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Tuesday, November 11, 2014

Making an orthonormal basis from a unit vector

Almost all ray tracing programs have a routine that creates an orthonormal basis from a unit vector n.  I am messing with my Swift ray tracer and just looked around for the "best known method".  This routine is always annoyingly inelegant in my code so I was hoping for something better.  Some googling yeilded a lovely paper by Jeppe Revall Frisvad.   He builds the basis with no normalization required:

It's not obvious, but those are orthonormal.  The catch is the case where 1-nz = 0 so unfortunately an if is needed:

Still even with the branch it is very nice.  I was hoping that a careful use of IEEE floating point rules would allow the branch to be avoided but I don't see it.  But perhaps some clever person will see a reconstructing.   The terms 0*0/0 should be a zero in principle and then all works out.  The good news is that for n.z near 1.0 the stability looks good.

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