Reading the Shape Around a Point
A contour map records the heights of a surface using level curves: each curve consists of points where the function has one fixed value. The shape of those curves matters, but the height labels matter just as much. Together they can show whether the surface locally rises toward a peak, falls toward a basin, or changes direction like a saddle.
The word “locally” is important. A point can be higher than nearby points while still being lower than parts of the surface farther away. Likewise, a basin need not be the lowest point on the entire surface. This tutorial uses nearby contours to describe local shape, not to make claims about the whole graph.
A strict local maximum has \(f(x,y)<f(a,b)\) at every other point in a sufficiently small neighborhood; a strict local minimum has \(f(x,y)>f(a,b)\). These strict cases are the clearest examples of a peak and a basin. For contour reading, the central question is: as the labeled height changes, are the nearby contours arranged around a point in a way consistent with all surrounding heights being lower, all being higher, or with heights occurring on both sides?
What Contour Labels Tell You
A contour labeled \(c\) is the level curve \(f(x,y)=c\). If a point has height \(c_0\), then a contour map with smaller labels as you move outward from a closed contour around that point suggests a peak at height \(c_0\). If the labels increase outward from the point, they suggest a basin. At a saddle, nearby values occur both above and below the point’s height, so the contour pattern does not form a simple sequence of enclosing levels that all lie on the same side of that height.
These are clues, not a substitute for checking the heights. A contour map usually shows only selected levels. The absence of a drawn contour between two labels does not mean there are no heights between them. Nor does the shape of a single contour determine whether its interior is higher or lower: the labels, or other height information, are needed to establish that.
Worked Example: A Circular Peak
Worked Example: A Circular Peak
Consider \(f(x,y)=12-x^2-y^2\). We will identify the height at the origin, find its contours, and check nearby values.
Step 1: Find the height at the candidate point. At \((0,0)\),
Step 2: Find the contour equation. Set \(f(x,y)=c\):
When \(c<12\), this is a circle centered at the origin with radius \(\sqrt{12-c}\). For \(c=12\), it reduces to the single point \((0,0)\). For \(c>12\), there are no points on that level, because \(x^2+y^2\) cannot be negative.
Step 3: Compare the labeled heights. The contours \(c=11\), \(c=8\), and \(c=3\) have radii \(\sqrt{1}=1\), \(\sqrt{4}=2\), and \(\sqrt{9}=3\), respectively. Moving outward from the origin, the labels decrease from \(12\) to \(11\), then \(8\), then \(3\).
Step 4: Check actual nearby points. At \((1,0)\), \(f(1,0)=12-1-0=11\). At \((0,1)\), \(f(0,1)=12-0-1=11\). At \((1,1)\), \(f(1,1)=12-1-1=10\). These are all below \(12\). More generally, if \((x,y)\neq(0,0)\), then \(x^2+y^2>0\), so \(f(x,y)=12-(x^2+y^2)<12\). Thus the origin is a strict local maximum—in fact, a global maximum. The formula for every nearby point confirms what the nested contours suggest.
Worked Example: An Elliptical Basin
Worked Example: An Elliptical Basin
Consider \(g(x,y)=2x^2+y^2\). The contours are not circles, but their heights still reveal the local shape.
Step 1: Evaluate the function at the origin.
Step 2: Write the contour equation. For a level \(c\),
Since both squared terms are nonnegative, this equation has no solutions if \(c<0\). If \(c=0\), both terms must be zero, giving only \((0,0)\). If \(c>0\), divide by \(c\) to express the contour as
This is an ellipse centered at the origin. For \(c=2\), its horizontal and vertical semiaxes are \(1\) and \(\sqrt{2}\). For \(c=8\), they are \(2\) and \(\sqrt{8}\). The ellipses expand as their height labels increase.
Step 3: Check points on and near the contours. At \((1,0)\), \(g(1,0)=2(1)^2+0^2=2\), so the point lies on the \(c=2\) contour. At \((0,\sqrt{2})\), \(g(0,\sqrt{2})=2\), as well. At \((1,1)\), \(g(1,1)=2+1=3\), which is greater than the origin’s height \(0\).
Step 4: Confirm the classification for all nearby points. If \((x,y)\neq(0,0)\), at least one of \(x^2\) and \(y^2\) is positive. Since both coefficients in \(2x^2+y^2\) are positive, \(2x^2+y^2>0=g(0,0)\). Therefore every other point, near or far, is higher than the origin. The origin is a strict local minimum, or basin.
Worked Example: A Saddle
Worked Example: A Saddle
Consider \(h(x,y)=x^2-y^2\). The origin has height \(0\), but nearby heights do not all lie above or all lie below it.
