Introduction
A contour map shows where a surface has selected heights. Each curve is drawn in the \(xy\)-plane, but its label gives the corresponding vertical coordinate \(z\). Reading the labels in their spatial order lets us sketch whether the surface rises or falls as we move across the domain.
The map does not usually specify a complete formula for the surface. A finite collection of contours gives information at selected heights; many different surfaces can agree with those contours and behave differently between them. A careful sketch therefore separates what the labels establish from what they merely suggest.
What a Labeled Contour Tells Us
For a surface that is the graph of \(z=f(x,y)\), the contour at height \(c\) consists of the domain points where \(f(x,y)=c\). It is drawn in the horizontal \(xy\)-plane. The corresponding points on the surface have vertical coordinate \(z=c\).
For example, if two contours are labeled \(1\) and \(3\), every point on the first has surface height \(1\), and every point on the second has height \(3\). The surface is higher at the second contour. To infer how height changes across the region, follow a route in the domain and read the contour labels in the order that the route encounters them.
A route need not be straight. It may be useful to follow a fixed-\(x\) or fixed-\(y\) direction, or to trace another path across the map. The conclusion is about height change along that route; it does not automatically describe every direction at each point.
Reading Rise, Fall, and Flattening
If the labels encountered along a route increase, the surface is higher at the later contour crossings than at the earlier ones. If they decrease, it is lower at the later crossings. This comparison of heights is direct. Saying that the surface rises steadily or falls steadily between crossings is a stronger claim: a finite map may omit intermediate behavior.
Contour spacing adds information about steepness when the height intervals are equal. If contours labeled \(1,2,3\) are close together in one region and farther apart in another, a similar horizontal distance produces a larger height change in the crowded region. This supports the conclusion that the surface is steeper there, on average, in the direction being considered.
Conversely, equally spaced height contours that spread farther apart suggest that the surface is flattening in that direction. This is a comparison based on the map’s scale and the height intervals shown. It does not give an exact rate of change or prove what happens at every point between contours.
Worked Example: Parallel Contours and Height Order
Worked Example: Parallel Contours and Height Order
Suppose a map shows the lines \(2x-y=0\), \(2x-y=2\), and \(2x-y=4\), labeled with heights \(0\), \(2\), and \(4\), respectively. Determine how the labels are ordered along the positive \(x\)-direction with \(y=0\), and identify one function consistent with the displayed contours.
On the route \(y=0\), the contour equation \(2x-y=c\) becomes \(2x=c\). Thus the three displayed contours meet this route at \(x=0\), \(x=1\), and \(x=2\), respectively. Their labels increase in that order: \(0,2,4\). The sampled heights rise as we move right.
One function consistent with these contours is \(f(x,y)=2x-y\). Substituting the three contour equations into this formula gives the indicated heights: \(f=0\) when \(2x-y=0\), \(f=2\) when \(2x-y=2\), and \(f=4\) when \(2x-y=4\). In particular, on the route \(y=0\), the points \((0,0)\), \((1,0)\), and \((2,0)\) have heights \(0\), \(2\), and \(4\).
This is one compatible model, not a uniquely determined formula. To see why, define \(\phi(t)=t+0.1\sin(\pi t/2)\) and \(g(x,y)=\phi(2x-y)\). The derivative of \(\phi\) is \(1+0.1(\pi/2)\cos(\pi t/2)\), which is at least \(1-0.1(\pi/2)>0\). Therefore \(\phi\) is strictly increasing. Also, \(\phi(0)=0\), \(\phi(2)=2\), and \(\phi(4)=4\). Since \(\phi\) is one-to-one, the contours of \(g\) at heights \(0,2,4\) are exactly the three displayed lines. Yet at \((1/2,0)\), \(f=1\), while \(g=1+0.1\sin(\pi/2)=1.1\). The two surfaces agree on the displayed contour heights but not at this point.
Check: Substituting any point from \(2x-y=2\) into \(f=2x-y\) gives \(f=2\); for \(g\), it gives \(\phi(2)=2\). Thus both models match the height-\(2\) contour, even though their heights between displayed contours can differ.
Why Contour Order Constrains a Continuous Surface
Continuity explains why a surface cannot move from below a height to above it without passing through that height. The relevant one-variable fact is the Intermediate Value Theorem: if a function \(h\) is continuous on a closed interval and \(h(a)<c<h(b)\), then \(h(s)=c\) for at least one \(s\) between \(a\) and \(b\).
Apply this fact to a route \(\boldsymbol{\gamma}(s)=(x(s),y(s))\) through the domain of a continuous surface \(f\). The height along the route is \(h(s)=f(x(s),y(s))\). If the route and surface are continuous, then \(h\) is continuous. If its height changes from below \(c\) to above \(c\), the route must meet the level curve \(f(x,y)=c\).
Track a continuous path through the domain, and consider the surface height along it.
If one height is below \(c\) and another is above \(c\), the height changes across \(c\).
The Intermediate Value Theorem guarantees a point on the route where the height equals \(c\); that point lies on the height-\(c\) contour.
This argument justifies treating labeled contours as boundaries between height levels. It does not imply that height changes monotonically between two contours. A surface may rise, dip, and rise again without crossing a particular level that is absent from the map. The contour labels constrain the surface, but do not display every detail of its behavior.
