13.6
Cylinders and Quadric Surfaces
Cylinders
Definition 13.6.1
A cylinder is a surface obtained by taking all of the lines parallel to a given line that
intersect a given (plane) curve.
Admittedly, this definition is at first confusing. Let’s try to illustrate the meaning of the definition.
Consider the plane curve
y
=
x
2
. In the
xy
plane the graph is just the “standard” parabola which opens
up and has vertex (0
,
0). However, if we were to graph this equation in 3space, we would obtain much
more than a plane curve. Why? Well if we look at the trace of this surface in any plane parallel to the
xy
plane, we will always obtain the plane curve
y
=
x
2
. So what does this look like?
Note:
If one of the 3 variables is missing in an equation, then the graph of the equation will be a cylinder.
The “missing” variable is the variable that will have free range. Thus the cylinder will be the lines
parallel to the axis belonging to the missing variable that intersect the plane curve with the equation
in the two given variables.
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 Spring '07
 Chung
 Multivariable Calculus, Conic section, Quadric, ) curve.

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