1.8: Cylinders

CLP-3 Multivariable Calculus · LibreTexts

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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.

    This page titled 1.8: Cylinders was last modified on Thu, 27 Jan 2022 16:55:17 GMT and is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform.