Re: Custom formulas & transforms - Release

[fractalrebel]

As Dave Makin mentioned earlier there are alternate ways to calculate a quaterionion power since quat multiplication does not commute. Here are the alternate methods I have in my quaternion math library:

; Quaternian power calculated as exp(ln(Q1)*Q2)
  static func Qpower(Vector a, Vector b, Vector c)
;    Q1^Q2 = exp(ln(Q1)*Q2)
;   Find the log of Q1
    float norm = sqrt(a.m_y*a.m_y + a.m_z*a.m_z + a.m_w*a.m_w)
    if norm == 0
      norm = 1e-100
    endif
    complex z = a.m_x + flip(norm)
    z = log(z)
    float at = imag(z)/norm
    float x1 = real(z)
    float y1 = at*a.m_y
    float z1 = at*a.m_z
    float w1 = at*a.m_w
;   Multiply log by Q2
    float dot = y1*b.m_y + z1*b.m_z + w1*b.m_w
    ; calc V x V
    float y2 = z1*b.m_w - b.m_z*w1
    float z2 = -(y1*b.m_w - b.m_y*w1)
    float w2 = y1*b.m_z - b.m_y*z1

    float x2 = x1*b.m_x-dot
    y2 = y2 + x1*b.m_y + b.m_x*y1
    z2 = z2 + x1*b.m_z + b.m_x*z1
    w2 = w2 + x1*b.m_w + b.m_x*w1
;   Take the exponential
    norm = sqrt(y2*y2 + z2*z2 + w2*w2)
    if norm == 0
      norm = 1e-100
    endif
    z = x2 + flip(norm)
    z = exp(z)
    at = imag(z)/norm
    c.m_x = real(z)
    c.m_y = at*y2
    c.m_z = at*z2
    c.m_w = at*w2
  endfunc

; Quaternian power calculated as exp(Q2*ln(Q1))
  static func Qpower2(Vector a, Vector b, Vector c)
 ;    Q1^Q2 = exp(Q2*ln(Q1))
 ;   Find the log of Q1
     float norm = sqrt(a.m_y*a.m_y + a.m_z*a.m_z + a.m_w*a.m_w)
     if norm == 0
       norm = 1e-100
     endif
     complex z = a.m_x + flip(norm)
     z = log(z)
     float at = imag(z)/norm
     float x1 = real(z)
     float y1 = at*a.m_y
     float z1 = at*a.m_z
     float w1 = at*a.m_w
 ;   Multiply log by Q2
     float dot = y1*b.m_y + z1*b.m_z + w1*b.m_w
     ; calc V x V
     float y2 = -z1*b.m_w + b.m_z*w1
     float z2 = (y1*b.m_w - b.m_y*w1)
     float w2 = -y1*b.m_z + b.m_y*z1

     float x2 = x1*b.m_x-dot
     y2 = y2 + x1*b.m_y + b.m_x*y1
     z2 = z2 + x1*b.m_z + b.m_x*z1
     w2 = w2 + x1*b.m_w + b.m_x*w1

 ;   Take the exponential
     norm = sqrt(y2*y2 + z2*z2 + w2*w2)
     if norm == 0
       norm = 1e-100
     endif
     z = x2 + flip(norm)
     z = exp(z)
     at = imag(z)/norm
     c.m_x = real(z)
     c.m_y = at*y2
     c.m_z = at*z2
     c.m_w = at*w2
  endfunc

[DarkBeam]

  Thanks for the suggestions! ps. are you "reb" of the UF database? You are a genius

The quats in UF database seems to use the bicomplex theory. I use the classic Hamilton theory and for now I prefer to use it for all computations.

Now I succeeded in coding log() and exp(). The problem is that exp(log(Q)) = Q only in a zone near to origin.

So to plot the arbitrary power fractal I must cut off everything outside. Else, an ugly noise will surround the fractal and the computations will be slowed down. I hope that this will be enough  

[Kali]

I remembered that I asked Jesse for making a test with a complex-power formula, and he had problems because the distance estimation algorithm didn't work well with that. Look: http://www.fractalforums.com/index.php?topic=5252.msg26722#msg26722

[DarkBeam]

That's the 2D section of the sequence. 3D image looks completely wrong, 

[DarkBeam]

This is much better, who knows why? But high powers are destructive

[DarkBeam]

yesssss!!! Now all works fine. I simply replaced some options at the start of each function and everything is correctly plotted

Now I only have to write the final formula combining everything in one!

[ericr]

Thank you very for all  this formulas Thank you

ERIC

[DarkBeam]

Thank you very for all  this formulas Thank you

ERIC

Without your suggestion of the source code of that old program I would never be able to write anything. The base code is very tricky...

[ericr]

can i give you other idea or have you not the time




example: cos(q)=(exp(qi)+exp(-qi))/2   the cosine in quaernions !
            sin(q)= (exp(qi)-exp(-qi))/2

etc .....      the moste difficulte is done
thank you eric

[DarkBeam]

can i give you other idea or have you not the time




example: cos(q)=(exp(qi)+exp(-qi))/2   the cosine in quaernions !
            sin(q)= (exp(qi)-exp(-qi))/2

etc .....      the moste difficulte is done
thank you eric

Yesterday I found some quat math libraries. In reality;

sin(a+bi) = sin a * cosh b + i ( cos a * sinh b ) (sine for a complex number)

But in our case;
m = sqrt( w^2+y^2+z^2 )
so x = sin x * cosh m (real part is simple)
but
y = ( cos x * sinh m ) / m * y
z = ( cos x * sinh m ) / m * z
w = ( cos x * sinh m ) / m * w
The division is the tricky part to code
 
(I hope I understood well the cryptic library )
I have all the codings for sinh and cosh already done in assembly ... Will see! (sin S)^2 + C must be a wonderful fractal!

[skyzyk]

Thank you very much Luca. No words to express the many thanks I have for your dedication to show us The Possibilities Are Infinite. Sorry if I've said that before but it's the same every revision. 

[DarkBeam]

Thank you very much Luca. No words to express the many thanks I have for your dedication to show us The Possibilities Are Infinite. Sorry if I've said that before but it's the same every revision. 

You are too kind

[DarkBeam]

Finally Jesse told me how to use transforms as pre-transforms! 

I wrote this in the first post, opens an immense field of new possibilities.

I will write new options for (hopefully) all transforms for complete / partial transforms now!

[visual.bermarte]

Hello, I'm testing inversions and things on GPU..interessante!!
dodeca kifs with invcylindrical-hack

[DarkBeam]

Hello, I'm testing inversions and things on GPU..interessante!!
dodeca kifs with invcylindrical-hack
<Quoted Image Removed>

why "hack"?