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Re: WTD: point intersection in space

• To: mathgroup at smc.vnet.net
• Subject: [mg48374] Re: WTD: point intersection in space
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Wed, 26 May 2004 02:41:23 -0400 (EDT)
• Organization: Universitaet Leipzig
• References: <c8vcl8$afs$1@smc.vnet.net>
• Reply-to: kuska at informatik.uni-leipzig.de
• Sender: owner-wri-mathgroup at wolfram.com

Hi,

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 ...

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.

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.

Regards
  Jens


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