Reading a Contour Map as a Surface
A contour map shows selected horizontal slices of a surface, projected onto the \(xy\)-plane. Each curve carries a height label. The map therefore records where the surface has particular heights, but it does not directly show the surface’s vertical appearance.
To interpret a map, connect three kinds of information: the labels tell you which heights occur, the curve shapes and arrangement suggest the surface’s form, and the spacing gives qualitative evidence about how rapidly the height changes across the plane. A formula can then test whether a proposed surface is consistent with the contours.
A contour is not a curve that the surface follows horizontally. Every point on the contour corresponds to a point on the surface at the same height. For instance, a circular contour labeled \(c=4\) means that the surface has height \(4\) above every point on that circle.
A Reliable Interpretation Method
Begin with the labels, not just the shapes. Ask which direction corresponds to increasing height. Then look for repeated patterns: nested closed curves often indicate a peak or basin; curves that change orientation around a point can indicate a saddle; and evenly spaced parallel lines can indicate a plane. Finally, use a formula to check the proposed interpretation by setting \(f(x,y)=c\) and seeing whether the resulting level sets match the map.
Order the contour labels and identify where heights increase or decrease.
Note whether curves are nested, open, parallel, or arranged around a notable point.
Describe the likely rise, fall, or change of direction, while treating the map as evidence rather than a complete picture.
Solve \(f(x,y)=c\) for representative labels and compare the resulting curves and values with the map.
Contour spacing is meaningful when you compare curves whose labels have the same height difference. If equal changes in height occur across a shorter horizontal distance, the surface changes height more rapidly there. Spacing alone does not determine an exact slope or the height of an unmarked point.
What Can Be Said Between Neighboring Contours?
Suppose two contours are labeled \(c_1\) and \(c_2\), with \(c_1<c_2\). A point in the band between them is often understood to have height between those values. That conclusion needs the map to be interpreted as complete in that band: the function must be continuous, the band must be connected, and there must be no omitted contour or disconnected region that changes the interpretation.
The reason is continuity. If a point in the band had height greater than \(c_2\), a path through the band from that point to its \(c_1\) boundary would have to pass through height \(c_2\). That would create another \(c_2\) contour inside the band, contrary to the assumption that the displayed contours bound a complete band. The same argument rules out a height below \(c_1\). This reasoning gives a bound, not an interpolation rule: being halfway across the band does not in general mean being halfway between its labeled heights.
Worked Example: Nested Circles Around a Basin
Worked Example: Nested Circles Around a Basin
Suppose a map shows concentric circles centered at the origin, labeled \(1,4,9,\) and \(16\), with labels increasing outward. What surface is plausible, and what height does a matching formula give at \((1.5,0)\)?
Step 1: Interpret the labels. Moving outward from the center, the height increases. The nested contours suggest a low central region rising in every direction. A plausible shape is a bowl.
Step 2: Check a candidate formula. Consider \(f(x,y)=x^2+y^2\). Its contour at height \(c\) satisfies:
For \(c>0\), this is a circle of radius \(\sqrt{c}\). The labeled heights \(1,4,9,16\) therefore give radii \(1,2,3,4\), respectively, matching the described map.
Step 3: Find the point’s contour band. At \((1.5,0)\), the distance from the origin is \(1.5\), which lies between radii \(1\) and \(2\). Thus the map places the point between contours labeled \(1\) and \(4\). The band supports the bound \(1<f(1.5,0)<4\).
Step 4: Compute and check using the formula.
The exact formula value \(2.25\) is indeed between \(1\) and \(4\), as the contour map requires. The formula also shows why the map alone would not justify that exact value: the labels establish the interval, while the formula determines the height within it.
In this example, equal height increments do not correspond to equal radial distances. The contour labels rise by \(3,5,\) and \(7\), while the radii rise by \(1\) each time. For a fair spacing comparison, use equal label increments: the change from height \(1\) to \(4\) spans one radial unit, while the change from \(4\) to \(9\) also spans one radial unit, but those height changes are not equal. Spacing comparisons must account for both horizontal distance and the height difference between labels.
Worked Example: Contours of a Saddle
Worked Example: Contours of a Saddle
Suppose a map contains contours shaped like \(x^2-y^2=c\), with labels \(-4,-1,0,1,\) and \(4\). What surface does this suggest, and what height does a matching formula assign to \((2,1)\)?
