill or summit region, and the highest displayed contour is \(260\) m. The map does not show whether the actual peak is exactly \(260\) m: it could lie above \(260\) m inside that contour, unless the map supplies further information.
Check: Moving outward reverses the sequence, giving \(260, 240, 220, 200\) m, so the land decreases in elevation away from the center of the rings.
Closed Contours: Hills and Depressions
A set of nested closed contours is often associated with a local high or local low. To decide which one, read the elevation labels in order. If the values increase toward the center, the contours enclose a hill. If the values decrease toward the center, they enclose a depression or basin. The closed shape by itself is not enough to distinguish them.
Some maps mark a depression with short inward-pointing hachures on the contours. Those marks are useful, but the elevation labels still provide the essential height information. For example, if the outer contour is \(300\) m and inner contours descend to \(260\) m, the center is lower than the surrounding land. It is not a hill simply because the contours are nested.
Worked Example: Interpreting a Depression
Suppose nested closed contours are labeled \(300\) m on the outside, then \(280\) m, then \(260\) m on the inside. Determine the direction of change toward the center and classify the feature.
Compare each successive pair:
The elevation drops by \(20\) m at each inward step. Thus, moving toward the center leads downhill, and the closed contours indicate a depression. If the map instead showed values \(300\), \(320\), and \(340\) m toward the center, the same nested geometry would indicate a hill.
Check: Starting at the inner contour and moving outward gives \(260, 280, 300\) m, so the ground rises outward, consistent with a central low.
Ridges and Valleys
A ridge is an elongated region of relatively high ground. A valley is an elongated region of relatively low ground, often associated with a channel or drainage line. On a contour map, their contours can bend into U- or V-like shapes. Those shapes alone do not settle whether the feature is a ridge or valley: the elevation labels tell which direction is uphill.
For a valley containing a stream, contour bends commonly point upstream, toward higher ground. For a ridge, bends often point downhill. This is a helpful pattern, not a substitute for reading the labels: local terrain can be more complicated, and the map’s elevations decide the direction.
Worked Example: Contours Along an Elongated Ridge
Consider contours described in map coordinates by \(y=10x^2\) at elevation \(0\) and \(y=10+10x^2\) at elevation \(10\). Here, larger \(y\) is north. Compare their vertices and their opening directions, and interpret the elevation pattern.
The first contour has vertex \((0,0)\), since \(y=10(0)^2=0\) at \(x=0\). The second has vertex \((0,10)\), since \(y=10+10(0)^2=10\). Thus the higher contour’s bend is farther north. Both equations have a positive coefficient on \(x^2\), so both parabolas open north. Their bends point south toward lower ground; the curves open north.
The elevation increase from \(0\) to \(10\) m occurs northward between these contours. At a fixed \(x\), the second contour is exactly \(10\) units north of the first, since
The contours are consistent with higher ground to the north and lower ground to the south. In this example, the repeated parabolic pattern is associated with an elongated raised feature; the labels establish the uphill direction, while the bending pattern helps show its shape.
Check: Substituting \(x=0\) gives the two vertex locations \(y=0\) and \(y=10\), and the positive quadratic coefficients confirm that both curves open north.
Contour Spacing and Slope
Contour spacing indicates how quickly elevation changes across horizontal distance, provided the contours represent equal elevation intervals and the map scale is accounted for. Closely spaced contours mean that a given elevation change occurs over a short horizontal distance, so the terrain is steep. Widely spaced contours mean the same change occurs over more horizontal distance, so the terrain is gentler.
A useful estimate is average slope along a path between two contours:
The result is a rise-to-run ratio; it can also be written as a percentage after multiplying by \(100\). The distance must be a horizontal map distance converted using the map scale, not the length measured along the sloping ground. If a path is not perpendicular to the contours, this calculation describes the average change along that particular path, not necessarily the steepest possible direction.
Worked Example: Comparing Two Slopes
Two routes each cross contours whose elevations differ by \(40\) m. On Route A, the contours are \(100\) m apart horizontally. On Route B, they are \(400\) m apart. Find each average slope as a ratio and percentage, then compare them.
For Route A, divide the rise by the horizontal run:
For Route B:
Route A has the greater average slope: \(40\%\) compared with \(10\%\). Its contours are closer together because the same \(40\) m rise takes place in one-quarter the horizontal distance. This is an average over each route, not a claim that the slope is constant at every point.
Check: Reversing the calculation, \(0.4(100\text{ m})=40\text{ m}\) and \(0.1(400\text{ m})=40\text{ m}\), recovering the stated elevation change.
Why Spacing Reflects Slope
The spacing interpretation follows from the definition of average rate of change. Suppose two points along a horizontal route have elevations \(h_1\) and \(h_2\), separated by horizontal distance \(d\). The average elevation change per unit distance is \((h_2-h_1)/d\). If the elevation difference is fixed, a smaller \(d\) makes the magnitude of this ratio larger.
For neighboring contours with a fixed interval \(\Delta h\), let \(\Delta s\) be the horizontal distance between them along the route being considered. Then the magnitude of the average slope across that interval is
When \(\Delta h\) is the same from one contour pair to another, reducing \(\Delta s\) increases this ratio. That is why close contours indicate steeper terrain. This comparison is local only to the extent that the chosen contour spacing and route represent the terrain of interest; it does not establish the exact slope at every location.
A Common Mistake: Guessing from Shape Alone
A frequent error is to call any closed set of contours a hill, or to label every U-shaped bend a valley. Both conclusions can be wrong if the elevation labels are ignored. A closed set with elevations falling inward is a depression, and a bend’s orientation must be interpreted alongside the values on either side.
Worked Example: Correcting a Shape-Only Guess
A student sees closed contours labeled \(500\) m on the outside, \(480\) m next, and \(460\) m on the inside, and writes, “The innermost ring is a summit because the contours close around it.” The conclusion does not follow from the shape.
Check the values in order:
The elevation decreases inward by \(20\) m at each step. Therefore the center is a depression relative to its surroundings, not a summit. The student’s mistake was treating closed geometry as proof of a high point without checking the labels.
Check: Moving outward from \(460\) m gives \(480\) m and then \(500\) m, confirming that elevation rises away from the center.
What a Contour Map Can and Cannot Tell You
A contour map supports useful conclusions about relative heights, terrain shape, and approximate steepness. It does not usually specify every detail between the drawn lines. A narrow peak or hollow can lie between contours, and two different surfaces can pass through the same set of displayed contours while differing between them.
Keep the map’s scale, contour interval, and labels in view. A reliable reading proceeds from elevation values to terrain interpretation, not from a shape guess to a presumed elevation pattern.
Identify the contour interval and determine which nearby contours are higher.
Check whether elevations rise or fall inward, along a bend, or along a route.
Use the combined elevation and shape evidence to identify a hill, depression, ridge, or valley.
For equal contour intervals, closer spacing indicates steeper ground; wider spacing indicates gentler ground.
Check Your Understanding
Use the contour labels and patterns in each question, rather than relying on shape alone.
- Nested closed contours have elevations \(150\) m, \(170\) m, and \(190\) m from outside to inside. What feature do they indicate?
- Nested closed contours have elevations \(420\) m, \(400\) m, and \(380\) m from outside to inside. What feature do they indicate, and why?
- Two routes each climb \(30\) m. One crosses the contours over \(150\) m horizontally and the other over \(300\) m. Which route has the greater average slope?
- Why is the bend direction of a contour not enough by itself to identify a ridge or valley?
- What does a topographic map leave undetermined when only selected contour elevations are shown?