Cylinder (geometry)

In Mathematics , a cylinder is a Quadric , i.e. a three-dimensional surface, with the following equation in Cartesian Coordinates :

:

\left(rac{x}{a}
ight)^2 + \left(rac{y}{b}
ight)^2 = 1

This equation is for an elliptic cylinder, a generalization of the ordinary, '''circular cylinder''' (a = b). Even more general is the '''generalized cylinder''': the Cross-section can be any curve.

The cylinder is a ''degenerate quadric'' because at least one of the coordinates (in this case ''z'') does not appear in the equation. By some definitions the cylinder is not considered to be a quadric at all.

In common usage, a ''cylinder'' is taken to mean a finite section of a right circular cylinder with its ends closed to form two circular surfaces, as in the figure (right). If the cylinder has a Radius ''r'' and length ''h'', then its Volume is given by

:

V = \pi r^2 h \,

and its Surface Area is

:

A = 2 \pi r ( r + h ) \,

For a given volume, the cylinder with the smallest surface area has ''h'' = 2''r''. For a given surface area, the cylinder with the largest volume has ''h'' = 2''r''.

There are other more unusual types of cylinders. These are the ''imaginary elliptic cylinders'':

:

\left(rac{x}{a}
ight)^2 + \left(rac{y}{b}
ight)^2 = -1

the ''hyperbolic cylinder'':

:

\left(rac{x}{a}
ight)^2 - \left(rac{y}{b}
ight)^2 = 1

and the ''parabolic cylinder'':

:

x^2 + 2y = 0 \,

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Cylinder (geometry)