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MathGroup Archive 2004

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Re: WTD: point intersection in space

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48406] Re: WTD: point intersection in space
  • From: dont at agora.rdrop.com (Don Taylor)
  • Date: Fri, 28 May 2004 00:50:36 -0400 (EDT)
  • References: <c8vcl8$afs$1@smc.vnet.net> <c944nt$r3s$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Jens-Peer Kuska <kuska at informatik.uni-leipzig.de> writes:
>there must not be a unique solution, your lines in 2d
>can be parallel and you will not find a intersection and
>two spheres in 3d must also not have a point in common, more
>over the two spheres can have a common curve ...

My apologies if I was not clear when I said:
>> For all this I can ensure that a unique solution does exist

I re-read what I wrote several times, trying to make it clear.

I can guarantee you that *for what I am doing* there will always
be a unique solution, as long as the tiny errors that result
from describing vectors using floating point numbers are handled.

As an example of these tiny errors, "intersecting" vectors in 3-space
can be skewed by very tiny amounts when their origin and direction
is given by floating point coefficients.  If Solve[] is given a set
of equations it can easily conclude that there is no solution that
satisfies this.  Wickham-Jones deals with problems like this in his
book neatly, for the examples he provides.

>So you can't ensure that a unique solution exist with real
>coordinates. You can have complex solutions but this is nonsense
>if you seach a intersection point on real coordinate axis.

With completely arbitrary vectors almost anything is possible.
Fortunately I have a simpler problem where a unique solution will
always exist.

>Suppose you have two n-1 dimensional surfaces, every with
>n-1 parameters S1[p[1],...,p[n-1]] and S2[q[1],...,q[n-1]]
>and S1 and S2 map the R^(n-1) -> R^(n) finding the intersection
>mean S1[p]==S2[q] and this yields n equations to find the 
>2(n-1) parameter p[1],..,p[n-1] and q[1],..,q[n-1]
>For 2d you have two equations for two parameters, for 3d
>3 equations for 4 parameters ...
>Only in the plane, the system is not underdetermined.

Perhap an overly simplified example will help here.

In 3-d let the 3 planes be x==0, y==4, z==2.
Find the unique point where they all intersect.
That is not underdetermined.

Now a similar example, closer to what I'm really trying to do.  In
3-d I have 3 planes, specified by a point on each plane and by a
normal vector to each plane or by an angle to an axis.  I guarantee
there will be a single unique point where all these intersect for
this set of planes.  The points for the planes can be given exactly
but the normals or angles are the result of inverse trig functions
that have no exact algebraic representation, so N[] can bring in
small errors.

In 4-d there are 4 3-d simplexes... and repeat the problem.

etc.

At the moment for 3-d I am using NMinimize[Norm[vect1]+Norm[vect2]]
as a complicated substitute for Solve[], trying to find the two
vectors that are as close to a precise solution as possible.  I've
written this as 2 vector equations in R3 with 6 unknown scalars,
and in a couple of other ways, but there are always just enough
equations for the number of unknowns.

But all these are a messy arrangement with precision problems and
lots of fiddling to get it to work, what I do for 3-d doesn't look
like it will apply for 4-d, and are not nearly as elegant as what
Wickham-Jones sees how to do.  I'm hoping there is a general solution,
with perhaps even beautiful simplicity.

>Regards
>  Jens

Thank you for any suggestion or attempt to help
don

>Don Taylor wrote:
>> I've looked carefully at Wickham-Jone's "Mathematica Graphics",
>> based on recommendations from old newsgroup postings, and it is
>> a good resource.  But I'm looking for a method that he doesn't
>> seem to include.
>> 
>> For all this I can ensure that a unique solution does exist,
>> but would like to handle tiny errors as are neatly handled by
>> Wickham-Jones with his methods.
>> 
>> In n-dimensional euclidian space n (n-1)-dimensional objects
>> intersect at a unique point.  For example, in 2-space 2 1-d
>> lines intersect in a point, in 3-space 3 2-d planes intersect
>> in a point.
>> 
>> I'm trying to find the general pattern that solves for this
>> in n-space.  At the moment I'd settle for seeing very similar
>> developments for 2-space and 3-space.  Maybe from that I could
>> guess what this would look like for n-space.
>> 
>> Any pointers or suggestions would be greatly appreciated.
>> Thank you


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