[TheForge] More trig (Re: lamp shade)

[Steve Smith]

Hey Bill,

I've got enough to do!

It was really good to bump into you at the conference. I keep thinking 
there is something I should ask you about. It will either surface...or 
it won't.

Steve

April & Bill Clemens wrote:
> Mike,
> 
> My humble apologies for doubting you.
> 
> Yes I was using the HOP as H.  
> I translated "measured at right angles to the upper and lower edges" from
> your diagram to mean the height of the lamp(HOL) not the perpendicular
> distance between the top and bottom of the side.  At that point I was too
> far gone to ever recognize my error.
> 
> I think this is the place we start discussing which way to point the horn of
> our anvils.  I tend to dance around the anvil so anywhere in a horizontal
> plane works for me.  
> 
> I think we should leave it as an exercise for all the other trig wizards out
> there to prove that all 3 formulas are identical when you take into account
> the relationships between HOP, H, and B-T.
> 
> Bill
> 
> 
> 
> 
> -----Original Message-----
> From: theforge-bounces at mailman.qth.net
> [mailto:theforge-bounces at mailman.qth.net] On Behalf Of Mike Spencer
> Sent: Thursday, November 09, 2006 1:56 AM
> To: theforge at mailman.qth.net
> Subject: [TheForge] More trig (Re: lamp shade)
> 
> 
> 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
>