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MathGroup Archive 2002

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Re: Generating Two Unit Orthogonal Vectors to a 3D Vector

  • To: mathgroup at smc.vnet.net
  • Subject: [mg36360] Re: Generating Two Unit Orthogonal Vectors to a 3D Vector
  • From: Tom Burton <tburton at brahea.com>
  • Date: Tue, 3 Sep 2002 01:41:02 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Adding my two cents to:

On 9/2/02 1:30 AM, in article akv7s5$fl8$1 at smc.vnet.net, "David Park"
<djmp at earthlink.net> wrote:

> There are many cases in graphics, and otherwise, where it is useful to
> obtain two orthogonal unit vectors to a given vector. I know a number of
> ways to do it, but they all seem to be slightly inelegant. I thought I would
> pose the problem to MathGroup. Who has the most elegant Mathematica
> routine...

To this I would like to add a criterion of smoothness. Armed with a second
vector b not parallel to the given vector a, it's a trivial matter to
orthogonalize b WRT a by Gram-Schmidt and then form the third vector c = a x
b. (Normalize as needed.)

I don't need more elegance that this, but I would like a scheme to select
the vector b that results in a triad {a,b,c} that various smoothly as "a"
varies over all possible directions. Each of my attempts to date involve a
branched algorithm and jumps in the resulting triad for certain small
changes in "a".

To David's call for elegance I add a call for smoothness.

Tom Burton



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