1.8: Cylinders
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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 x2+y2=1.
In Mathematics, the word “cylinder” is given a more general meaning.
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.
Here are sketches of three cylinders. The familiar cylinder on the left below
is called a right circular cylinder, because the given fixed curve (x2+y2=1, 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). For that cylinder, the given fixed curve is once again the circle x2+y2=1, z=0, but the given line is y=z, x=0.
We have already seen the 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, x=0 and the given line is the x-axis.