One Equation, Repeated Through Space
A surface equation in three variables may use only two of the coordinates. When that happens, the coordinate that does not appear is free to vary: changing it cannot change whether a point satisfies the equation. Geometrically, the same planar curve is therefore repeated throughout space.
This gives a useful way to recognize the direction in which a surface extends. If an equation omits \(z\), for example, its curve in the \(xy\)-plane extends parallel to the \(z\)-axis. The resulting surface is a cylinder in the broad sense that it is made of lines, called generators, parallel to one fixed direction. The curve need not be a circle.
Definition and Geometric Meaning
Let \(F\) be a function of two variables. The planar curve described by \(F(x,y)=0\) is the set of all pairs \((x,y)\) satisfying that equation. The corresponding three-dimensional surface has equation \(F(x,y)=0\), interpreted as an equation for points \((x,y,z)\) in space.
Here “base curve” identifies the planar curve used to describe the surface; it does not mean that the surface has a bottom or a preferred starting height. The extension runs in both directions along the axis. The same idea applies if \(x\) or \(y\), rather than \(z\), is missing: the surface extends parallel to the corresponding coordinate axis.
Why the Missing Coordinate Produces Lines
The conclusion follows directly from what it means for a point to satisfy the equation. Suppose \(F(a,b)=0\), so \((a,b)\) is on the planar curve. For any real number \(t\), the point \((a,b,t)\) satisfies the surface equation, because the equation evaluates only \(F(a,b)\). Thus the surface contains the entire line through \((a,b,0)\) parallel to the \(z\)-axis.
Conversely, if \((x,y,z)\) is on the surface, then \(F(x,y)=0\), so its first two coordinates form a point on the planar curve. These two directions show that the surface contains exactly the full lines through points of that curve, not just some of those lines or a portion of each one.
The set notation expresses the entire geometric argument: choose any point on the curve, then choose any real height. Alternatively, fix any height \(z=c\). The cross-section in the plane \(z=c\) is always the original curve \(F(x,y)=0\). It does not change as \(c\) changes.
Worked Examples
Worked Example: A Curve Extended Parallel to the \(z\)-Axis
Describe the surface \(x^2+y^2=25\) and verify that the point \((3,4,-2)\) lies on it.
The equation involves \(x\) and \(y\) but not \(z\). Its planar curve is \(x^2+y^2=25\) in the \(xy\)-plane. To check the proposed point, substitute its \(x\)- and \(y\)-coordinates:
The equation is satisfied, regardless of the point’s \(z\)-coordinate. In particular, \((3,4,-2)\) is on the surface. More generally, keeping \(x=3\) and \(y=4\) while letting \(z=t\) gives the full line \((3,4,t)\), where \(t\in\mathbb R\).
Check the line: substituting \((3,4,t)\) gives \(3^2+4^2=9+16=25\) for every real \(t\). At any fixed height \(z=c\), the cross-section is \(x^2+y^2=25\), the same planar curve. Thus the surface is that curve extended parallel to the \(z\)-axis.
Worked Example: A Missing \(y\)-Coordinate
For \(x^2+z^2=9\), identify the direction of extension and determine whether \((\sqrt{5},7,2)\) lies on the surface.
The variable \(y\) is absent, so any solution pair \((x,z)\) gives a line parallel to the \(y\)-axis. The planar curve is described using \(x\) and \(z\). Substitute the proposed \(x\)- and \(z\)-coordinates:
The equation is satisfied, so the point lies on the surface; its \(y\)-coordinate \(7\) does not affect the result. In fact, keeping \(x=\sqrt{5}\) and \(z=2\) gives the line \((\sqrt{5},t,2)\), with \(t\in\mathbb R\).
Substitution checks every point on this line: \(x^2+z^2=(\sqrt{5})^2+2^2=5+4=9\). Fixing \(y=7\), or any other value, leaves the same equation \(x^2+z^2=9\) in the cross-section. Therefore the extension is parallel to the \(y\)-axis, not to either axis that appears in the equation.
Worked Example: A Curve with a Coordinate That Cannot Be Zero
Describe the surface \(xz=6\), determine whether \((3,-4,2)\) lies on it, and explain what happens in the plane \(z=0\).
