Building a Cone

Date: 01/28/2002 at 02:43:43
From: Surendra Kumar Chordia
Subject: Building a larger radius cone

My question is about cones.  I am trying to draw a larger radius size 
cone (frustum). The details are as follows: 

Example:

   x = diameter of small end of cone   x is 6.5 meter
   y = diameter of large end of cone   y is 7.0 meter
   z = height of cone                  z is 5.0 meter

For the above cones the the inner and outer radius comes out to:

   R =  larger radius of developed cone   R is 70.087 meter
   r =  smaller radius of developed cone  r is 65.081 meter
   
the above radii are difficult to draw to 1:1 scale, so please gives 
some alternative.

Any help or suggestions would be greatly appreciated.  Thank you.
S.K.Chordia

Date: 01/28/2002 at 12:58:34
From: Doctor Peterson
Subject: Re: Building a larger radius cone

Hi, Surendra.

It appears that you are forming a frustum by cutting out a portion of 
an annulus and rolling it into the lateral surface of the frustum. You 
have not told me how you found the radii; I presume you have a formula 
such as those explained here:

   Pattern for Lampshade
   http://mathforum.org/dr.math/problems/loraine.02.17.01.html   

   Flattening the Frustum of a Cone
   http://mathforum.org/dr.math/problems/scholl.10.15.00.html   

I think what you are saying is that you want a convenient way to 
draw an arc with a 70 meter radius without having to find a center 
70 meters away and use a 70-meter string. The simplest solution I can 
think of is to plot a set of points on the arc, through which you can 
draw a smooth curve. One way to do this uses the properties of a 
segment of a circle:

   Circle Radius from Chord Length and Depth
   http://mathforum.org/dr.math/problems/duranleau.6.16.99.html   

The right triangle in the picture on that page shows that

    r-h
    --- = cos(theta/2)
     r

Assuming you know the angle theta of your arcs (as given in the 
earlier references), you can solve this for h:

    h = r(1 - cos(theta/2))

The same formula is found here:

   Segments of Circles - Dr. Math FAQ
   http://mathforum.org/dr.math/faq/faq.circle.segment.html#13   

So draw the chord of the appropriate length, find its center, and 
measure h units perpendicular to the chord to locate the midpoint of 
the arc. Now repeat the process using the two new chords you've found, 
repeatedly bisecting the arc until you have points close enough to 
form an accurate arc.

There are probably some quick methods you can develop for doing these 
calculations more efficiently; I seem to recall a compass and 
straightedge construction that can accomplish this, but the 
calculations are probably good enough, and more accurate in practice.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/