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

(*)

The curve of order defined by equation (*) is sometimes called the base of the cylindrical surface.