How Coordinate-Plane Slices Reveal a Surface
A surface in three-dimensional space can be difficult to picture from its equation alone. One useful way to inspect it is to look at the curves where it meets the three coordinate planes. Each such intersection is a planar cross-section, or trace, of the surface.
For an implicit surface, finding a coordinate-plane trace is often a direct calculation: the plane sets one coordinate equal to zero. The resulting equation describes the points of intersection within that plane. Interpreting all three traces together gives evidence about the surface’s shape, orientation, and extent.
Definition and Coordinate-Plane Conditions
Let \(F(x,y,z)\) be a function defined on a domain \(D\subseteq\mathbb{R}^3\), and fix a constant \(c\). The implicit surface is the level set of points in the domain satisfying \(F(x,y,z)=c\). The coordinate planes are the planes \(z=0\), \(y=0\), and \(x=0\).
The coordinate-plane condition matters. In the \(xy\)-plane, the trace is not every point whose coordinates happen to satisfy \(F(x,y,0)=c\) when evaluated using some other triple. It is the set of points whose third coordinate is actually zero and which also satisfy the surface equation.
Written explicitly, the three traces are described by:
These equations are planar equations: the \(xy\)-trace is drawn using \(x\) and \(y\) with \(z=0\), for instance. A trace may be an ellipse, parabola, pair of lines, a point, or the empty set. The result depends on the particular surface and the chosen level \(c\).
Why Setting a Coordinate to Zero Works
The method follows directly from what it means to belong to an intersection. Consider the \(xy\)-plane trace. A point belongs to it precisely when it belongs both to the surface and to the plane \(z=0\). Requiring membership in the plane first gives a point of the form \((x,y,0)\). Requiring that this point lie on the surface then gives \(F(x,y,0)=c\).
where \(S_c=\{(x,y,z)\in D:F(x,y,z)=c\}\) and \(P_{xy}=\{(x,y,z)\in\mathbb{R}^3:z=0\}\). This is a complete characterization: every point satisfying the right-hand conditions is in the surface and in the plane, and every point in the intersection has those conditions. The same reasoning applies to the other coordinate planes, with the appropriate coordinate set to zero.
In practice, the calculation is short, but the interpretation deserves care. After substituting zero, use the remaining equation to identify the curve within the correct plane. Then compare the three curves: their orientations and intersections with the axes provide clues about the whole surface.
Write its equation: \(z=0\), \(y=0\), or \(x=0\).
Keep the level equation \(F=c\) and replace only the coordinate fixed by the plane.
Use the remaining variables to recognize its shape, or note if it is only a point or is empty.
Use all three traces to guide the sketch, and check that points on each trace satisfy the original equation.
Worked Example: An Ellipsoid
Worked Example: An Ellipsoid
Find the three coordinate-plane traces of
Trace in the \(xy\)-plane: Set \(z=0\). The equation becomes
This is an ellipse with semiaxis lengths \(2\) along the \(x\)-axis and \(3\) along the \(y\)-axis. Its axis intercepts are \((2,0,0)\), \((-2,0,0)\), \((0,3,0)\), and \((0,-3,0)\).
Trace in the \(xz\)-plane: Set \(y=0\). Then
This ellipse has semiaxis lengths \(2\) along \(x\) and \(4\) along \(z\). Its intercepts are \((2,0,0)\), \((-2,0,0)\), \((0,0,4)\), and \((0,0,-4)\).
Trace in the \(yz\)-plane: Set \(x=0\). Then
This ellipse has semiaxis lengths \(3\) along \(y\) and \(4\) along \(z\), with intercepts \((0,3,0)\), \((0,-3,0)\), \((0,0,4)\), and \((0,0,-4)\).
Each trace checks against the original equation. For example, the \(xz\)-trace point \((0,0,4)\) gives \(0/4+0/9+16/16=1\); the \(xy\)-trace point \((2,0,0)\) gives \(4/4+0+0=1\); and the \(yz\)-trace point \((0,3,0)\) gives \(0+9/9+0=1\). Together, the traces indicate a closed ellipsoid, longer along the \(z\)-axis than along the \(y\)-axis, and longer along the \(y\)-axis than along the \(x\)-axis.
