Re: [math-fun] perpendicular planes
If two lines are given parametrically by {u0 + su} and {v0 + tv} for (s,t) in R^2, vectors u0, v0, and unit vectors u, v, then the distance between the lines is just (setting w0 = u0-v0) |w0 * u x v|, or am I missing something? --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx
In case you haven't read the paper below, you should give a look: "Packing lines, planes ..." by Conway, Hardin and Sloane. http://www2.research.att.com/~njas/doc/grass.pdf Victor
It's certainly interesting and generally relevant; however they discuss exclusively elliptic space --- accounting for (elliptic angular) separation, but ignoring (parabolic linear) distance. Coping with distance in Euclidean space (or separation in more general quadratic spaces) ratchets up the difficulty, as the example of parallel subspaces illustrates. It's noteworthy too that they run into a complication right at the start, over the geodesic definition of angular separation failing to be differentiable. This occurs because their lines (aka points in elliptic space) are undirected, and the problem would go away if they worked instead with directed lines (in spherical space); though it's not clear that the original application would permit that model. WFL On 3/27/11, Victor Miller <victorsmiller@gmail.com> wrote:
In case you haven't read the paper below, you should give a look:
"Packing lines, planes ..." by Conway, Hardin and Sloane.
http://www2.research.att.com/~njas/doc/grass.pdf
Victor _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Sun, March 27, 2011 12:20:32 AM Subject: Re: [math-fun] perpendicular planes If two lines are given parametrically by {u0 + su} and {v0 + tv} for (s,t) in R^2, vectors u0, v0, and unit vectors u, v, then the distance between the lines is just (setting w0 = u0-v0) |w0 * u x v|, or am I missing something? --Dan ________________________________________________________________________________________ Let's divide that expression by |u x v|. -- Gene
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Victor Miller