Why the Signs Reveal the Shape
A standard quadric equation can look like a short list of squared terms, but the signs of those terms contain geometric information. They tell us which coordinate-plane traces are ellipses, which are hyperbolas, and whether a coordinate-plane trace is only a point or has no real points at all.
Here we consider centered, axis-aligned standard equations with positive denominators. The goal is to diagnose the surface from its signs and its traces in the coordinate planes \(xy\), \(xz\), and \(yz\). A trace is found by restricting the equation to a plane: for example, the \(xy\)-plane has \(z=0\).
The Standard Sign Patterns
In the equations below, \(a\), \(b\), and \(c\) are positive numbers. The coordinate associated with the lone sign in a hyperboloid or cone helps identify its axis. The letters can be permuted if the roles of the coordinates are different.
| Surface | Standard form | Sign pattern |
|---|---|---|
| Ellipsoid | \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\) | Three positive terms; right side \(1\) |
| Elliptic cone | \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0\) | Two terms of one sign, one of the other; right side \(0\) |
| Hyperboloid of one sheet | \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\) | Two positive terms, one negative; right side \(1\) |
| Hyperboloid of two sheets | \(\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\) | One positive term, two negative; right side \(1\) |
For an equation with right side \(-1\), multiplying the entire equation by \(-1\) may put it into a standard form with right side \(1\). Make that comparison before counting signs. In the cone form, the right side is \(0\), so the significant information is the balance of squared terms and their signs.
How to Read a Coordinate-Plane Trace
Setting a coordinate to zero removes its squared term. What remains determines the trace. A sum of two positive squared terms equal to a positive constant gives an ellipse. A difference of two squared terms equal to a positive constant gives a hyperbola. If a sum of squares equals zero, both squares must be zero, so the trace is a single point. If a sum of negative squares is required to equal \(1\), there are no real points.
For a cone, a difference of two squares equal to zero factors into a pair of lines. For instance, \(u^2-v^2=0\) means \((u-v)(u+v)=0\), so the trace consists of the two lines \(u=v\) and \(u=-v\). That pair of lines meets at the cone’s vertex.
Worked Example: Identifying an Ellipsoid
Worked Example: Identifying an Ellipsoid
Classify the surface and find its three coordinate-plane traces:
All three squared terms have positive signs, and the right side is \(1\). This is the ellipsoid sign pattern.
For the \(xy\)-trace, set \(z=0\):
This is an ellipse. For the \(xz\)-trace, set \(y=0\):
This is also an ellipse. For the \(yz\)-trace, set \(x=0\):
This is the third ellipse. Check that the traces lie on the original surface: the point \((2,0,0)\) lies on the \(xy\)-trace, and substitution into the original equation gives \(2^2/4+0+0=1\). The point \((0,0,4)\) lies on the \(xz\)-trace and gives \(0+0+4^2/16=1\). Thus both trace calculations agree with the surface equation.
A Sign-and-Trace Result
- An ellipsoid has an ellipse in each coordinate plane.
- An elliptic cone has a single point in the plane perpendicular to its axis and a pair of intersecting lines in each of the other two coordinate planes.
- A hyperboloid of one sheet has an ellipse in the plane perpendicular to its axis and a hyperbola in each of the other two coordinate planes.
- A hyperboloid of two sheets has no points in the plane perpendicular to its axis and a hyperbola in each of the other two coordinate planes.
The coordinate perpendicular to the axis is the coordinate whose term is the lone-sign term for the cone and hyperboloids. For the displayed forms, that coordinate is \(z\) for the cone and one-sheet hyperboloid, and \(x\) for the two-sheet hyperboloid.
Why the Trace Test Works
The trace test follows directly from setting the appropriate coordinate equal to zero. Consider the one-sheet form with its lone negative term in \(z\). In the plane \(z=0\), the negative term disappears, leaving a sum of positive squares equal to \(1\), an ellipse. In the plane \(x=0\), one positive and one negative squared term remain, giving a hyperbola. Setting \(y=0\) produces the other hyperbola.
For a two-sheet hyperboloid, the one positive term is the \(x\)-term. Setting \(x=0\) leaves a negative sum equal to \(1\), which cannot happen for real \(y\) and \(z\). Setting either of the other coordinates to zero leaves a positive-minus-negative equation equal to \(1\), a hyperbola. The same substitutions establish the cone and ellipsoid trace patterns.
For a cone or hyperboloid, identify which squared term has the sign different from the other terms.
