Re: [Fractint] Re: quaternion product correction

4 Nov 2005

      Gerald,

An amazing image... Hell of a job! It'll probably take me a while to fathom the mechanisms you employ in your formula to accomplish this. Could you say a few words about the theory involved? It's been about three years since we began to explore the 'T-set' idea on this list, and it comes to a fascinating fruition in Fractint at last.

Ever since Paul Bourke's POVray renditions, I've had some variants on this 3-D M-set in mind, and I've included a par file of the complementary 'C-set' below. Some 2-D renderings of this object are facilitated by a simple toggle in certain M-set formulas. E.g.,

MsetCset1 {;toggle fn1 between 'ident' and 'conj'
z=0, c=(pixel):
  z=fn1(z^2+c)
z < 4 }

MsetCset2 {;toggle p1 between '1' and '-1'
x=real(pixel),y=imag(pixel)
x1=y1=0:
  a=x1^2-y1^2
  b=p1*(2*x1*y1)
  x1=a+x,y1=b+y
z=x1^2+y1^2
z < 4 }

To get this effect in the 'T-set_in_3d' formula, I changed the line
'x1 = a + cx, y1 = b + cy, z1 = c + cz'
to read
'x1 = c + cx, y1 = b + cy, z1 = a + cz'.
Any two-element transposition of a b c gives the same effect, just different orientations.

Thanks again for your innovations.

Ciao, Russell

---------------------------- Parameters below ------------------------------

C-set_in_3d {;Greyscale depth image
              ;
  reset=2004 type=formula formulafile=test.frm
  formulaname=3d_C-set passes=g
  center-mag=-0.68046/-0.24858/0.9363296
  params=0/0/-0.75/0.75/0/0/0/4/150/253 float=y maxiter=2000000000
  outside=summ periodicity=0
  colors=000zzzyzzzyzzyyyyyxyyyxzyxxxxxwxxxwyxwwwwwvwwwvxwvvvvvuvvvuwvuuuu\
  utuuutvutttttstttsutsssssrsssrtsrrrrrqrrrqsrqqqqqpqqqprqpppppopppoqpoooo\
  onooonponnnnnmnnnmonmmmmmlmmmlnmlllllklllkmlkkkkkjkkkjlkjjjjjijjjikjiiii\
  ihiiihjihhhhhghhhgihgggggfgggfhgfffffefffegfeeeeedeeedfedddddcdddcedcccc\
  cbcccbdcbbbbbabbbacbaaaaa`aaa`ba`````_```_a`_____Z___Z`_ZZZZZYZZZY_ZYYYY\
  YXYYYXZYXXXXXWXXXWYXWWWWWVWWWVXWVVVVVUVVVUWVUUUUUTUUUTVUTTTTTSTTTSUTSSSS\
  SRSSSRTSRRRRRQRRRQSRQQQQQPQQQPRQPPPPPOPPPOQPOOOOONOOONPONNNNNMNNNMONMMMM\
  MLMMMLNMLLLLLKLLLKMLKKKKKJKKKJLKJJJJJIJJJIKJIIIIIHIIIHJIHHHHHGHHHGIHGGGG\
  GFGGGFHGFFFFFEFFFEGFEEEEEDEEEDFEDDDDDCDDDCEDCCCCCBCCCBDCBBBBBABBBACBAAAA\
  A9AAA9BA9999989998A98888878887987777767776876666656665765555545554654444\
  4344435433333233324322222122213211111011102100000000PPP
  }

frm:3d_C-set {;periodicity=no, outside=summ
              ;maxit > p5real*(p5imag+1)
  ;--------------------------------------------
  ;p1real: Rotation about x-axis (1st rotation)
  ;p1imag: Rotation about y-axis (2nd rotation)
  ;p2real: Far clipping plane
  ;p2imag: Near clipping plane
  ;p3real: x1(0)
  ;p3imag: y1(0)
  ;p4real: z1(0)
  ;p4imag: Bailout
  ;p5real: Maxiter per slice
  ;p5imag: Number of slices - 1
  ;--------------------------------------------
  ;
  bailout = imag(p4), tiefnum = imag(p5)
  delta = (real(p2)-imag(p2))/tiefnum
  tmp = pi/180
  rotXax = exp(flip(real(p1)*tmp)), rotYax = exp(flip(imag(p1)*tmp))
  ;
  HPixXY = rotYax
  VPixZ = real(rotXax)
  VPixXY = flip(conj(rotYax))
  NXY = VPixZ*VPixXY
  NZ = imag(conj(rotXax))
  VPixXY = -NZ*VPixXY
  ;
  tmp = NXY*imag(p2) + HPixXY*real(pixel) + VPixXY*imag(pixel)
  cx = cx0 = real(tmp), cy = cy0 = imag(tmp)
  cz = cz0 = NZ*imag(p2) + VPixZ*imag(pixel)       ;HPixZ -> 0
  tmp = NXY*delta, dcx = real(tmp), dcy = imag(tmp)
  dcz = NZ*delta
  x1 = real(p3)
  y1 = imag(p3)
  z1 = real(p4)
  j = m = i = 0:
    a = sqr(x1) + 2*y1*z1
    b = sqr(z1) + 2*x1*y1
    c = sqr(y1) + 2*x1*z1
    x1 = c + cx, y1 = b + cy, z1 = a + cz
    IF (bailout >= (sqr(x1)+sqr(y1)+sqr(z1)))
      i = i + 1
    ELSE
      i = 0
      m = m + 1
      cx = cx0 = cx0 + dcx
      cy = cy0 = cy0 + dcy
      cz = cz0 = cz0 + dcz
      x1 = real(p3), y1 = imag(p3), z1 = real(p4)
    ENDIF
    z = m - j
    j = j + 1
  tiefnum >= m && p5 >= i
  }

