intersecting lines in higher than 3D

[qsa]

I typically run a simulation that draws a line between two random point in 3d space. then I throw two random numbers that represent two lines ( see red lines in attachment) starting from the two previuos points and sit on top of the original line. if the lengths of those two lines are such that they cross I update a register. I repeat that J times.

 

I want to generalize this to higher than 3d. Is that possible and is it unique. Please provide any links to such information.

 

 

 

Please check this thread, the last post, it says the concept of line (or I guess he means line intersection ) does not work in higer than 3d. I wonder why.

 

My link

 

 

Thank you for your help.

Edited January 10, 2012 by qsa

[DrRocket]

I typically run a simulation that draws a line between two random point in 3d space. then I throw two random numbers that represent two lines ( see red lines in attachment) starting from the two previuos points and sit on top of the original line. if the lengths of those two lines are such that they cross I update a register. I repeat that J times.

 

I want to generalize this to higher than 3d. Is that possible and is it unique. Please provide any links to such information.

 

 

 

Please check this thread, the last post, it says the concept of line (or I guess he means line intersection ) does not work in higer than 3d. I wonder why.

 

My link

 

 

Thank you for your help.

 

The concept of a line works in a vector space, or affine space, of any dimension, including infinite-dimensional spaces. The styatement made in thevdiscussion at your linked site is just plain wrong.

 

But any two lines can always be found in a 3-dimensional subspace so there is not a lot to be gained from going to the higher dimensions to study two lines at a time.

[qsa]

Thanks for the reply. But there is also this link, see last post. it is saying in 4D "it is very special situation" what is meant by that.

 

My link

 

 

 

I am specificly asking about 4D and higher. would two lines intersect as shown in attachment. Is there a reference I can study. I don't seem to find much on google.

[DrRocket]

Thanks for the reply. But there is also this link, see last post. it is saying in 4D "it is very special situation" what is meant by that.

 

My link

 

 

 

I am specificly asking about 4D and higher. would two lines intersect as shown in attachment. Is there a reference I can study. I don't seem to find much on google.

 

Your link doesn't seem to be working.

 

There is nothing special about 4D or higher with respect to the notion of lines and their intersection. As I said, given two lines in n-dimensions you can always find a 3-dimensional subspace that contains those two lines, so anything that you care to say about two lines can be said and studied in dimension 3 without losing anything.

 

There are topological questions -- for instance the Poincare conjecture, now the Poincare theorem -- that are relatively easily solvable in higher dimensions (though Smale received the Fields Medal for solutions in dimension 5 and above), but very difficult in dimension 3 and 4 (Friedman received the Fields Medal for the solution in dimension 4 and Pereleman was awarded, but did not accept, the medal for the solution in dimension 3). There is also the issue of things like "exotic differentiable structures" that arise only in special dimensions -- and 4 happens to be special for this particular problem. But to even understand the problem you need to understand what is meant by a differentiable structure and the difference between a homeomorphism and a diffeomorphism.

 

There are references on topology in dimension 4. The Geometry of Four-Manifolds by Donaldson and Kronheimer and

Topology of 4-Manifolds by Friedman and Quinn are two such references. However, these are very specialized and advanced graduate texts and not for the faint of heart. Unless you have extensive education in topology and geometry you are likely to find these books unreadable.

 

As far as the question of lines in dimension 4 and above there are no specific references simply because the problem reduces, rather trivially, to dimension 3 and that is covered in any decent text on analytic geometry or calculus.

[qsa]

Thanks for the reply.

 

 

 

ok,here is the link, not sure why it is not working.

 

http://www.physicsfo...ad.php?t=431256

 

 

Also studying this one

 

 

http://www.cs.indiana.edu/pub/hanson/Siggraph01QuatCourse/ggndgeom.pdf