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 $S$ is a ruled surface with the given condition:

If $p,q$ are two distinct points in $S$, and $l$ and $m$ are the rulings passing through $p$ and $q$ respectively, then  $l\parallel m$ (this includes the case when  $l=m$).

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

Figure 1: A closed cylindrical surface

The solid 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 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 of the cylinder.  The volume ($V$) of the cylinder equals the product of the base area ($A$) and the height ($h$):

$$V=Ah$$

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 $n\ge 3$, the following are equivalent statements about a prism $P$:

1. $P$ has a base that is an $n$-gon;

2. $P$ has $n+2$ faces;

3. $P$ has $2n$ vertices;

4. $P$ has $3n$ edges.

Note.  The notion of the prism (or cylinder) of a polygon in ${ℝ}^3$ has a higher-dimensional analogue.  Given any polytope $P$, the prism of P is the polytope  $\text{Prism}(P):=P\times [0,\mathrm{\hspace{0.17em}1}]$.  The vertices of $\text{Prism}(P)$ are the points  $(x,\mathrm{\hspace{0.17em}0})$ and  $(x,\mathrm{\hspace{0.17em}1})$, where $x$ over the vertices of $P$.  In other words, we drag $P$ a short distance through a vector orthogonal to everything in $P$, just as we would to obtain the prism of a polygon.

Title	cylinder
Canonical name	Cylinder
Date of creation	2013-03-22 15:29:21
Last modified on	2013-03-22 15:29:21
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	12
Author	stevecheng (10074)
Entry type	Topic
Classification	msc 51M20
Classification	msc 51M04
Related topic	Parallelotope
Defines	cylindrical surface
Defines	lateral surface
Defines	mantle
Defines	base
Defines	side line
Defines	right cylinder
Defines	skew cylinder
Defines	prism
Defines	parallelepiped
Defines	right prism
Defines	skew prism