Frango com Nata <odo### [at] yahoocombr> wrote:
>
> Since them, I've read a little about quaternions, namely a page titled
> "Doing Physics with Quaternions"
> (http://world.std.com/~sweetser/quaternions/qindex/qindex.html), ...

interesting link, thank you. Haven't got enough time to browse it seriously
just now, too much RealLife what with christmas suddenly being so close
again, but I will try to look into it in the near future.

>
>  > One reason might be, that the "classical" Mandelbrot sets of formulas
>  > pow(z,n)+c should look quite boring because any 3d-cut through them
>  > orthogonal to the real axis is just a sphere ...
>
>
> Well, I hadn't taken the care to realize that :) But of course, it
> wouldn't be the case of the fractal defined by the same iteration
> formula q[n+1]=f(q[n])+c as Mandelbrot's, but where you could specify an
> arbitrary quaternion q[0] as the initial term, instead of just setting
> it to zero, so achieving the same single quaternion degree of freedom
> we've got when selecting a Julia fractal for some given function.
>

Well, that might give some interesting pictures, but there is mathematical
reason for "THE" Mandelbrot Set being the one with z(0) = 0; it's simply
the only one with meaning - but you probably know that, don't you ;-)

Talking about shortcomings of POV-Ray's incorporation of quaternions there
is the additional problem that you can't choose the 3d-cut you want to see
freely ( see the docs; w != 0  !!!), and that the pictures are distorted !
Last point is easily seen by depicting sqr Julia with c=0; as with complex
numbers, this defines a 4D-sphere regardless of iteration-depth, and any
"plane" 3d-cut through a 4D-sphere is a 3D-sphere ( or a point or empty ).
BUT you get ellipsoids ...

Ah well, I fear that if we say too much, we will be asked to implement that
ourselves or shut up - that has happened before to very small minorities
asking for a lot of programmer's work - and rightfully so, I fear.

Have a nice weekend

Karl

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