Worked Example: A Paraboloid with a Point Trace
Worked Example: A Paraboloid with a Point Trace
Find the coordinate-plane traces and use them to describe the surface
Trace in the \(xy\)-plane: Setting \(z=0\) gives
For real \(x\) and \(y\), both squares are nonnegative, so their sum is zero only when \(x=0\) and \(y=0\). The trace is the single point \((0,0,0)\), not a circle.
Trace in the \(xz\)-plane: Setting \(y=0\) gives
This is an upward-opening parabola in the \(xz\)-plane. Its vertex is \((0,0,0)\); for example, \(x=1\) gives \(z=1\), and \(x=-1\) also gives \(z=1\).
Trace in the \(yz\)-plane: Setting \(x=0\) gives
This is an upward-opening parabola in the \(yz\)-plane, also with vertex at the origin. At \(y=1\), \(z=1\), and at \(y=-1\), \(z=1\).
The trace equations satisfy the original equation: on the \(xz\)-trace, \(z=x^2\) and \(y=0\), so \(z-x^2-y^2=x^2-x^2-0=0\); on the \(yz\)-trace, the same expression is \(y^2-0-y^2=0\); and the origin also gives zero. These traces support a sketch of a bowl opening in the positive \(z\)-direction. The point trace in the \(xy\)-plane records that the surface touches that plane only at its vertex.
Worked Example: A Saddle
Worked Example: A Saddle
Find the three traces of the surface
Trace in the \(xy\)-plane: Setting \(z=0\) gives
The trace consists of two intersecting lines through the origin in the \(xy\)-plane.
Trace in the \(xz\)-plane: Setting \(y=0\) gives \(z=x^2\), an upward-opening parabola. For instance, \(x=2\) or \(x=-2\) gives \(z=4\).
Trace in the \(yz\)-plane: Setting \(x=0\) gives \(z=-y^2\), a downward-opening parabola. For instance, \(y=2\) or \(y=-2\) gives \(z=-4\).
Check the traces in the original equation. If \(z=0\) and \(y=x\), then \(x^2-y^2=x^2-x^2=0\); if \(y=-x\), then \(x^2-y^2=x^2-(-x)^2=0\). On the \(xz\)-trace, \(y=0\) and \(z=x^2\), so the equation holds; on the \(yz\)-trace, \(x=0\) and \(z=-y^2\), so it holds as well. The opposite curvature of the two parabolas, together with the crossing lines in the \(xy\)-plane, indicates a saddle shape.
Using Traces Carefully
Coordinate-plane traces are useful evidence, not a complete description of every point on a surface. A surface can curve in ways that are not fully determined by just its three coordinate-plane intersections. Use the traces to establish the surface’s behavior in those planes, then make a sketch consistent with all of them rather than treating one trace as the entire surface.
A common error is to substitute zero but then forget which points belong to the plane. For example, if \(F(x,y,z)=z\) and \(c=0\), the \(xy\)-trace is the set of points \((x,y,0)\) satisfying \(F(x,y,0)=0\). It is the whole \(xy\)-plane. But the statement \(F(x,y,0)=0\) does not say that an arbitrary point \((x,y,z)\) lies in that trace: for \((0,0,1)\), \(F(0,0,0)=0\), yet \((0,0,1)\) is not in the \(xy\)-plane. The correct point description retains the zero coordinate: \((x,y,0)\in S_0\cap P_{xy}\) precisely when \(F(x,y,0)=0\) and the point is in the domain.
Also, inspect the simplified planar equation before naming its shape. In the paraboloid example, \(x^2+y^2=0\) is not a circle of radius zero with multiple points; over the real numbers it has exactly one solution, the origin. In other cases, a substituted equation may have no real solutions, so the corresponding trace is empty.
Check Your Understanding
Use the coordinate-plane conditions and trace method to answer these questions.
- For \(x^2+y^2+z^2=9\), what equation describes the trace in the \(yz\)-plane?
- For \(z=x^2+y^2\), why is the \(xy\)-plane trace a single point rather than a circle?
- Find the \(xz\)-plane trace of \(x^2+z^2-y^2=1\).
- What coordinate must be set to zero to find a trace in the \(xz\)-plane?
- Why is the equation \(F(x,y,0)=c\) not, by itself, a condition that an arbitrary point \((x,y,z)\) lies in the \(xy\)-plane trace?