Worked Example: Contour Spacing and Flattening
Worked Example: Contour Spacing and Flattening
On the strip \(0\leq x\leq3\), suppose the contours are vertical lines at \(x=0\), \(x=1\), and \(x=3\), labeled \(0\), \(1\), and \(2\). Compare the average height change per unit horizontal distance across the two bands. Then check the comparison using one function consistent with the data.
From \(x=0\) to \(x=1\), the height increases by \(1-0=1\) over a horizontal distance of \(1-0=1\). The average height change per unit distance is \(1/1=1\). From \(x=1\) to \(x=3\), the height increases by \(2-1=1\) over a distance of \(3-1=2\). The average change is \(1/2\). Thus the map suggests a less steep rise in the second band: the same height increase is spread over twice the horizontal distance.
For a model check, let \(f(x,y)=-x^2/6+7x/6\) on this strip. At \(x=0\), \(f=0\). At \(x=1\), \(f=(-1+7)/6=1\). At \(x=3\), \(f=(-9+21)/6=2\). Its contours at these heights are the stated lines within the strip: solving \(f=0\) gives \(x=0\) or \(7\), so only \(x=0\) is in the strip; solving \(f=1\) gives \(x=1\) or \(6\), so only \(x=1\) is in the strip; solving \(f=2\) gives \(x=3\) or \(4\), so only \(x=3\) is in the strip.
The model also verifies the average changes in smaller steps. At \(x=2\), \(f=(-4+14)/6=5/3\). From \(x=1\) to \(x=2\), the height increases by \(5/3-1=2/3\); from \(x=2\) to \(x=3\), it increases by \(2-5/3=1/3\). These values illustrate a rise that becomes less steep in this particular model. The contour data alone supports the average comparison, not this exact behavior at every intermediate point.
Check: Substituting \(x=0,1,3\) into the model returns \(0,1,2\), respectively, exactly matching the contour labels. Restricting to the strip is essential: the same formula has other solutions to those level equations outside the displayed domain.
Worked Example: A Sampled Change from Rising to Falling
Worked Example: A Sampled Change from Rising to Falling
A route across a contour map meets contours at the following ordered locations: at \(s=0\), height \(0\); at \(s=1\), height \(2\); at \(s=2\), height \(1\); and at \(s=3\), height \(-1\). Describe the sampled height changes and state what can be concluded.
Between the first and second locations, the height change is \(2-0=2\) over distance \(1-0=1\), an average increase of \(2\) per unit route parameter. Between the second and third, it is \(1-2=-1\) over distance \(2-1=1\), an average decrease of \(1\) per unit. Between the third and fourth, it is \(-1-1=-2\) over distance \(3-2=1\), an average decrease of \(2\) per unit.
The sampled heights first increase and then decrease. This supports a sketch with an upward trend along the first part of the route and a downward trend across the later sampled locations. It does not tell us exactly where the change in trend occurs, whether there are smaller intervening rises and falls, or what happens off the route. Those details require additional information.
Check: The computed changes telescope: \(2+(-1)+(-2)=-1\), which equals the final height \(-1\) minus the initial height \(0\). The arithmetic agrees with the endpoint labels.
Common Pitfall: Treating a Few Contours as a Formula
A frequent mistake is to rearrange one or more contour equations and announce that the resulting expression must be the surface’s formula. In the first example, \(2x-y=c\) for the displayed labels makes \(f(x,y)=2x-y\) a convenient model. But the map shows only the levels \(0,2,4\). It does not require the surface to equal \(2x-y\) at every other point.
The alternative \(g(x,y)=\phi(2x-y)\) in that example is a concrete correction: it has the same three displayed contours, yet gives height \(1.1\), rather than \(1\), at \((1/2,0)\). The right conclusion is that \(f(x,y)=2x-y\) is one function consistent with the shown contours. Uniqueness would require substantially more information, such as a complete family of contours with specified heights and additional assumptions on the surface.
Another error is to infer steepness from contour spacing when the height intervals differ. A map with contours \(0,1,2\) close together in one region and contours \(2,10,18\) farther apart elsewhere does not permit a simple spacing comparison: the height increments are not equal. Always compare horizontal spacing for equal changes in height, and account for the map’s scale.
Using Contours to Build a Surface Sketch
To sketch from a contour map, first mark the labeled curves and imagine each as a horizontal trace of the surface at its stated height. Then choose one or more routes across the map. Along each route, place the corresponding heights in order and connect them with a plausible profile. Use close equal-height contours to show a relatively steep section and wider spacing to show a gentler section, without adding more precision than the map supports.
Finally, keep the sketch consistent in all directions. A contour is a curve of constant height, so the surface must stay at that height above the entire curve. Nearby contours impose neighboring height constraints. A surface sketch that rises correctly along one route but contradicts the labels along another is not compatible with the map.
Check Your Understanding
Use the ideas in this tutorial to answer the following questions.
- What does a contour labeled \(c\) tell you about the height of the surface above each point on that curve?
- Along a route, the labels encountered are \(4,2,0\). What height trend do the sampled locations show?
- For equal height intervals, what does increasing distance between successive contours suggest about the surface in that direction?
- Why can a finite set of labeled contours fail to determine a unique formula for the surface?
- State the continuity condition that ensures a route moving from below height \(c\) to above height \(c\) meets the contour at height \(c\).