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Topic: Distance Between Lines in R^3 (fwd)
Replies: 15   Last Post: Sep 13, 2013 1:25 PM

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Posts: 1,412
Registered: 12/3/04
Re: Distance Between Lines in R^3 (fwd)
Posted: Jul 28, 2013 5:01 PM
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In article <kste74$gue$>,
Thomas Nordhaus <> wrote:

> Am 26.07.2013 05:33, schrieb William Elliot:
> > How do we find the shortest distance between two lines L, L' in R^3 ?
> >
> > 2.;task=show_msg;msg=0792
> >

> How about doing it straight forward from scratch? Let P1(s) = v0 +s*v,
> P2(t) = w0 + t*w be arbitrary points on the line L, L' resp.
> ....

Yes. Then just make the vector (P1(s) - P2(t)) perpendicular to
both lines:

v.((v0 +s*v) - (w0 + t*w)) = 0
w.((v0 +s*v) - (w0 + t*w)) = 0.

(v.v)s - (v.w)t = - v.v0 + v.w0
(w.v)s - (w.w)t = - w.v0 + w.w0.

Solve those linear equations for s and t,
then find ||(P1(s) - P2(t))||.

That's all. It's a traditional method in old text-books which aren't
read much any more, and it doesn't need any calculus.

Ken Pledger.

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