Cylindrical surface (cylinder)

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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 is a curve of the second order. Depending on the form of the directrix one distinguishes an elliptic cylinder, the canonical equation of which is


an imaginary elliptic cylinder:


a hyperbolic cylinder:


and a parabolic cylinder:


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 $n$ is an algebraic surface given in some affine coordinate system $x,y,z$ by an algebraic equation of degree $n$ not containing one of the coordinates (for example, $z$):


The curve of order $n$ defined by equation \eqref{*} is sometimes called the base of the cylindrical surface.

How to Cite This Entry:
Cylindrical surface (cylinder). Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article