On 7 Apr., 12:21, Laurence Reeves <l...@[EMAIL PROTECTED]
> wrote:
> Jon G. wrote:
> > Suppose you have 3 timbers of odd lengths, and plant them on the
ground to
> > form a pyramid. If you know the lengths of the 3 timbers, and where
you
> > plant them on the ground, this Excel worksheet will find the apex of
the
> > pyramid where the 3 timbers meet. You only need use Sheet 1 for
instant
> > answers.
>
> >http://www.freefileserver.com/index/p_download/hash_Ihywe3KYeyzT/
>
> > The math behind this is in my web site,
>
> >http://mypeoplepc.com/members/jon8338/math/index.html
>
> I fail to see quite why you would want to approach this problem in such
> a complex way. Given your static A, B and C point, with the distances to
> your unknown point D, it is fairly trivial 3D geometry to locate D (if
> there exists a solution).
>
> E.g. you could note that D lies on the circular locus implied by its
> distances from A and B, plus it also lies on the appropriate sphere
> about C. The circle may intersect the sphere in two points, reflections
> in ABC (or coincident in that plane). A somewhat "lop-sided" route, but
> it works.
>
> I think that the conditions for a solution to exist are simply that each
> of the three triangles formed above ABC must be possible individually.
No, this is not enough.
If ABC is an equilateral triangle of sidelength 1 and each "log"
has length 0.5 + eps then the individual triangles are possible but
not the pyramid.
A good condition however is: Produce the two possible triangles ABX
and AYB
above AB lying in the ABC plane.
The pyramid construction is possible iff the third length is between
CX and CY -- just note that X and Y are the points closest to /
farthest
from C on the circular locus.
> However, I'm not certain about that being necessary (nine
> inequalities sounds too many) and sufficient (although maybe it is).
>
> --
> Lau AS! d-(!) a++ c++++ p++ t+ f-- e++ h+ r--(+) n++(*) i++ P- m++
> ASC Decoder at <http://www32.brinkster.com/ascdecode/>
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