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 $x^2+y^2=1\text{.}$

$cylinderA.svg$

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

$cylinderR.svg$ $cylinderO.svg$

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 the third cylinder

$hyperbolicCylinderD.svg$

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.