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From: bungi@milton.u.washington.edu (Timothy J. Wood)
Newsgroups: alt.fractals,sci.math,comp.graphics
Subject: Log of Quaternion / 3d Fractal raytracer / distance estimator
Message-ID: <1991May3.183944.14124@milton.u.washington.edu>
Date: 3 May 91 18:39:44 GMT
References: <1991Apr30.074427.29894@milton.u.washington.edu> <1991May3.085302.18527@zorch.SF-Bay.ORG>
Reply-To: bungi@u.washington.edu
Organization: University of Washington, Seattle
Lines: 38



   In 'Ray Tracing Deterministic 3-D Fractals' (John C. Hart, ACM
SIGGRAPH 89) a formula for estimating a lower bound on the distance from a
point in the space of quaternions to the surface of the 4-D fractal is
given as : 
            n
          |f (z)|                        n
  d(z) = ---------  log(f^n(z));  where f (z) is f(f(...(f(z))..)  with n "f's"
            'n
         2|f  (z)|


   I can handle the magnitudes, derivatives, and compositions, but what is
the log of a quaternion?  I have Hamilton's 'Elemente der Quaternionen'
(1882, and _very_ German), which has something like:

  lq = lTq + lUq

   which really doesn't tell _me_ much :)  What are T and U?  And why is it
define recursively?

   Am I to assume that the log of a quaternion is a real?  If not, then is
d(z) going to be a quaternion rather than a real?  How would that be a
'distance'?

   Any help on logs of quaternions or the distance estimator function would
be excellent... i'll even give you a copy of the software (distributed
fractal raytracer) when i'm done.  Oh drats.  it's going to be public
domain  :)

  Well, send me help anyway.

thanks
-- 
-------------------------------------------------------------------------------
        Timothy Wood : bungi@u.washington.edu, tjwood@cs.washington.edu
                   ... alice everyday brings sunshine ...