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Section 1.8 Cylinders
There are some classes of relatively simple, but commonly occurring, surfaces that are given their own names. One such class is cylindrical surfaces. You are probably used to thinking of a cylinder as being something that looks like \(x^2+y^2=1\text{.}\)
In Mathematics, the word “cylinder” is given a more general meaning.
Definition 1.8.1 . Cylinder.
A cylinder is a surface that consists of all points that are on all lines that are
parallel to a given line and
pass through a given fixed curve, that lies in a fixed plane that is not parallel to the given line.
Example 1.8.2 .
Here are sketches of three cylinders. The familiar cylinder on the left below
is called a right circular cylinder, because the given fixed curve (\(x^2+y^2=1\text{,}\) \(z=0\) ) is a circle and the given line (the \(z\) -axis) is perpendicular (i.e. at right angles) to the fixed curve.
The cylinder on the left above can be thought of as a vertical stack of circles. The cylinder on the right above can also be thought of as a stack of circles, but the centre of the circle at height \(z\) has been shifted rightward to \((0,z,z)\text{.}\) For that cylinder, the given fixed curve is once again the circle \(x^2+y^2=1\text{,}\) \(z=0\text{,}\) but the given line is \(y=z\text{,}\) \(x=0\text{.}\)
We have already seen the third cylinder
in Example
1.7.3 . It is called a hyperbolic cylinder. In this example, the given fixed curve is the hyperbola
\(yz=1\text{,}\) \(x=0\) and the given line is the
\(x\) -axis.
