Changes between Version 3 and Version 4 of BoundingRegionTransformation

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Timestamp:
10/01/10 04:52:30 (4 years ago)
Author:
byorgey (IP: 207.138.174.254)
Comment:

add notes on computing orthonormal bases

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  • BoundingRegionTransformation

    v3 v4  
    6868  2. Work out the details of step (4).  Does it really work? 
    6969 
     70For step (4), perhaps see [http://en.wikipedia.org/wiki/Gram–Schmidt_process Gram-Schmidt process], and also this exchange from the [http://tunes.org/~nef/logs/haskell/10.09.30 #haskell IRC channel on 30 Sep 2010]: 
     71 
     72{{{ 
     7318:18:27 <byorgey> given a vector v in an inner product space, how might I go about computing a basis for the hyperplane orthogonal to v? 
     7418:18:45 <Veinor> ask #math? :D 
     7518:19:02 --- join: dnolen (~davidnole@cpe-68-173-254-181.nyc.res.rr.com) joined #haskell 
     7618:19:09 * byorgey shudders 
     7718:19:11 <ddarius> byorgey: for all basis vectors b, keep b if b ^ v /= 0 
     7818:20:00 --- join: c_wraith (~c_wraith@209.237.247.90) joined #haskell 
     7918:20:01 <byorgey> ddarius: ^ is inner product? 
     8018:20:10 <Cale> byorgey: Well, pick random vectors and subtract off the orthogonal projection onto the subspace spanned by v 
     8118:20:11 <benmachine> isn't ^ usually used for cross product 
     8218:20:11 <ddarius> byorgey: Wedge product. 
     8318:20:13 <benmachine> or is that v 
     8418:20:24 * benmachine always just used, you know, cross 
     8518:20:30 <ddarius> Of course, the dual of v will be a blade that represents the hyperplane orthogonal to it already... 
     8618:20:50 <aristid> so google found IOSpec for me http://www.cse.chalmers.se/~wouter/repos/IOSpec/index.html 
     8718:20:53 <ddarius> b ^ v = b . dual(v) 
     8818:21:08 <byorgey> ddarius: where can I read about this? 
     8918:21:28 <Veinor> byorgey: for each basis vector b, compute b - proj_v b 
     9018:21:36 <byorgey> can I apply affine transformations to such duals? 
     9118:21:47 <Veinor> (I don't know anything about vector spaces besides R^n) 
     9218:22:42 <ddarius> http://geocalc.clas.asu.edu/  (mathematical/physics), http://www.mrao.cam.ac.uk/~clifford/ (physics), http://www.science.uva.nl/ga/ (computer science) 
     9318:22:47 <ddarius> byorgey: ^ 
     9418:23:36 <benmachine> haskell-src-exts' fixity resolution is super-brokwn 
     9518:23:39 <benmachine> *broken 
     9618:23:45 <benmachine> well, fairly broken anyway 
     9718:24:12 <ddarius> byorgey: I recommend starting here: http://geocalc.clas.asu.edu/html/Intro.html and here: http://www.science.uva.nl/ga/publications/index.html 
     9818:25:14 <ddarius> byorgey: This deceptively named paper is rather useful a bit later on: http://www.science.uva.nl/ga/publications/content_publications.html#leo_inner 
     99}}} 
     100 
    70101== Inverse transpose == 
    71102