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From: hart@uicbert.eecs.uic.edu (John Hart)
Newsgroups: alt.fractals,sci.math,comp.graphics
Subject: Re: Log of Quaternion / 3d Fractal raytracer / distance estimator
Message-ID: <1991May10.145516.28956@uicbert.eecs.uic.edu>
Date: 10 May 91 14:55:16 GMT
References: <1991Apr30.074427.29894@milton.u.washington.edu> <1991May3.085302.18527@zorch.SF-Bay.ORG> <1991May3.183944.14124@milton.u.washington.edu>
Organization: University of Illinois at Chicago
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Oops! Looking at equation (7) of Ray Tracing Deterministic 3-D Fractals
(Computer Graphics 23, 3 (1989) pp. 289-296) we have

d(z) = log G(z) sinh(G(z))/(2 e^G(z) |G'(z)|                          (7)

which is approximated using sinh(G(z)) = G(z) and e^G(z) = 1 for small G(z)
(values of z close to the set). Furthermore we replace G(z) with |f^n(z)|
giving us

d(z) = log |f^n(z)| |f^n(z)|/(2 |f'^n(z)|)                            (8)

which is the same as in the paper except that I forgot the absolute value
symbols for the log. Essentially, this is the same formula as given in The
Science of Fractal Images for complex values. I just tried it in the
quaternions and expected it to work. It does but I still remain clueless
as to why or how.

Good luck on your ray tracer. I never published the source code because it
was written and optimized for the Pixel Machine and is quite ugly. Its rather
straight forward and shouldn't be too difficult to prototype. To check your
results, try z^2 + 0.2809 + 0.53i (figure 1a) and z^2 - 1 (a surface of
revolution).

Alan Norton will be talking about these sets this year at SIGGRAPH at the
Fractal Models in Computer Graphics advanced course. His notes have some
pictures of quaternion Julia sets and a 3-D Mandelbrot set that have never
before been published.


-John C. Hart
 Electronic Visualization Laboratory
 EECS Dept. M/C 154
 University of Illinois at Chicago
 Chicago, IL 60680-4348
 office: (312)996-3002 lab: (312)996-5909
 hart@uicbert.eecs.uic.edu