For a cone, the remaining sum of squares equals zero and gives only the vertex. For a one-sheet hyperboloid, it gives an ellipse. For a two-sheet hyperboloid, it gives an impossible negative sum equal to \(1\).
These traces retain one positive and one negative squared term for the cone and hyperboloids; the resulting equation is a line pair for the cone and a hyperbola for either hyperboloid.
Worked Example: Recognizing an Elliptic Cone
Worked Example: Recognizing an Elliptic Cone
Classify the surface and determine its coordinate-plane traces:
The right side is \(0\), and there are two positive terms and one negative term. This is an elliptic cone, with its axis along the \(z\)-coordinate direction.
In the plane \(z=0\), the trace equation is
Both terms are nonnegative, so their sum is zero only when \(x=0\) and \(y=0\). The trace is the single point \((0,0,0)\).
In the plane \(x=0\), the equation becomes
Thus the trace consists of the two lines \(z=\frac{4}{3}y\) and \(z=-\frac{4}{3}y\). In the plane \(y=0\),
so the lines are \(z=2x\) and \(z=-2x\). Check the first line pair using the point \((0,3,4)\): the original left side is \(0+9/9-16/16=0\). Check the second using \((2,0,4)\): it is \(4/4+0-16/16=0\). Both points satisfy the original cone equation.
Worked Example: Distinguishing the Hyperboloids
Worked Example: Distinguishing the Hyperboloids
Classify the surface and calculate its coordinate-plane traces:
There is one positive term, two negative terms, and the right side is \(1\). This is a hyperboloid of two sheets, with its axis along the \(x\)-coordinate direction.
For the \(yz\)-trace, set \(x=0\):
The left side is at most zero, so it cannot equal \(1\). The trace is empty. For the \(xy\)-trace, set \(z=0\), giving \(\frac{x^2}{4}-\frac{y^2}{9}=1\), a hyperbola. For the \(xz\)-trace, set \(y=0\), giving \(\frac{x^2}{4}-\frac{z^2}{16}=1\), also a hyperbola.
As a check, the point \((2,0,0)\) belongs to both hyperbola traces and gives \(2^2/4-0-0=1\) in the original equation. The point \((0,0,0)\), which lies in the \(yz\)-plane, gives \(0\) on the left, not \(1\); more generally, no point in that plane can satisfy the equation because both remaining terms are nonpositive.
Compare this with a one-sheet form, \(\frac{x^2}{9}+\frac{y^2}{4}-\frac{z^2}{16}=1\). Its \(xy\)-trace at \(z=0\) is the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\), while setting \(x=0\) or \(y=0\) gives a hyperbola. The ellipse in the plane perpendicular to the lone-sign coordinate is the key distinction.
A Common Mistake
A frequent error is to classify a hyperboloid by counting signs while ignoring which side of the equation equals \(1\), or to confuse the lone positive and lone negative cases. For example, the equation \(\frac{x^2}{4}-\frac{y^2}{9}-\frac{z^2}{16}=1\) has one positive term, not two. Calling it a one-sheet hyperboloid because it has both positive and negative signs is not enough: both hyperboloid types have mixed signs.
The correction is to count the signs after putting the equation in standard form, then confirm the classification with the trace perpendicular to the lone-sign coordinate. Here that is \(x=0\), which gives \(-\frac{y^2}{9}-\frac{z^2}{16}=1\), an empty trace. That is the two-sheet pattern, not the one-sheet pattern.
Why Coordinate Traces Matter
Trace shapes provide a practical check on a sign-based classification. A calculation that predicts an ellipse in the wrong coordinate plane may signal that a sign was copied incorrectly or that the wrong coordinate was set to zero. In particular, comparing the plane perpendicular to the axis distinguishes the two hyperboloids: it contains an ellipse for one sheet and no real points for two sheets.
This test applies to the centered standard forms stated here. It does not by itself classify every possible quadric equation. For these forms, however, the method is direct: read the signs, set coordinates to zero, and verify that each trace has the shape the signs predict.
Check Your Understanding
Use the sign pattern and coordinate-plane traces to answer the following questions.
- What sign pattern and right side identify an ellipsoid in standard form?
- For \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0\), what is the \(xy\)-trace, and why is it only a point?
- Which coordinate-plane trace distinguishes a one-sheet hyperboloid from a two-sheet hyperboloid when their lone-sign term is in \(z\)?
- What kind of trace results from setting one of the repeated-sign coordinates to zero in a standard elliptic cone equation?
- For \(\frac{x^2}{9}-\frac{y^2}{4}-\frac{z^2}{25}=1\), which coordinate plane has an empty trace?