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CYLINDER

Cartesian equation:   (for a cylinder with axis Oz).
Cartesian parametrization: .
First fundamental quadratic form: .
Second fundamental quadratic form: .

The cylinders are the ruled surfaces the generatrices of which have a fixed direction .
A curve traced on the cylinder which meets all the generatrices is called a directrix of the cylinder; there exists a unique cylinder with given direction and directrix.

Necessary and sufficient condition: surface globally invariant under the action of any translation in the direction .
We can also consider that a cylinder is a cone the vertex of which is at infinity.
Cylinders are translation surfaces and developable surfaces.

The word cylinder is also used in a topological sense and refers to any surface homeomorphic to the cylinder of revolution, or, which amounts to the same thing, to a sphere minus two points. For example, an open strip with an even number of half-turns is topologically equivalent to a cylinder.

See also the cylindrical catenaries.
 
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© Robert FERRÉOL  2017