Cylinder

When a straight line moves in the space without changing its direction, the ruled surface it sweeps is called a cylindrical surface (or, in some special cases, simply a cylinder). Formally, a cylindrical surface is a ruled surface with the given condition:

If are two distinct points in , and and are the rulings passing through and respectively, then (this includes the case when ).

If the moving line returns to its starting point, the cylindrical surface is said to be closed. In other words, if we take any plane perpendicular to any of its rulings, and observe the curve of intersection of and , then is closed if is a closed curve.

The solid bounded by a closed cylindrical surface and two parallel planes is a cylinder. The portion of the surface of the cylinder belonging to the cylindrical surface is called the lateral surface or the mantle of the cylinder and the portions belonging to the planes are the bases of the cylinder.

The bases of any cylinder are congruent. The line segment of a generatrix between the planes is a side line of the cylinder. All side lines are equally long. If the side lines are perpendicular to the planes of the bases, one speaks of a right cylinder, otherwise of a skew cylinder.

The perpendicular distance of the planes of the bases is the height of the cylinder. The volume () of the cylinder equals the product of the base area () and the height ():

If the base is a polygon, the cylinder is called a prism (which is a polyhedron). The faces of the mantle of a prism are parallelograms. If also the bases of a prism are parallelograms, the prism is a parallelepiped. If the faces of the mantle of a prism are rectangles, one speaks of a right prism, otherwise of a skew prism.

For any integer , the following are equivalent statements about a prism :

1. has a base that is an -gon;
2. has faces;
3. has vertices;
4. has edges.

Note. The notion of the prism (or cylinder) of a polygon in has a higher-dimensional analogue. Given any polytope , the prism of P is the polytope Prism. The vertices of Prism are the points and , where ranges over the vertices of . In other words, we drag a short distance through a vector orthogonal to everything in , just as we would to obtain the prism of a polygon.