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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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Thomas Nordhaus

Posts: 433
Registered: 12/13/04
Re: Distance Between Lines in R^3 (fwd)
Posted: Jul 28, 2013 6:30 PM
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Am 28.07.2013 23:01, schrieb Ken Pledger:
> In article <kste74$gue$1@news.albasani.net>,
> Thomas Nordhaus <thnord2002@yahoo.de> 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. http://at.yorku.ca/cgi-bin/bbqa?forum=calculus;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:


That's what I meant "by scratch": You start by minimizing distances
between points on the lines (variational principle) and end up with
orthogonality conditions:

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


I think implicitly it does.

>
> Ken Pledger.
>



--
Thomas Nordhaus



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