[TheForge] More trig (Re: lamp shade)

[Mike Spencer]

If the formulas, algebra and trigonometry is boring, just "move along,
nothing to see here" as the story-book English cops say.

Bill wrote:

> 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.

The problem is that I didn't make any explicit constraints on what
numbers one might plug into the formula.

First, lets refer again to the diagram of one panel of the truncated
pyramid again, with your labels (T & B) one new one, angle q:

				
                      T
          ------------------------        ---+--
  Side   /                        \          |
   S--->/                          \         |   Height H measured at right
       /                            \        |   angles to the upper and 
      /                              \       |   lower edges.
     / q                              \      |
     ----------------------------------    --+---
Angle of this       B  
corner is q


Now consider the extreme cases of a truncated pyramid:

    Extremely steep: The "pyramid" is a vertical square tube.  Angle q
    is 90 deg, angle between sides is 90 deg, S = H.

    Extremely flat: The "pyramid" lies all in a plane, essentially one
    square inside another with corresponding corners connected by
    lines.  Angle q is 45 deg, angle between sides is 180 deg, S =
    sqrt(2) H .

For all real truncated pyramid lamp shades we might build [1], angle q
will be between 45 deg. and 90 deg.  If we make angle q smaller than
45 deg. then the four pieces can't be assembled into a square-based
pyramid!

Back to the hentracks:

    If angle q can never be larger than 90 deg or smaller than 45 deg,
    then:

       sin 90  >=  sin q  >=  sin 45  =  sqrt(2) / 2  =  1/sqrt(2)

        1      >=  sin q  >=  0.707   =  sqrt(2) / 2  =  1/sqrt(2)

    But (refer to diagram):

        sin q  =  H/S

    So to make a real pyramidal object, the ratio H/S has to be
    greater than (or equal to) sqrt(2)/2.

        H / S  >=  1 / sqrt(2)         Constraint on values in my formula

   But your counter-example is any case where:

        H < S / sqrt(2)                 [Bill]
   or
       
        H / S < 1/sqrt(2)               [Bill, rearranged]

    Values of H and S that satisfy your counter-example inequality are
    excluded by the constraint and by observation that you can't make
    a real truncated pyramid from panels with angle q < 45.

So, you can plug any numbers into the formula, even numbers such that
H < S / sqrt(2) but those numbers don't correspond to pyramid panels.

> For the case where H = S / sqrt(2) slope of 45 degrees you get 
>
> A = 180 degrees 

Just so.  That's a slope of 45 deg. for angle q, not the angle between the
pyramid's side and the plane it sits on.  A = 180 is correct for the
completely flat "pyramid".

> http://www.josephfusco.org/Articles/Dihedral/Dihedral.html
>
> Using this site I came up with the following solution 
>
> A= 2 * arctan (S/H)  

Well, it looks like we're at the point where we ought to be able to
swap scribbles faster that by email.  But lessee...

On that web page, he has a pyramid (not truncated, but that doesn't
matter) with a 24x24 base and a 24 height.  That's the height measured
from THE CENTER OF THE BASE TO THE APEX.  That makes:

    height of the      26.833  Corresponds to H in our formulas
    triangular panel 

    edge of the panel  29.394  Corresponds to S in out formulas

Using my formula, A = 2  arcsin( sqrt(2) S / 2 H), I get 101.54 for
the dihedral angle.

Using your formula, A = 2 arctan( S/H), I get 95.22 for the dihedral
angle.

Hmm, what's the conflict?

If, instead of using the height of the triangular panel, you use the
HEIGHT OF THE PYRAMID measured from the center of the base to the apex
(call that HOP or 24) and use S, the length of the side of one of the
triangular panels (29.294), then:

    2 arctan( S / HOP ) = 101.54

which agrees with the web author's calculation and with my formula.

So I think our disagreement is in being precise about what is "height".

Puff puff pant.....

Jeez, if exercising the little grey cells protects from Alzheimer's,
I'm staying safe as can be. :-)


[1] We're ignoring the possibility that our lamp shade might be
    funnel-shaped (infundibular, if you're a Kurt Vonnegut fan :-),
    i.e. with its base smaller than its top and appropriate for a
    floor lamp.  But why add computational mess?  Figure out the
    truncated pyramid "right side up" and then turn it over.



- Mike

-- 
Michael Spencer                  Nova Scotia, Canada       .~. 
                                                           /V\ 
mspencer at tallships.ca                                     /( )\
http://home.tallships.ca/mspencer/                        ^^-^^