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

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1;$$

an imaginary elliptic cylinder:

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=-1;$$

a hyperbolic cylinder:

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1;$$

and a parabolic cylinder:

$$y^2=2px.$$

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, $z$):

$$f(x,y)=0.\label{*}\tag{*}$$

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