Learning Objectives
- Identify a cylinder as a type of three-dimensional surface.
The first surface we’ll examine is the cylinder. Although most people immediately think of a hollow pipe or a soda straw when they hear the word cylinder, here we use the broad mathematical meaning of the term. As we have seen, cylindrical surfaces don’t have to be circular. A rectangular heating duct is a cylinder, as is a rolled-up yoga mat, the cross-section of which is a spiral shape.
In the two-dimensional coordinate plane, the equation [latex]x^{2}+y^{2}=9[/latex] describes a circle centered at the origin with radius 3. In three-dimensional space, this same equation represents a surface. Imagine copies of a circle stacked on top of each other centered on the [latex]z[/latex]-axis (Figure 1), forming a hollow tube. We can then construct a cylinder from the set of lines parallel to the [latex]z[/latex]-axis passing through circle [latex]x^{2}+y^{2}=9[/latex] in the [latex]xy[/latex]-plane, as shown in the figure. In this way, any curve in one of the coordinate planes can be extended to become a surface.
DEFINITION
A set of lines parallel to a given line passing through a given curve is known as a cylindrical surface, or cylinder. The parallel lines are called rulings.
From this definition, we can see that we still have a cylinder in three-dimensional space, even if the curve is not a circle. Any curve can form a cylinder, and the rulings that compose the cylinder may be parallel to any given line (Figure 2).
Example: graphing cylindrical surfaces
Sketch the graphs of the following cylindrical surfaces.
- [latex]x^{2}+z^{2}=25[/latex]
- [latex]z=2x^{2}-y[/latex]
- [latex]y=\sin x[/latex]
try it
Sketch or use a graphing tool to view the graph of the cylindrical surface defined by equation [latex]z=y^{2}[/latex].
Watch the following video to see the worked solution to the above Try IT.
DEFINITION
The traces of a surface are the cross-sections created when the surface intersects a plane parallel to one of the coordinate planes.
Traces are useful in sketching cylindrical surfaces. For a cylinder in three dimensions, though, only one set of traces is useful. Notice, in Figure 7, that the trace of the graph of [latex]z=\sin x[/latex] in the [latex]xz[/latex]-plane is useful in constructing the graph. The trace in the [latex]xy[/latex]-plane, though, is just a series of parallel lines, and the trace in the [latex]yz[/latex]-plane is simply one line.
Cylindrical surfaces are formed by a set of parallel lines. Not all surfaces in three dimensions are constructed so simply, however. We now explore more complex surfaces, and traces are an important tool in this investigation.