View source for Cylindrical surface (cylinder)

The surface formed by the motion of a line (the generator) moving parallel to itself and intersecting a given curve (the directrix).

The directrix of a cylindrical [[Surface of the second order|surface of the second order]] is a curve of the second order. Depending on the form of the directrix one distinguishes an [[Elliptic cylinder|elliptic cylinder]], the canonical equation of which is

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an imaginary elliptic cylinder:

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a [[Hyperbolic cylinder|hyperbolic cylinder]]:

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and a [[Parabolic cylinder|parabolic cylinder]]:

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If the directrix is a degenerate curve of the second order (i.e. a pair of lines), then the cylindrical surface is a pair of planes (intersecting, parallel or coincident, real or imaginary, depending on the corresponding property of the directrix).

A cylindrical surface of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027650/c0276506.png" /> is an algebraic surface given in some affine coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027650/c0276507.png" /> by an algebraic equation of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027650/c0276508.png" /> not containing one of the coordinates (for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027650/c0276509.png" />):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027650/c02765010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

The curve of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027650/c02765011.png" /> defined by equation (*) is sometimes called the base of the cylindrical surface.