cylinder

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

∥

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

π

perpendicular

to any of its rulings, and observe the curve

c

of

intersection

of

π

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

=

A

⁢

h

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

≥

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

2

⁢

n

vertices;

4.

P

has

3

⁢

n

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

Prism

⁢

(

P

)

:=

P

×

[

0

,

1

]

.  The
vertices of

Prism

⁢

(

P

)

are the points

(

x

,

0

)

and

(

x

,

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

Generated on Fri Feb 9 22:08:29 2018 by

LaTeXML