Reading a Surface One Slice at a Time
A surface in three-dimensional space can be difficult to understand from its equation alone. A useful approach is to examine its intersections with carefully chosen planes. Each intersection gives a two-dimensional slice of the surface, which can often be recognized and sketched more easily than the full three-dimensional object.
For example, fixing the \(x\)-coordinate means looking only at points in a plane of the form \(x=a\), where \(a\) is a chosen constant. The remaining coordinates, \(y\) and \(z\), can vary. The resulting intersection can reveal how the surface looks at that particular location. Repeating this at several values of \(a\) gives a more complete picture.
Definition: A Trace at a Fixed Coordinate
Let a surface \(S\) be a set of points \((x,y,z)\) in three-dimensional space. Fixing \(x=a\) means intersecting \(S\) with the plane \(x=a\). The result is called the trace of \(S\) in the plane \(x=a\). The same construction applies when fixing \(y=b\) or \(z=c\).
The plane \(x=a\) is parallel to the \(yz\)-plane, which is \(x=0\). It is the \(yz\)-plane itself only when \(a=0\). Similarly, the plane \(y=b\) is parallel to the \(xz\)-plane and equals it only when \(b=0\); the plane \(z=c\) is parallel to the \(xy\)-plane and equals it only when \(c=0\).
When \(x=a\) is fixed, \(y\) and \(z\) are the coordinates that can vary. When \(y=b\) is fixed, \(x\) and \(z\) can vary; when \(z=c\) is fixed, \(x\) and \(y\) can vary. These facts tell us which two-dimensional plane contains each trace.
Why Substitution Gives the Trace
Suppose a surface is specified by an equation \(F(x,y,z)=0\). A point belongs to its trace at \(x=a\) exactly when two conditions hold: the point lies on the surface, and its \(x\)-coordinate equals \(a\). Thus it must satisfy \(F(x,y,z)=0\) and \(x=a\). Substituting \(a\) for \(x\) expresses the first condition using only the coordinates that vary in the plane.
This rule is a direct consequence of the definition, not a separate geometric assumption. For example, the set described by \(F(a,y,z)=0\) in the plane \(x=a\) consists precisely of points \((a,y,z)\) for which \(F(a,y,z)=0\). Those are exactly the points in both the original surface and the plane \(x=a\). If there are no such points, the trace is empty; if every point of that plane satisfies the equation, the intersection is the whole plane.
Worked Example: A Sphere Cut by a Parallel Plane
Worked Example: A Sphere Cut by a Parallel Plane
Find the trace of \(x^2+y^2+z^2=25\) in the plane \(x=3\), and interpret the result.
Step 1: Fix the coordinate. The plane is \(x=3\), so set \(x=3\) in the surface equation:
Step 2: Simplify. Since \(3^2=9\), subtract \(9\) from both sides:
This is the equation of the trace in the plane \(x=3\), which is parallel to the \(yz\)-plane. In the \(yz\)-coordinates, it is a circle centered at \((y,z)=(0,0)\) with radius \(4\).
Check: Any point on this trace has \(x=3\) and \(y^2+z^2=16\). Substitution into the original equation gives \(x^2+y^2+z^2=9+16=25\), so every such point lies on the sphere. Conversely, any point on the sphere with \(x=3\) must satisfy \(y^2+z^2=25-9=16\). Thus the equation describes exactly the intersection.
The plane label matters: \(y^2+z^2=16\) describes the shape of the trace using the two varying coordinates, but the trace itself lies in the plane \(x=3\), not in the \(yz\)-plane \(x=0\).
Some Traces Are Not Curves
Substitution also makes it clear why a trace need not be a curve. Its form depends on both the surface and the chosen plane. A fixed coordinate may produce a familiar curve, a single point, no points at all, or an entire plane.
Worked Example: A Trace That Is a Circle, a Point, or Empty
Worked Example: A Trace That Is a Circle, a Point, or Empty
For the surface \(z=x^2+y^2\), find the traces in the planes \(z=4\), \(z=0\), and \(z=-1\).
At \(z=4\): Substitute \(4\) for \(z\):
This is a circle of radius \(2\) in the plane \(z=4\), which is parallel to the \(xy\)-plane.
At \(z=0\): Substitute \(0\) for \(z\):
Both squares are nonnegative, so their sum is zero only if \(x=0\) and \(y=0\). The trace is therefore the single point \((0,0,0)\).
