Am 28.07.2013 23:01, schrieb Ken Pledger: > In article <email@example.com>, > Thomas Nordhaus <firstname.lastname@example.org> 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.