The surface generated by a variable line which remains parallel to a fixed line and intersects a given curve (or touches a given surface) is called a cylinder.
The variable line is called the generator, the fixed straight line is called the axis of the cylinder and the given curve is called the guiding curve of the cylinder.
Let us find the equation of the cylinder whose generators are parallel to the line
and whose guiding curve is the conic
Let (α, β, γ) be any point on the cylinder. Then the equations of a generator are