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Definition For Cylinder without Big Words


Date: 11/03/98 at 21:58:28
From: Teresa Bowden
Subject: Geometry Definition

I just need to know a good definition for cylinder that I can 
UNDERSTAND.  The ones that I find are all impossible and use huge words 
that make no sense to me.


Date: 11/04/98 at 13:01:47
From: Doctor Peterson
Subject: Re: Geometry Definition

Hi, Teresa. There are a lot of different answers I could give you, 
depending on what you are going to do with the definition. A young 
child will be satisfied by just being told that a cyclinder is a "can 
shape." Later in math you use more precise definitions of the same 
shape. Then you may learn a more general definition that allows it to 
be "tilted" (an oblique cylinder, rather than a right cylinder), or 
even to have some other shape than a circle. Another way the definition 
can vary is by whether you want to work with a cylinder that has a top 
and bottom, or just the lateral (side) surface of an infinitely tall 
cylinder with no top or bottom. The more general you want it to be, the 
more complicated the definition is.

Our FAQ on cylinders gives a relatively complicated, general 
definition, which defines most of the terms involved, but requires you 
to understand the ideas of "generating" a surface:

  http://mathforum.org/dr.math/faq/formulas/faq.cylinder.html   

My dictionary says the same thing without the big words, defining a 
circle as:

  (a) A surface generated by a straight line moving parallel to a 
      fixed straight line and intersecting a plane curve. 
  (b) The portion of such a surface bounded by two parallel planes 
      and the regions of the planes bounded by the surface.

For a normal "right circular cylinder," I can rephrase this to say

    A cylinder is the surface formed by the set of lines perpendicular 
to a plane, which pass through a given circle in that plane.

               |
               |
               |
               | ------>
               |
               |
               |
               |
            ***|****************
      ******   |                ******
    **         |                      **
     *         |                      *
      ******   |                ******
            ***+****************
               |
               |

This says that you make a cylinder by dragging a line around in a 
circle. The "generatrix" is the axis, and the "directrix" is the circle 
forming the base, around which you move a line parallel to the axis to 
form the cylinder.

Here's a page that gives a more restricted definition of a circular 
cylinder:

  http://www.astro.virginia.edu/~eww6n/math/Cylinder.html   

   "In common usage, the term 'cylinder' refers to a Solid of circular
    Cross-Section in which the centers of the Circles all lie on a 
    single Line. In mathematical usage, 'cylinder' is commonly taken to 
    refer to only the lateral sides of this solid, excluding the top 
    and bottom caps. ... A cylinder is called a right cylinder if it is 
    'straight' in the sense that its cross-sections lie directly on top 
    of each other; otherwise, the cylinder is called oblique."

I can rewrite this for a right circular cylinder to say

    A cylinder is the surface formed by all circles of a given radius,
    in planes perpendicular to a given line (the axis), whose centers 
    are on that line.

                     |
                     |
            *********|**********
      ******         |          ******
    **               +                **
     *               |                *
      ******         |          ******
            ********************
                     |               ^
                     |               |
                     |               |
            *********|**********
      ******         |          ******
    **               +                **
     *               |                *
      ******         |          ******
            ********************
                     |
                     |
                     |

This says that you make a cylinder by dragging a circle in a straight 
line.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Definitions
High School Geometry
High School Higher-Dimensional Geometry
Middle School Definitions
Middle School Geometry
Middle School Higher-Dimensional Geometry

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