RE: [Algorithms] transforming vectors...
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From: Greg S. <gr...@st...> - 2003-03-31 15:56:26
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A good resource is here: http://www.gignews.com/realtime020100.htm -----Original Message----- From: gda...@li... [mailto:gda...@li...]On Behalf Of Per Vognsen Sent: Monday, March 31, 2003 9:00 AM To: gda...@li... Subject: Re: [Algorithms] transforming vectors... ----- Original Message ----- From: "Jim Offerman" <j.o...@cr...> To: <gda...@li...> Sent: Monday, March 31, 2003 1:13 PM Subject: [Algorithms] transforming vectors... > While we're all in math-theory mode... I just want to check something: > > When transforming vectors (i.e. directions, normals, etc.) - as opposed > to points - you need to use inverse-transpose of the original No. When transforming a vector you simply use the matrix and not its inverse transpose. However, note that normals are not vectors; that's why we need to give them special treatment! In the very special case of three-dimensional Euclidean space, it is okay to pretend that normals are vectors as long as we observe certain rules concerning their behavior under transformations. Physicists call these gadgets pseudovectors. Clifford-algebraists call them bivectors. Anyway, let's try to work out how the normal "vector" transforms when the points on the plane are transformed. My claim is that it is transformed by the inverse transpose. Here's a proof: Let's assume for convenience that the plane in consideration goes through the origin; then its equation is n.x = 0 where n is the normal vector. Now a linear transformation given by a matrix A is applied to all points on the plane, i.e. all points x for which n.x = 0. Now we want to figure out how n transforms. Thus we seek a matrix B so that n.x = 0 if and only if (B n).(A x) = 0. Note that the dot product on the left-hand side can be written as (B n)^t (A x) = n^t B^t A x. Thus n.x = 0 = n^t x if and only if B^t = A^(-1) which implies B = (A^(-1))^t. Done. Hope this helps. Per ------------------------------------------------------- This SF.net email is sponsored by: ValueWeb: Dedicated Hosting for just $79/mo with 500 GB of bandwidth! No other company gives more support or power for your dedicated server http://click.atdmt.com/AFF/go/sdnxxaff00300020aff/direct/01/ _______________________________________________ GDAlgorithms-list mailing list GDA...@li... https://lists.sourceforge.net/lists/listinfo/gdalgorithms-list Archives: http://sourceforge.net/mailarchive/forum.php?forum_ida88 |