Talk:Skew lines

Untitled

The following formula doesn't make sense; I'll find a replacement:

The distance D between two skew lines is given by:

$D={\sqrt {{(\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} ))^{2}} \over {(\mathbf {a} \times \mathbf {b} )^{2}}}}$

I've made the discussion valid for n dimensions, and replaced

(v_{1}-v_{3})\wedge (v_{2}-v1)\wedge (v_{4}-v_{3})

with something equivalent but much nicer. Gene Ward Smith 20:44, 22 April 2006 (UTC)[reply]

The current formula is Ok, but I'll me making some style changes and explanation. Although x is a valid line, deliberately choosing x for a line, and y for a separate line is likely to confuse many students. I will also mention that b & d are the direction vectors, lambda and mu are the parameters. I'll look up the related pages & shoot for consistency to make it easier to understand.Nickalh50 (talk) 02:35, 26 February 2011 (UTC)[reply]

Technical

The {{technical}} tag was added 20 Oct 2006 by SkerHawx, without further explanation. I've performed some major cleanups to the article, which I hope leave it more accessible — the main formulas are there, but with less auxiliary manipulation and more English explanation of what's being calculated, and I also included more nontechnical introductory material. So I think I'm justified in removing the tag. —David Eppstein 18:15, 24 October 2006 (UTC)[reply]

Parallelity

The article currently says

As in lower dimensions, surfaces are non-parallel if and only if their surface normals are non-parallel.

I don't think that makes sense. Consider, for example, in R⁴ the plane A generated by (0,0,0,0), (1,0,0,0), and (0,1,0,0); A is the xy-plane. Then consider the plane B generated by (0,0,0,1), (1,0,0,1), and (0,0,1,1). Here, B is the xz-plane translated in the direction of the fourth coordinate to make the planes A and B non-intersecting. Clearly A and B should be considered skew planes.

The "normal" to A through some point on A is a new two-dimensional surface. Therefore, the quotation above does not define anything.

/83.221.140.244 (talk) 18:45, 11 December 2007 (UTC)[reply]

I removed the whole section. There were two problems: the one you mention and a dimensional one (in four dimensions, planes in general meet in points and are not skew; in five dimensions, they can be skew). But defining parallelism is problematic: e.g. with planes, they can be totally unparallel (do not contain any parallel tangent vectors), they can contain some parallel tangent vectors and some non-parallel, or all tangent vectors from one plane may be parallel to tangent vectors of the other (in which case one plane is a translate of the other). My guess at the correct definition of skew is that there are no parallel tangent vectors and they do not intersect (which first happens in five dimensions), but until I or someone else tracks down reliable sources in the literature to help us determine how the word is actually used in practice I think we would be best not to say anything. —David Eppstein (talk) 19:07, 11 December 2007 (UTC)[reply]

I guess you could define two affine subsets with the same dimension to be parallel iff one is obtained from the other by a pure translation. Similarly, an affine subset with a smaller dimension could be defined to be parallel with one of a higher dimension iff the first one by a translation is made into a subset of the other one. If you understand what I mean. This is a "natural" definition to me. /83.221.140.244 (talk) 19:50, 11 December 2007 (UTC)[reply]

It's a plausible definition, but it's not the only plausible definition. We shouldn't be making up definitions here, we should be using definitions that can be supported by reliable sources elsewhere. —David Eppstein (talk) 20:47, 11 December 2007 (UTC)[reply]

Ratings Template

I'm rating this article as mid priority and assigning it to the "geometry" field. Bryanrutherford0 (talk) 16:35, 17 October 2013 (UTC)[reply]

"Skew line segments"?

On the line segments page, it says "A pair of line segments can be any one of the following: intersecting, parallel, skew, or none of these", linking to this page. This page does not talk about skew line segments. Somethings wrong somewhere on the Internet, I got confused by the use of the word Ray on the Vector page, and irritated enough to try to fix it, only to discover that I stuck my nose into a hornet nest. As I am not even a matematician, I'll leave the remaining fixing of the messes to someone else.