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.x-z.y-z.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
|
|
|
Logged
|
|
|
|
DarkBeam
Global Moderator
Fractal Senior
Posts: 2512
Fragments of the fractal -like the tip of it
|
|
« 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.x-z.y-z.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 W-N standard theory
|
|
|
Logged
|
No sweat, guardian of wisdom!
|
|
|
kram1032
|
|
« Reply #2 on: March 26, 2012, 07:48:51 PM » |
|
Divisions have been defined... According to bugmans's original formulation:
x/y=[x1,x2,x3]/[y1,y2,y3] = [x1,x2,x3]*[y1,y2,y3]-1 = 1/Ry2[(x1y1+x2y2)(1+(x3y3)/(rxry)),(x2y1-x1y2)(1+(x3y3)/(rxry)),ryx3-rxy3]
with
Ry = Sqrt(y12+y22+y32) rx = Sqrt(x12+x22) ry = Sqrt(y12+y22)
By putting x=(1,0,0), you should get exactly what you want.
|
|
« Last Edit: March 26, 2012, 09:19:39 PM by kram1032 »
|
Logged
|
|
|
|
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
|
|
|
Logged
|
|
|
|
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/Ry2 [y1,-y2,-y3]
Actually kind of a weird inverse. But I guess, it's a weird algebra.
|
|
|
Logged
|
|
|
|
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/Ry2 [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.x-z.y-z.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]
|
|
|
Logged
|
|
|
|
DarkBeam
Global Moderator
Fractal Senior
Posts: 2512
Fragments of the fractal -like the tip of it
|
|
« Reply #6 on: March 26, 2012, 11:46:38 PM » |
|
not a good inversion trust me
|
|
|
Logged
|
No sweat, guardian of wisdom!
|
|
|
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 pseudo-inverse 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?
|
|
|
Logged
|
|
|
|
DarkBeam
Global Moderator
Fractal Senior
Posts: 2512
Fragments of the fractal -like the tip of it
|
|
« 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.
|
|
|
Logged
|
No sweat, guardian of wisdom!
|
|
|
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 x-axis,
some more ideas anyone?
|
|
|
Logged
|
|
|
|
DarkBeam
Global Moderator
Fractal Senior
Posts: 2512
Fragments of the fractal -like the tip of it
|
|
« Reply #10 on: March 27, 2012, 10:45:21 AM » |
|
That is Talis formula, already implemented in Mb "ages" ago, not good in 3D
|
|
|
Logged
|
No sweat, guardian of wisdom!
|
|
|
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 "ages-ago" 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 "brute-force" fractalist rather than a mathematician
|
|
|
Logged
|
|
|
|
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...
|
|
|
Logged
|
---
divide and conquer - iterate and rule - chaos is No random!
|
|
|
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 off-topic I guess... will open a new thread for related posts, sorry Chris.
|
|
|
Logged
|
|
|
|
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 ...!
|
|
|
Logged
|
---
divide and conquer - iterate and rule - chaos is No random!
|
|
|
|