Step 1: Find the contours. Setting \(h(x,y)=c\) gives
The zero contour factors as \(x^2-y^2=(x-y)(x+y)=0\). It consists of the two lines \(y=x\) and \(y=-x\), crossing at the origin. For \(c=1\), the contour is \(x^2-y^2=1\); for \(c=-1\), it is \(x^2-y^2=-1\). The positive and negative levels occupy different orientations around the crossing.
Step 2: Check points in different directions. On the \(x\)-axis, \(h(x,0)=x^2\), which is positive for \(x\neq0\). For instance, \(h(1,0)=1\). On the \(y\)-axis, \(h(0,y)=-y^2\), which is negative for \(y\neq0\); for instance, \(h(0,1)=-1\). Thus even points the same distance from the origin can have values on opposite sides of the origin’s height.
Step 3: Show this happens in every neighborhood. Let \(r>0\) be any radius, and choose \(t=r/2\). Then \(0<t<r\). The points \((t,0)\) and \((0,t)\) both lie within distance \(r\) of the origin. Their values are
Because this works for every \(r>0\), every neighborhood contains points both higher and lower than the origin. Therefore the origin is saddle-like, not a local maximum or minimum.
Why the Algebra Justifies the Contour Reading
The examples illustrate a general reason the height comparisons work. Suppose a function has the form \(F(x,y)=M-(x-a)^2-(y-b)^2\). At \((a,b)\), its value is \(M\). For any other point, the sum of squares is positive, so \(F(x,y)<M\). Its contours satisfy \((x-a)^2+(y-b)^2=M-c\); as the radius increases, the level \(c\) decreases. The algebra proves both the peak classification and the outward ordering of the labels.
For \(G(x,y)=m+(x-a)^2+(y-b)^2\), the sum of squares is positive at every point other than \((a,b)\), so \(G(x,y)>m\). Its contours satisfy \((x-a)^2+(y-b)^2=c-m\); the level increases with the radius. This proves the basin classification and explains why the labels rise outward. In both cases, the proof depends on comparing actual function values, not merely on recognizing a circular pattern.
For \(H(x,y)=(x-a)^2-(y-b)^2\), choose any radius \(r>0\) and \(t=r/2\). The points \((a+t,b)\) and \((a,b+t)\) are both within that radius. Their function values are \(t^2>0\) and \(-t^2<0\), while \(H(a,b)=0\). This proves the saddle-like property in every neighborhood. The crossing zero contour is consistent with that behavior, but the two explicit height comparisons establish it.
A Common Mistake: Following Only One Direction
A common error is to inspect a surface along one line and use that behavior to classify the whole neighborhood. For \(h(x,y)=x^2-y^2\), the \(x\)-axis gives \(h(x,0)=x^2\), so heights rise from \(0\) as you move away from the origin. One might incorrectly call the origin a basin. But on the \(y\)-axis, \(h(0,y)=-y^2\), so the heights fall below \(0\). The origin is neither a basin nor a peak; the two directions together show a saddle.
Another mistake is to treat contour spacing as if it alone determines whether the surface rises or falls. Spacing can indicate how quickly height changes across the map, but the labels determine which direction is upward. Always read the numerical levels: contours labeled \(10,8,6\) in order away from a point indicate falling height, whereas labels \(2,4,6\) indicate rising height.
A Practical Reading Method
Read its height from the graph, formula, or labeled contour information.
Check whether nearby contours enclose the point and whether their labels increase or decrease as you move outward.
For a possible saddle, look for nearby points in different directions with heights on opposite sides of the candidate height.
Use the function values or the stated height information to confirm that the nearby comparisons hold. Do not infer more than the contours show.
Why These Patterns Matter
Recognizing peaks, basins, and saddles gives a useful geometric summary of a surface near a point. A peak is locally high, a basin is locally low, and a saddle has competing directions of rise and fall. Contour maps make these patterns visible without requiring a three-dimensional drawing. The same discipline applies in practical settings: read the height labels, compare multiple directions, and distinguish a visual suggestion from a conclusion supported by nearby values.
Check Your Understanding
Use contour labels and nearby height comparisons to answer these questions.
- A point is surrounded by closed contours whose labels decrease as you move outward. What local shape does this suggest, and what additional height comparison would confirm it?
- For \(q(x,y)=x^2+y^2\), what is the height at the origin, and are all other points higher or lower?
- For \(p(x,y)=y^2-x^2\), compare the values on the \(x\)-axis and \(y\)-axis near the origin. What does this show?
- Why is a single contour’s shape, without its height label or other height information, insufficient to identify a peak or basin?
- Can a surface rise away from a point along one direction and fall along another while the point is still a local maximum? Explain using the definition.