From: Robin Chapman <rjc@maths.ex.ac.uk>
Newsgroups: sci.math
Subject: Re: Shortest distance between 3d lines
Date: Wed, 11 Mar 1998 10:10:35 -0600

In article <01bd4b90$f1815240$66ad6ccf@richards.s-vision.com>,
  "Richard Shields" <richards@s-vision.com> wrote:
>
> This might be a frequently ask question but couldn't find any reference to
> it in the FAQ for this group so...
>
> Given two lines defined by four points (p1 and p2 define line 1 and p3 and
> p4 define line 2) in three dimensional space how do you
> 1) determine the shortest distance between the two lines,
> 2) determine the point on line 1 that is closest to line 2.

This is a standard exercise in vector algebra. One can give the
equations of the lines in vector form as r = p_1 + t a_1 and
r = p_3 + u a_2 where t and u are real parameters and a_1 = p_2 - p_1 and
a_2 = p_4 - p_3. Let's assume the lines are not parallel. If the closest
points on the lines are Q_1 and Q_2 then the segment Q_1 Q_2 is perpendicular
to both lines. Thus c = (a_1 x a_2)/|a_1 x a_2| is a unit vector in the
direction Q_1 Q_2. (Here x denotes the vector product.)
Then line 1 is contained in the plane r.c = p_1.c
and line 2 in r.c = p_3.c. The distance between these planes, and between
the lines 1 and 2 is |p_1.c - p_2.c|.

To determine Q_1 and Q_2 set Q_1 = p_1 + t a_1 and Q_2 = p_3 + u a_2.
Then Q_1 - Q_2 is perpendicular to a_1 and a_2 and so a_1.Q_1 = a_1.Q_2.
and a_2.Q_1 = a_2.Q_2. These give two linear equations for Q_1
and Q_2 which can be solved to find t and u and so Q_1 and Q_2.

Robin Chapman                           + "They did not have proper
Department of Mathematics               -  palms at home in Exeter."
University of Exeter, EX4 4QE, UK       +
rjc@maths.exeter.ac.uk                  -    Peter Carey,
http://www.maths.ex.ac.uk/~rjc/rjc.html +      Oscar and Lucinda

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