At \(z=-1\): Substitution gives
For real \(x\) and \(y\), \(x^2+y^2\geq 0\), so this equation has no solutions. The trace is empty.
Check: In each case, points in the proposed trace satisfy both the chosen plane equation and \(z=x^2+y^2\). For \(z=4\), any point on \(x^2+y^2=4\) gives \(z=4\); for \(z=0\), the only possible point is the origin; for \(z=-1\), nonnegativity rules out any point. These cases confirm that the intersection can change type as the fixed coordinate changes.
Using Several Cross-Sections
A single trace shows only one slice. Comparing several traces can show how a surface changes as the fixed coordinate moves. The resulting collection is useful for building a spatial description, but each equation still refers to a particular plane. Keep track of the fixed coordinate and the plane label as you compare the slices.
For example, choosing \(z=c\) means taking a horizontal plane parallel to the \(xy\)-plane.
Replace \(z\) by \(c\) in the equation for the surface.
Determine whether it is a curve, a point, empty, or a higher-dimensional intersection, and state the plane in which it lies.
Compare the traces to see how their size, position, or existence changes with the fixed coordinate.
Worked Example: Comparing Three Horizontal Traces
Worked Example: Comparing Three Horizontal Traces
For the surface \(\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1\), find and compare the traces at \(z=0\), \(z=2\), and \(z=4\).
At \(z=0\): Substitute \(z=0\). The \(z\)-term becomes \(0^2/16=0\), so
This is an ellipse in the \(xy\)-plane. Its axis intercepts are \(x=\pm2\) when \(y=0\), and \(y=\pm3\) when \(x=0\).
At \(z=2\): Substitute \(z=2\), so \(z^2/16=4/16=1/4\):
Dividing the equation by \(3/4\) gives
This is an ellipse in the plane \(z=2\). Its intercepts are \(x=\pm\sqrt{3}\) when \(y=0\), and \(y=\pm\sqrt{27/4}=\pm 3\sqrt{3}/2\) when \(x=0\).
At \(z=4\): Substitute \(z=4\), so \(z^2/16=16/16=1\):
Each term on the left is nonnegative, so both must be zero: \(x=0\) and \(y=0\). The trace is the single point \((0,0,4)\) in the plane \(z=4\).
Check: At \(z=0\), the original equation reduces to \(x^2/4+y^2/9=1\), as stated. At \(z=2\), the trace equation gives \(x^2/4+y^2/9=3/4\), and adding \(z^2/16=1/4\) restores \(1\). At \(z=4\), the only trace point has \(x=y=0\), and \(0+0+16/16=1\). The three traces therefore satisfy the original surface equation, including the limiting point at \(z=4\).
The first two slices are ellipses of different sizes, while the third consists of one point. Considering these intersections together conveys more than any one slice by itself. The same process works for other coordinate directions: fix \(x\) or \(y\), substitute, and interpret the remaining equation in its corresponding plane.
A Common Substitution Mistake
A frequent error is to set a coordinate to its chosen value but forget to account for its contribution in the equation. For the sphere \(x^2+y^2+z^2=25\), the incorrect calculation for the trace at \(x=3\) might be:
This drops the term \(x^2=9\). The correction is to substitute first, then simplify:
A second, related error is to call the plane \(x=3\) the \(yz\)-plane. The \(yz\)-plane is \(x=0\); \(x=3\) is a parallel plane. A correct trace description names both the resulting equation and its actual plane.
Why Traces Matter
Cross-sections provide a disciplined way to interpret equations in three variables. They reduce a spatial question to a two-variable equation while preserving the geometric location of the slice. Comparing traces at several values can reveal whether sections appear, shrink, grow, or disappear. Traces are not, by themselves, a complete description of every point of a surface, but they are an effective way to gather and organize geometric evidence.
Check Your Understanding
Answer these questions using the definition of a trace and the substitution rule.
- In a trace at \(y=b\), which coordinates can vary, and in what plane does the trace lie? How is that plane oriented relative to the \(xz\)-plane?
- For \(x^2+y^2+z^2=9\), find the trace at \(z=0\). State the plane and describe the resulting curve.
- For \(z=x^2+y^2\), why is the trace at \(z=-2\) empty?
- Can the intersection of a surface with a plane be a single point? Give a trace from this tutorial that demonstrates this.
- Why must the equation \(y^2+z^2=16\) at \(x=3\) be described as lying in a plane parallel to the \(yz\)-plane rather than in the \(yz\)-plane itself?