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