[TheForge] Re: lamp shade

[April & Bill Clemens]

Mike,

1.  I don't think your formula is correct.
  
For any case where S  > H  * sqrt(2) 
( or conversely H < S / sqrt(2) )

then ( sqrt(2) S / 2 H ) is greater than one 

and arcsin is undefined.

For the case where H = S / sqrt(2) slope of 45 degrees you get 

A = 180 degrees 

2.  I looked at the websites Charles gave and found that:

http://www.josephfusco.org/Articles/Dihedral/Dihedral.html

seemed to fit best although it's drawings are not completely labeled.

Using this site I came up with the following solution 

A= 2 * arctan (S/H)  


using S and H as you defined them in your drawing:



          ------------------------        ------
  Side   /                        \          ^
   S--->/                          \         |   Height H measured at right
       /                            \        |   angles to the upper and 
      /                              \       |   lower edges.
     /                                \      v
     ----------------------------------    ------ 


(In case your mail reader messes up the ASCII sketch: S is the length
of one of the angled sides.  H is the right-angle distance between the
two parallel sides.)




3.  Also if you start with a given  trapezoid lamp side: 
				
                      T
          ------------------------        ------
  Side   /                        \          ^
   S--->/                          \         |   Height H measured at right
       /                            \        |   angles to the upper and 
      /                              \       |   lower edges.
     /                                \      v
     ----------------------------------    ------
                      B  

where S and the Top (T) and Bottom (B) are defined  you can determine the
corner angle using the following formula:

A= 2* arctan (1 / sin ( arccos( (B-T )/ (sqrt(2) * S))))

4.  I checked both of these against the example given in the website and get
the same result as they did(101.54 degrees). They also work for the case
where S = H (and B=T) where A = 90 degrees(Square Box) and also as H
approached zero ( B-T approaches sqrt(2) * S ) the angle A approaches 180
degrees (flat frame)


for the case where H = S / sqrt(2) the result is A = 109.47 degrees



Bill