Why Sketches Go Wrong
A sketch of a function of two variables can show a three-dimensional graph, curves in the input plane, or intersections of the graph with planes. These objects are related, but they are not interchangeable. A common error is to draw the right-looking shape in the wrong space or to read a slice as if it were a different kind of slice.
For a function \(f:A\to\mathbb{R}\), where \(A\) is a subset of the \(xy\)-plane, its graph is the set of points \((x,y,z)\) satisfying \(z=f(x,y)\). A level set at height \(c\) is the set of input points \((x,y)\) satisfying \(f(x,y)=c\). A contour sketch displays such level sets in the \(xy\)-plane, while a surface sketch displays graph points in three-dimensional space.
Error 1: Treating a Level Curve as the Surface
A contour such as \(x^2+y^2=4\) is an equation in the input coordinates. By itself, it describes a circle in the \(xy\)-plane, not the graph of a function of two variables. If it is a level set for \(f(x,y)=x^2+y^2\), the corresponding graph points also have height \(z=4\). The circle is then represented on the surface by the points \((x,y,4)\) satisfying \(x^2+y^2=4\).
Worked Example: Separating a Surface from Its Contours
Consider \(f(x,y)=x^2+y^2\). Its graph is \(z=x^2+y^2\). At level \(c=4\), solve the level-set equation:
This is a circle of radius \(2\) in the \(xy\)-plane. For instance, \((2,0)\), \((0,2)\), and \((-2,0)\) satisfy the equation. Lift each point to the graph by using \(z=f(x,y)\): the graph points are \((2,0,4)\), \((0,2,4)\), and \((-2,0,4)\). In fact, every point on the circle lifts to a point with height \(4\), so the lifted intersection is \(\{(x,y,4):x^2+y^2=4\}\).
Check: at \((2,0,4)\), \(x^2+y^2=2^2+0^2=4=z\), as required. The circle in the input plane and its lifted version on the graph are related, but they are not the entire surface.
Another error is to call every level set a curve. A level set can be empty, a single point, a curve, several curves, or even a two-dimensional region. Its shape depends on both the function and the chosen level. The word “contour” is useful when the level set is being represented as a curve, but the equation \(f(x,y)=c\) defines the set regardless of its shape.
Error 2: Misreading Horizontal Intersections
A horizontal plane has equation \(z=c\). Its intersection with the graph of \(f\) consists precisely of graph points whose heights equal \(c\). Thus the intersection is \((x,y,c)\) for those input points satisfying \(f(x,y)=c\). It is a curve when the corresponding level set is a curve, but it need not always be one.
Worked Example: The Shape of a Horizontal Slice Can Vary
First let \(f(x,y)=x^2+y^2\) and take \(c=0\). The level equation is \(x^2+y^2=0\). Since both squares are nonnegative, their sum is zero only if \(x=0\) and \(y=0\). The horizontal plane \(z=0\) therefore meets this graph at the single point \((0,0,0)\).
For \(c=1\), the level equation is \(x^2+y^2=1\), a circle. The horizontal intersection is the circle lifted to height \(1\), namely \(\{(x,y,1):x^2+y^2=1\}\). For example, \((1,0,1)\) lies on the graph because \(1^2+0^2=1\).
Now let \(g(x,y)=0\) for every \((x,y)\), and take \(c=0\). Every input point satisfies \(g(x,y)=0\), so the graph is the whole plane \(z=0\). The horizontal intersection is a plane, not a curve. These cases show why a sketch should describe the actual level set rather than assume its shape.
Error 3: Confusing Vertical and Horizontal Traces
A vertical trace is obtained by intersecting a surface with a vertical plane, such as \(x=a\) or \(y=b\). The resulting points vary in one input coordinate and in height. A horizontal trace is obtained by fixing the height, \(z=c\), and then finding the input points whose function values equal \(c\). The two procedures fix different coordinates and answer different questions.
Worked Example: Reading the Slices of a Saddle
Consider \(f(x,y)=x^2-y^2\), so the graph is \(z=x^2-y^2\). In the vertical plane \(y=0\), substitute \(y=0\):
This vertical trace is an upward-opening parabola in the plane \(y=0\). In the vertical plane \(x=0\), substitute \(x=0\):
This trace is a downward-opening parabola in the plane \(x=0\). Both traces pass through \((0,0,0)\), as the surface equation requires.
For the horizontal plane \(z=1\), instead solve \(x^2-y^2=1\). The resulting input set is a hyperbola, and its points lift to height \(1\). For \(z=0\), solve \(x^2-y^2=0\), which factors as \((x-y)(x+y)=0\). Thus the level set consists of the lines \(y=x\) and \(y=-x\), and the horizontal intersection is those lines at height zero. Check a point on each: \((1,1,0)\) gives \(1-1=0\), and \((1,-1,0)\) also gives \(1-1=0\).