Step 1: Read the zero contour. For \(c=0\), the equation becomes \(x^2-y^2=0\). Factoring gives \((x-y)(x+y)=0\), so the zero contour consists of the two lines \(y=x\) and \(y=-x\).
Step 2: Compare signs in different directions. Along the \(x\)-axis, \(y=0\), so \(f(x,0)=x^2\), which is positive except at the origin. Along the \(y\)-axis, \(x=0\), so \(f(0,y)=-y^2\), which is negative except at the origin. The surface rises in one pair of directions and falls in the other, indicating a saddle rather than a peak or basin.
Step 3: Locate and calculate the requested height. The point \((2,1)\) lies in the region where \(x^2-y^2\) is positive. Using the matching formula \(f(x,y)=x^2-y^2\):
Since \(1<3<4\), this value lies between the contours labeled \(1\) and \(4\). The formula’s level set at height \(3\) is \(x^2-y^2=3\), consistent with the positive-valued family of hyperbolas in the map.
Check along another direction. At \((1,2)\), the formula gives \(1^2-2^2=1-4=-3\), which lies between \(-4\) and \(-1\). This is consistent with the negative-valued contours on the other side of the zero set.
When interpreting a saddle map, the zero contour is especially informative, but the labels on either side matter too. A crossing contour pattern alone is not enough to tell which regions are high and which are low; the signs or ordering of the labels provide that information.
Worked Example: Parallel Contours of a Tilted Plane
Worked Example: Parallel Contours of a Tilted Plane
A map shows parallel lines labeled \(-2,0,2,\) and \(4\). Suppose the contour at height \(c\) has equation \(2x-y=c\). Identify a matching surface and find the height at \((1,1)\).
Step 1: Identify a formula. The contour relation \(2x-y=c\) is obtained by setting \(f(x,y)=c\) for \(f(x,y)=2x-y\). The corresponding surface is the plane \(z=2x-y\).
Step 2: Verify the contour geometry. Rearranging the level equation gives \(y=2x-c\). For each fixed \(c\), this is a line with slope \(2\); changing \(c\) shifts the line without changing its slope. The parallel contour pattern therefore matches the formula.
Step 3: Locate the point and calculate its height. At \((1,1)\), the contour expression has value \(2(1)-1=1\), between the labeled levels \(0\) and \(2\). The formula gives:
The computed height \(1\) lies between \(0\) and \(2\), as expected from the contour band.
Step 4: Check the point against its level set. The height \(1\) contour is \(2x-y=1\). Substituting \((1,1)\) gives \(2(1)-1=1\), so the point lies exactly on that contour.
For this plane, equal height changes correspond to equal perpendicular distances between the parallel lines. For example, the normal direction to the lines is given by \((2,-1)\), whose length is \(\sqrt{2^2+(-1)^2}=\sqrt{5}\). A change of \(2\) in the contour label corresponds to perpendicular spacing \(2/\sqrt{5}\). This is consistent with a constant rate of height change across the plane.
A Common Mistake: Linear Interpolation by Position
A common error is to treat the gap between two contours as though height changed linearly across it. In the bowl example, the point \((1.5,0)\) lies halfway in radius between the circles of radii \(1\) and \(2\). A naive interpolation between their labels gives:
But the matching formula gives \(f(1.5,0)=2.25\), not \(2.5\). The calculation \(2.5\) assumes a linear change in height with radius, an assumption that the contour labels do not supply. The correction is to report only the justified bound \(1<f(1.5,0)<4\) from the map, unless a formula or further information determines a more precise value.
Check Your Understanding
Use the contour labels and the stated formulas to answer the following questions.
- For \(f(x,y)=x^2+y^2\), what radius corresponds to the contour \(c=25\), and does the map suggest a peak or a basin as the labels increase outward?
- A point lies in a connected band between contours labeled \(2\) and \(7\). What height information follows from the map if the function is continuous and the displayed contours are complete in that band?
- For \(f(x,y)=x^2-y^2\), calculate the height at \((1,3)\), and identify the two labeled levels \(-9\) and \(-1\) that bound it.
- Why does a point halfway across a band between heights \(0\) and \(10\) not necessarily have height \(5\)?
- What feature of the contours \(2x-y=c\) indicates that the surface \(z=2x-y\) is a plane?