definition may be rephrased to say that a cylinder is any ruled surface spanned by a one-parameter family of parallel lines. A cylinder having a right section that is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder and hyperbolic cylinder, respectively. These are degenerate quadric surfaces. When the principal axes of a quadric are aligned with the reference frame (always possible for a quadric), a general equation of the quadric in three dimensions is given by with the coefficients being real numbers and not all of , and being 0. If at least one variable does not appear in the equation, then the quadric is degenerate. If one variable is missing, we may assume by an appropriate rotation of axes that the variable does not appear and the general equation of this type of degenerate quadric can be written as where Elliptic cylinder If this is the equation of an elliptic cylinder. Further simplification can be obtained by translation of axes and scalar multiplication. If has the same sign as the coefficients and , then the equation of an elliptic cylinder may be rewritten in Cartesian coordinates as: This equation of an elliptic cylinder is a generalization of the equation of the ordinary, circular cylinder (). Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid. If has a different sign than the coefficients, we obtain the imaginary elliptic cylinders: which have no real points on them. ( gives a single real point.) Hyperbolic cylinder If and have different signs and , we obtain the hyperbolic cylinders, whose equations may be rewritten as: Parabolic cylinder Finally, if assume, without loss of generality, that and to obtain the parabolic cylinders with equations that can be written as: Projective geometry In projective geometry, a cylinder is simply a cone whose apex (vertex) lies on the plane at infinity. If the cone is a quadratic cone, the plane at infinity (which passes through the vertex) can intersect the cone at two real lines, a single real line (actually a coincident pair of lines), or only at the vertex. These cases give rise to the hyperbolic, parabolic or elliptic cylinders respectively. This concept is useful when considering degenerate conics, which may include the cylindrical conics. Prisms A solid circular cylinder can be seen as the limiting case of a