The equation involves \(x\) and \(z\), but not \(y\). Its planar curve is \(xz=6\) in the \(xz\)-plane, and the curve extends along lines parallel to the \(y\)-axis. To test the point, multiply its \(x\)- and \(z\)-coordinates:
Thus \((3,-4,2)\) lies on the surface. Keeping \(x=3\) and \(z=2\) gives the whole line \((3,t,2)\), since for every real \(t\) the product of the first and third coordinates remains \(3\cdot2=6\).
In the plane \(z=0\), however, the surface equation would require \(x\cdot0=6\), or \(0=6\), which is impossible. So this particular cross-section contains no points. That does not prevent the surface from being a cylinder: the generating lines are parallel to the \(y\)-axis, and the original curve has no points with \(z=0\). The equation and the missing coordinate determine the extension direction; they do not guarantee that every coordinate-plane cross-section is nonempty.
Finding the Direction and Describing Sections
To use the rule reliably, first identify the variable absent from the equation. That variable gives the direction of the generators. Then regard the equation as a planar curve in the other two coordinates. This reverses a common instinct: the surface does not extend along one of the variables shown in the equation. It extends along the variable that is missing.
If the equation contains \(x\) and \(y\), but not \(z\), the generators are parallel to the \(z\)-axis.
Use the equation in the other two coordinates to identify which coordinate pairs can occur.
For each point of the curve, let the missing coordinate take every real value.
Fix the missing coordinate to see the repeated curve, or substitute a proposed point into the equation.
For a surface \(F(x,y)=0\), setting \(z=c\) changes none of the terms in the equation. The section in \(z=c\) is therefore \(F(x,y)=0\) for every real \(c\). By contrast, a section in \(x=c\) or \(y=c\) may be different, because it restricts a coordinate that does occur in the equation. A particular section may even be empty, as in the third example.
The same reasoning works for any omitted coordinate. If an equation uses \(y\) and \(z\) but omits \(x\), fixing \(x=c\) gives the same curve in the \(yz\)-plane for every \(c\). If it uses \(x\) and \(z\) but omits \(y\), fixing \(y=c\) gives the same curve in the \(xz\)-plane. The orientation follows from the coordinate axes, not from how complicated the curve equation looks.
A Common Mistake: Setting the Missing Coordinate to Zero
A frequent error is to treat a missing coordinate as if the equation required that coordinate to equal zero. For example, someone might reason that \(x^2+y^2=25\) means \(z=0\), and conclude that the surface is only a curve in the \(xy\)-plane. This confuses a coordinate absent from an equation with a coordinate explicitly restricted by an equation.
There is no \(z\) in the equation, so substituting \(z=0\) is an additional condition, not a consequence. For example, \((3,4,0)\) satisfies the equation because \(3^2+4^2=25\), but so does \((3,4,6)\), since the same calculation gives \(3^2+4^2=25\). The correct conclusion is that every point \((3,4,t)\) lies on the surface for real \(t\). Restricting to \(z=0\) would keep only one section of the surface and discard all its other points.
When the Cylinder Description Applies
The rule describes the geometry whenever one coordinate is genuinely absent from an equation in the other two coordinates. It does not require the planar curve to be a particular familiar shape. The planar curve might be disconnected, have gaps in its coordinate values, or even be empty; the set description still applies. If the planar equation has no solutions, the three-dimensional equation has no solutions either.
In introductory analytic geometry, “cylinder” often brings to mind a round tube. Here the useful geometric feature is more general: a curve is carried along parallel straight lines. The equation tells us the curve, while the missing variable tells us the direction in which it is extended. To identify a specific familiar type of cylinder, further analysis of the curve is needed; the missing-variable rule alone determines its orientation and repeated cross-sections.
This viewpoint is useful when interpreting surface equations and choosing cross-sections for a sketch. Rather than trying to picture every point at once, identify the planar curve and imagine translating it without changing its shape along the direction of the missing coordinate. Before drawing, check that the proposed generators are parallel to the correct axis and that the same curve appears in every plane perpendicular to those generators.
Check Your Understanding
For each question, use the missing coordinate to reason about the surface before attempting to sketch it.
- For \(y^2+z^2=16\), which coordinate is free, and parallel to which axis do the generators run?
- Describe the cross-section of \(x=z^2\) in each plane \(y=c\), where \(c\) is any real number.
- Does the point \((2,-5,3)\) satisfy \(xz=6\)? What line of the surface contains it?
- Why is it incorrect to conclude that \(x^2+y^2=9\) forces \(z=0\)?
- Can a surface described by an equation missing one coordinate have an empty section in some plane perpendicular to a different coordinate axis? Explain using \(xz=6\).