Kali


« on: March 26, 2012, 07:05:23 PM » 

I need some help on how to implement z^{1} (or 1/z) using triplex, I'm trying to render a formula and it requires this operation (z+1/z+c).
In complex, 1/z = conjugate(z)/z^{2}  I tried in 3D with conjugate(z)=z.xz.yz.z but the result is more like a quaternion (the "revolved brot" one)
I'll appreciate any clues on how to do it with tricomplex system.
Kali



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DarkBeam
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« Reply #1 on: March 26, 2012, 07:09:32 PM » 

I need some help on how to implement z^{1} (or 1/z) using triplex, I'm trying to render a formula and it requires this operation (z+1/z+c).
In complex, 1/z = conjugate(z)/z^{2}  I tried in 3D with conjugate(z)=z.xz.yz.z but the result is more like a quaternion (the "revolved brot" one)
I'll appreciate any clues on how to do it with tricomplex system.
Kali Me too, but I think it's impossible keeping it coherent with WN standard theory



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kram1032


« Reply #2 on: March 26, 2012, 07:48:51 PM » 

Divisions have been defined... According to bugmans's original formulation:
x/y=[x_{1},x_{2},x_{3}]/[y_{1},y_{2},y_{3}] = [x_{1},x_{2},x_{3}]*[y_{1},y_{2},y_{3}]^{1} = 1/R_{y}^{2}[(x_{1}y_{1}+x_{2}y_{2})(1+(x_{3}y_{3})/(r_{x}r_{y})),(x_{2}y_{1}x_{1}y_{2})(1+(x_{3}y_{3})/(r_{x}r_{y})),r_{y}x_{3}r_{x}y_{3}]
with
R_{y} = Sqrt(y1^{2}+y2^{2}+y3^{2}) r_{x} = Sqrt(x1^{2}+x2^{2}) r_{y} = Sqrt(y1^{2}+y2^{2})
By putting x=(1,0,0), you should get exactly what you want.


« Last Edit: March 26, 2012, 09:19:39 PM by kram1032 »

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Kali


« Reply #3 on: March 26, 2012, 09:00:59 PM » 

By putting x=(1,0,0), you should get exactly what you want.
Thanks. If I get it right it should simplify to: 1/Ry ^{2}[y1,y2/(rx.ry),rx.y3)] Also rx=Sqrt(1) I tried it with the formula I mentioned (z+1/z+c with initial z=1) and something that could be interesting emerged... but now I have to deal with the DE



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kram1032


« Reply #4 on: March 26, 2012, 09:21:25 PM » 

well... not quite, I did a small mistake with the ()s... It would be:
1/R_{y}^{2} [y1,y2,y3]
Actually kind of a weird inverse. But I guess, it's a weird algebra.



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Kali


« Reply #5 on: March 26, 2012, 10:12:35 PM » 

well... not quite, I did a small mistake with the ()s...
Oh, that's the cause of the DE going crazy It would be:
1/R_{y}^{2} [y1,y2,y3]
So... it's the same as I first posted: In complex, 1/z = conjugate(z)/z^{2}  I tried in 3D with conjugate(z)=z.xz.yz.z but the result is more like a quaternion (the "revolved brot" one)
conjugate(z)/z ^{2} = 1/z ^{2} [z.x,z.y,z.z]



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DarkBeam
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« Reply #6 on: March 26, 2012, 11:46:38 PM » 

not a good inversion trust me



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kram1032


« Reply #7 on: March 27, 2012, 09:38:53 AM » 

Well, it's the inversion that would make sense... You could aribitarily define something else but whatever you do, it wouldn't work as expected, most likely...
If you just take the pseudoinverse of a standard vector (x,y,z) , what you get is 1/r (x,y,z)... Not very exciting either but maybe that works better for you?



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DarkBeam
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« Reply #8 on: March 27, 2012, 10:12:59 AM » 

That's ok, it's coherent with the theory but the resulting fractals are not very good. That's what I meant with "impossible" Probably the best thing should be to study a new number set that works good with neg powers.



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Kali


« Reply #9 on: March 27, 2012, 10:39:06 AM » 

I was trying to do a 3D version of this brot (z+1/z+c), only works with initial z=1
See attached image... I was curious on how it could look as a Mandelbulb but no good results yet... with the previous methods only a revolved brot around the xaxis,
some more ideas anyone?



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DarkBeam
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« Reply #10 on: March 27, 2012, 10:45:21 AM » 

That is Talis formula, already implemented in Mb "ages" ago, not good in 3D



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Kali


« Reply #11 on: March 27, 2012, 11:14:14 AM » 

That is Talis formula, already implemented in Mb "ages" ago, not good in 3D Oh, I didn't know it was called Talis, and I don't even recall where I first saw this formula, but I think it was on fracmonk's megathread. But I won't give up just because your "agesago" MB3D implementation is not good Btw, maybe is a DE problem? in Fragmentarium, in order to render the "revolved brot" I mentioned, I must set the default mandelbulb DE to 0,5 pow and scale it by 4. Don't ask me why, you know, I'm more an intuitive "bruteforce" fractalist rather than a mathematician



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cKleinhuis


« Reply #12 on: March 27, 2012, 12:38:56 PM » 

is this really related to triplex math ? if not please consider opening up a new thread...



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Kali


« Reply #13 on: March 27, 2012, 07:41:17 PM » 

is this really related to triplex math ? if not please consider opening up a new thread...
It was at first, but it didn't work, and now we are a bit offtopic I guess... will open a new thread for related posts, sorry Chris.



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cKleinhuis


« Reply #14 on: March 27, 2012, 10:05:45 PM » 

i or you can split this off, i would like to lock this thread because it has really valuable information in it ...!



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