Why the Choice of Trace Matters
A trace is the curve formed where a surface meets a plane. A quadric may have many possible traces, but not all are equally useful. A coordinate plane can reveal symmetry or a principal direction; a parallel plane slightly away from the origin can show whether the surface widens, narrows, or changes shape. A good sketch uses a small number of informative traces that agree with one another.
The process is not to guess a three-dimensional picture from one equation. Instead, choose planes, find the corresponding curves, and use their locations and orientations to constrain the sketch. Each curve must be part of the same surface, so its points must satisfy both the plane equation and the original quadric equation.
A Practical Strategy for Selecting Planes
Begin with the coordinate planes \(x=0\), \(y=0\), and \(z=0\). These are useful because substitution often removes terms and leaves a recognizable curve. But a coordinate-plane trace can be a single point or a pair of lines, and that may not show the overall shape very well. If so, choose a parallel plane such as \(z=1\), \(x=2\), or \(y=-1\). The offset should be simple enough to keep the arithmetic clear and far enough from the origin to reveal a meaningful section.
Before sketching, record which plane contains each curve. For example, a trace in \(x=0\) lies in the \(yz\)-plane; it is not a curve in the \(xy\)-plane. Also record important points, such as vertices, intercepts, and where a trace crosses another selected plane. These facts help prevent the two-dimensional sections from being placed inconsistently.
Substitute one coordinate equal to zero and see whether the resulting curve is informative.
Use \(x=c\), \(y=c\), or \(z=c\) to see how the section changes away from the coordinate plane.
Keep track of the plane, orientation, and key points for every trace.
Substitute points from each trace into the original equation and compare any shared points.
Why Substitution Gives the Correct Trace
This method follows directly from the definition of an intersection. A point lies on the surface and in the plane \(x=c\) exactly when it satisfies the surface equation and has \(x\)-coordinate \(c\). Substituting \(c\) for \(x\) expresses both requirements in one equation for the remaining coordinates.
Worked Example: Traces of an Elliptic Paraboloid
Worked Example: Traces of an Elliptic Paraboloid
Consider the surface \(z=x^2+4y^2\). First take the coordinate-plane traces.
In the plane \(x=0\), substitution gives
This is a parabola in the \(yz\)-plane with vertex at \((0,0,0)\). In the plane \(y=0\),
a parabola in the \(xz\)-plane with the same vertex. In the plane \(z=0\),
Both terms on the right are nonnegative, so their sum is zero only if \(x=0\) and \(y=0\). The \(xy\)-plane trace is therefore just the origin.
The coordinate-plane cuts show the two upward-opening parabolas, but the \(z=0\) trace gives little information about the surface above the origin. Select the parallel plane \(z=1\):
This is an ellipse. Its intercepts are found by setting one of \(x,y\) to zero: for \(y=0\), \(x^2=1\), so \(x=\pm1\); for \(x=0\), \(4y^2=1\), so \(y=\pm\frac12\). Each intercept checks in the equation: \(1^2+4(0)^2=1\), and \(0^2+4(\frac12)^2=1\).
A second parallel section, \(z=4\), gives
Its intercepts are \(x=\pm2\) when \(y=0\), and \(y=\pm1\) when \(x=0\). Substitution checks them: \(2^2+4(0)^2=4\), and \(0^2+4(1)^2=4\). The higher section is wider than the \(z=1\) section. Together, the two coordinate-plane parabolas and the widening ellipses support a sketch that rises from the origin and expands as \(z\) increases.
Check against the original equation: the point \((1,0,1)\) lies on the \(z=1\) trace and satisfies \(1=1^2+4(0)^2\). The point \((0,1,4)\) lies on the \(z=4\) trace and satisfies \(4=0^2+4(1)^2\).
Choose Offset Planes to Reveal Orientation
A coordinate-plane trace can show the direction in which a surface bends, while an offset parallel trace can show how that bend sits elsewhere on the surface. The two pieces of information work together: the coordinate plane anchors the sketch, and the parallel plane tests whether the proposed shape continues in a compatible way.
Worked Example: Selecting Traces for a Saddle
Worked Example: Selecting Traces for a Saddle
Consider \(z=x^2-y^2\). In the plane \(y=0\), the trace is
an upward-opening parabola in the \(xz\)-plane. In the plane \(x=0\), it is
a downward-opening parabola in the \(yz\)-plane. The two directions bend oppositely, an important feature to carry into the sketch. In the plane \(z=0\),
so the trace consists of the lines \(y=x\) and \(y=-x\) in the \(xy\)-plane.
