[Jprogramming] Szilassi's toroidal heptahedron: interfacial angles? Last try.

[PMA]

Hi Pista!

This comes just in time to vet my results, got for better
or worse after a Bo > Spherical Trig > Wolfram search.

At first glance I see a slight difference in our formulae,
but won't ask about it yet (esp. lest it prove cosmetic).
Comparing further....

You've gone to a lot of trouble here.  Thank you!

Pete


Istvan Kadar wrote:
> Hi Pete,
> 
>          A  NUMERICAL EXAMPLE
> 
>                               NB.      A             B
>          known           NB.         \           /
>     edge-angles      NB.           \        /
>    a=:57 01 45.2     NB.           \   c  /
>    b=:87 42 41.1     NB.            \    /
>    c=:55 37 19.6     NB.              \ /
>                               NB.              D
>        unknown         NB.         a     |     b
>     face-angles       NB.                |
>          C=: ?           NB.                 |
>          A=: ?            NB.                |
>          B=: ?            NB.                |
>                               NB.               C
> 
>              a + b + c << 360
> (57 01 45.2+87 42 41.1+55 37 19.6=200 21 45.9<<360)
> 
>               A formula for the solution
> 
> cos_C =: ((cos c)-~(cos a)*cos b)%(-sin a)*sin b
>     C =: acos cos_C
> 
>    ]sin_a=:1&o.rfd((%60)p.~a)
> 0.838948
> 9j7":sin_a
> 0.8389482
>    ]cos_a=:2&o.rfd((%60)p.~a)
> 0.544211
> 9j7":cos_a
> 0.5442112
> 
>    ]sin_b=:1&o.rfd((%60)p.~b)
> 0.999202
> 9j7":sin_b
> 0.9992024
>    ]cos_b=:2&o.rfd((%60)p.~b)
> 0.0399327
> 9j7":cos_b
> 0.0399327
> 
>    ]cos_c=:2&o.rfd((%60)p.~c)
> 0.564649
> 9j7":cos_c
> 0.5646485
> 
>   ]cos_C =: ((cos_c)-~cos_a*cos_b)%-sin_a*sin_b
> 0.647656
>   9j7":cos_C =: ((cos_c)-~cos_a*cos_b)%-sin_a*sin_b
> 0.6476563
>    _2&o.0.6476563
> 0.866292
>    dfr=:rfd^:_1
>    9j7":_2&o.0.6476563
> 0.8662919 radian
>    dfr 0.8662919
> 49.6349
> 
>      C=:49.6349 degree (49° 38' 05.5" )
> 
> Best regards
>    Pista
> 
> 2009/8/22 PMA <PeterArmstrong at aya.yale.edu>
> 
>> Bo, Thank you for these leads!
>>
>> I felt the algebra would be just plain,
>> but hadn't the grasp to specify it.
>> Will pursue....
>>
>> Best regards,
>> Pete
>>
>>
>> Bo Jacoby wrote:
>>> Isn't it just plain linear algebra?  You have a 3-corner and knows
>>> the 3 cosines of angles between 3 vectors. Compute the cosines
>>> of angles between pairwise normals to the vectors.
>>>
>>> Venlig hilsen, Bo.
>>>
>>>
>>> --- Den fre 21/8/09 skrev Sherlock, Ric <R.G.Sherlock at massey.ac.nz>:
>>>
>>>> Fra: Sherlock, Ric <R.G.Sherlock at massey.ac.nz>
>>>> Emne: Re: [Jprogramming] Szilassi's toroidal heptahedron: interfacial
>> angles? Last try.
>>>> Til: "Programming forum" <programming at jsoftware.com>
>>>> Dato: fredag 21. august 2009 03.07
>>>> Hi Peter,
>>>> You could try providing some sample data and show what you
>>>> expect the result to be.
>>>> That might elicit a better response.
>>>>
>>>>> From: PMA
>>>>>
>>>>> Here is a final crack at posing my question -- this is
>>>> as whittled as
>>>>> I can whittle:
>>>>>
>>>>> Given a 3-D vertex of 3 faces with all edge-to-edge
>>>> angles known,
>>>>> how does one calculate its face-to-face angles (all,
>>>> one per edge)?
>>>>> If there come to mind other sites especially
>>>> interested in this sort
>>>>> of query, I'd appreciate hearing.
>>>>>
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