A frequent sketching mistake is to use a horizontal contour as though it were a vertical cross-section. A contour tells which input points have one specified height. A vertical trace tells how the height changes while one input coordinate is held fixed. Label the plane of the slice, or explicitly state the coordinate being fixed, to avoid swapping these interpretations.
Error 4: Omitting Contour Labels
A collection of nested curves is not a complete contour map unless the heights are identified. The curves may suggest whether a surface rises or falls, but without level values the reader cannot tell which curve corresponds to which height or how the levels are spaced. Labels such as \(c=0\), \(c=1\), and \(c=4\) state the defining equation for each contour: \(f(x,y)=c\).
For \(f(x,y)=x^2+y^2\), the contour \(x^2+y^2=1\) has height \(1\), while \(x^2+y^2=4\) has height \(4\). These labels are not radii: the corresponding radii are \(1\) and \(2\). Confusing the level value with the radius changes the interpretation of the map. A clear sketch should label contours with their function values and use a consistent convention for whether the label is written on the curve or in a legend.
Error 5: Drawing Slices That Cannot Belong to One Graph
Different slices of a proposed surface must agree wherever their planes intersect. This requirement follows directly from the definition of a graph: at any given input \((x,y)\), a function assigns only one height. If two stated slices assign different heights to the same input point, they cannot both be traces of the same function.
Worked Example: Detecting Incompatible Vertical Traces
Suppose a sketch claims that in the plane \(y=0\), the trace is \(z=x^2\), and in the plane \(y=x\), the trace is \(z=x^2+1\). The planes intersect where \(y=0\) and \(y=x\), so \(x=0\) and \(y=0\). Their shared vertical line contains the point \((0,0,z)\).
The first proposed trace assigns \(z=0^2=0\) at that input. The second assigns \(z=0^2+1=1\). Thus the same input \((0,0)\) would have to produce both \(f(0,0)=0\) and \(f(0,0)=1\), which is impossible for a function.
Check by substitution: the first claim requires the graph point \((0,0,0)\), while the second requires \((0,0,1)\). A graph of one function cannot contain both points, because their first two coordinates agree but their heights differ. Therefore at least one proposed slice must be corrected.
Why the Checks Work
The checks above are consequences of the definitions, not merely visual preferences. Here is the argument that connects a horizontal slice to a level set. A point on the graph has the form \((x,y,f(x,y))\). It lies in the horizontal plane \(z=c\) exactly when its third coordinate equals \(c\). Therefore it lies in the intersection exactly when \(f(x,y)=c\). This gives the set equality below.
This argument identifies the intersection without asserting that it must be a curve. Its shape is determined by the level set. The same definition explains the compatibility test: if two proposed traces contain graph points with the same \((x,y)\) but different \(z\)-coordinates, then the rule \(z=f(x,y)\) would assign two values to one input, contradicting the definition of a function.
A Practical Error-Checking Routine
Before accepting a surface or contour sketch, check what each drawn object represents and test the equations behind it. A disciplined review can catch both geometric and labeling errors:
State whether the drawing is a graph in three dimensions, a level set in the input plane, or an intersection with a specified plane.
For a vertical trace, specify a fixed input coordinate such as \(x=a\). For a horizontal trace, specify \(z=c\) and solve \(f(x,y)=c\).
Write the level value \(c\) for each contour, and distinguish that value from geometric quantities such as a circle’s radius.
Where proposed slices overlap, verify that they assign the same height to each shared input.
For any claimed point \((x,y,z)\) on the graph, verify \(z=f(x,y)\). For a claimed contour point, verify \(f(x,y)=c\).
These checks matter because surface sketches are often assembled from partial information. A contour map may show horizontal levels, while vertical traces show behavior in selected planes. Each view can be useful, but they must describe the same function and must be labeled so the reader can tell which information is being shown.
Check Your Understanding
For each question, identify the relevant set or slice before deciding what the sketch should show.
- For \(f(x,y)=x^2+y^2\), what does the equation \(x^2+y^2=9\) describe, and what are the coordinates of its points when lifted to the graph?
- Can the intersection of a graph with a horizontal plane be a single point or an entire plane? Give the function and height for each case.
- For \(z=x^2-y^2\), what equations describe the vertical traces in \(x=0\) and \(y=0\)?
- Why must contour curves carry level labels to identify their function values unambiguously?
- Two proposed vertical traces assign different heights at a shared input point. What does this imply about the proposed surface?