Now select the two parallel planes \(y=1\) and \(y=-1\). For \(y=1\),
This is an upward-opening parabola in the plane \(y=1\), with vertex \((0,1,-1)\). At \(x=\pm1\), it has \(z=0\); for example, \(0=1^2-1\). For \(y=-1\), the equation is also \(z=x^2-1\), because \((-1)^2=1\). Its vertex is \((0,-1,-1)\), and its points with \(x=\pm1\) have \(z=0\) as well.
As a second orientation check, take \(x=1\). Then \(z=1-y^2\), a downward-opening parabola in the plane \(x=1\), with vertex \((1,0,1)\). Its points at \(y=\pm1\) have \(z=0\), since \(1-(\pm1)^2=0\).
Check against the original equation: \((0,1,-1)\) gives \(0^2-1^2=-1\), and \((1,0,1)\) gives \(1^2-0^2=1\). The opposite bending seen in the coordinate planes is consistent with the offset sections: moving along \(y\) lowers the vertex of the \(x\)-parabola, while the section at \(x=1\) bends downward as \(y\) varies.
Use a Parallel Section When a Coordinate Trace Collapses
Some coordinate-plane sections reduce to a point or a few lines. That result is valid, but it may conceal the width or orientation of the nearby surface. A parallel-plane trace can turn the same equation into a more revealing curve. In a sketch, do not replace a collapsed coordinate trace with an imagined curve; instead, keep the point or lines and use an offset section to show the surface around them.
Worked Example: Two Nappes from Cone Traces
Worked Example: Two Nappes from Cone Traces
Consider \(z^2=x^2+y^2\). In the plane \(x=0\), the trace satisfies \(z^2=y^2\), so \((z-y)(z+y)=0\) and \(z=y\) or \(z=-y\). These are two lines in the \(yz\)-plane. In the plane \(y=0\), similarly, \(z^2=x^2\), so \(z=x\) or \(z=-x\), two lines in the \(xz\)-plane.
In the plane \(z=0\), the equation becomes
Since squares are nonnegative, this holds only when \(x=y=0\). Thus this coordinate trace is the vertex alone. To see a full curve, choose the parallel plane \(z=2\):
This is a circle of radius \(2\) in the plane \(z=2\). Its points \((2,0,2)\), \((-2,0,2)\), \((0,2,2)\), and \((0,-2,2)\) each satisfy the original equation: in every case \(z^2=4=x^2+y^2\). The parallel plane \(z=-2\) gives the same circle in a different location, because \((-2)^2=4\). The two circular sections and the lines in the coordinate planes fit together around the vertex to show both the orientation and the two directions in which the surface extends.
Check against the original equation: the vertex \((0,0,0)\) satisfies \(0^2=0^2+0^2\), and the point \((0,2,2)\) satisfies \(2^2=0^2+2^2\). Each selected trace therefore consists of points on the original surface.
A Common Mistake: Treating a Trace as a Projection
A trace is an intersection, not a shadow. For example, when finding the trace of \(z^2=x^2+y^2\) in \(z=0\), it is incorrect to describe the result as the entire \(xy\)-plane because \(z\) has been set to zero. The surface equation still has to hold. The correct substitution gives \(x^2+y^2=0\), which forces \(x=0\) and \(y=0\), so the trace is only the origin.
The reverse error is also possible: finding a valid curve in a plane and forgetting to check that its points satisfy the original surface equation. A curve is a trace only if both conditions hold. The safest habit is to state the plane, substitute into the original equation, and test any points used to guide the sketch.
Putting the Traces Together
Each trace constrains the three-dimensional picture in a different way. Coordinate-plane curves show how the surface meets the principal planes. Parallel-plane curves reveal what happens as one coordinate changes while the other two vary. They should not contradict one another: shared points must agree, and curves in different planes must be placed in those planes rather than treated as if they occupied one common sheet of paper.
When a trace is too simple to be informative, that is a reason to select another plane, not a reason to ignore the calculation. A point can locate a vertex; a pair of lines can indicate directions through it; and an offset ellipse, parabola, or circle can show the shape surrounding it. Together, carefully chosen sections provide a disciplined basis for a consistent sketch.
Check Your Understanding
For each question, identify the plane before interpreting the resulting curve.
- For \(z=x^2+4y^2\), what equation describes the trace in \(x=0\), and in which coordinate plane does it lie?
- For the same surface, what curve results in \(z=1\), and what are its two positive-axis intercepts?
- For \(z=x^2-y^2\), what is the trace in \(z=0\), and how can its equation be factored?
- For \(z^2=x^2+y^2\), why is the trace in \(z=0\) not the entire \(xy\)-plane?
- What two conditions must a point satisfy in order to belong to a trace in the plane \(y=c\)?