---------------------------- End of parameters -----------------------------

----- Original Message -----
From: "Gerald K. Dobiasovsky" <gerald.dob@aon.at>
To: "Fractint and General Fractals Discussion" <fractint@mailman.xmission.com>
Subject: [Fractint] Re: quaternion product correction
Date: Thu, 3 Nov 2005 02:45:56 +0100
...
[Russell Walsmith, regarding Triternions and the rendereing thereof]
Paul Bourke's rendering seems to use some density cloud function (probably
the number of iterations accumulated while stepping along the 
line-of-sight for each pixel), so the object's outside is structured
as well.
All I can offer for the moment is a greyscale depth rendering similar to
Fractint's Julibrot type (that is, the object is solid and the surrounding
space is transparent).
You can achieve a kind of outside density rendering by switching from
outside=summ to outside=iter, but the colors will wrap around many times,
which really doesn't look good.
Gerald
---------------------------- Parameters below ------------------------------
T-set_in_3d {;Greyscale depth image
              ;
  reset=2004 type=formula formulafile=test.frm
  formulaname=Rot3d_T-set passes=g
  center-mag=-0.68046/-0.24858/0.9363296
  params=0/0/-0.75/0.75/0/0/0/4/150/253 float=y maxiter=2000000000
  outside=summ periodicity=0
  colors=000zzzyzzzyzzyyyyyxyyyxzyxxxxxwxxxwyxwwwwwvwwwvxwvvvvvuvvvuwvuuuu\
  utuuutvutttttstttsutsssssrsssrtsrrrrrqrrrqsrqqqqqpqqqprqpppppopppoqpoooo\
  onooonponnnnnmnnnmonmmmmmlmmmlnmlllllklllkmlkkkkkjkkkjlkjjjjjijjjikjiiii\
  ihiiihjihhhhhghhhgihgggggfgggfhgfffffefffegfeeeeedeeedfedddddcdddcedcccc\
  cbcccbdcbbbbbabbbacbaaaaa`aaa`ba`````_```_a`_____Z___Z`_ZZZZZYZZZY_ZYYYY\
  YXYYYXZYXXXXXWXXXWYXWWWWWVWWWVXWVVVVVUVVVUWVUUUUUTUUUTVUTTTTTSTTTSUTSSSS\
  SRSSSRTSRRRRRQRRRQSRQQQQQPQQQPRQPPPPPOPPPOQPOOOOONOOONPONNNNNMNNNMONMMMM\
  MLMMMLNMLLLLLKLLLKMLKKKKKJKKKJLKJJJJJIJJJIKJIIIIIHIIIHJIHHHHHGHHHGIHGGGG\
  GFGGGFHGFFFFFEFFFEGFEEEEEDEEEDFEDDDDDCDDDCEDCCCCCBCCCBDCBBBBBABBBACBAAAA\
  A9AAA9BA9999989998A98888878887987777767776876666656665765555545554654444\
  4344435433333233324322222122213211111011102100000000PPP
  }
frm:Rot3d_T-set {;periodicity=no, outside=summ
              ;maxit > p5real*(p5imag+1)
  ;--------------------------------------------
  ;p1real: Rotation about x-axis (1st rotation)
  ;p1imag: Rotation about y-axis (2nd rotation)
  ;p2real: Far clipping plane
  ;p2imag: Near clipping plane
  ;p3real: x1(0)
  ;p3imag: y1(0)
  ;p4real: z1(0)
  ;p4imag: Bailout
  ;p5real: Maxiter per slice
  ;p5imag: Number of slices - 1
  ;--------------------------------------------
  ;
  bailout = imag(p4), tiefnum = imag(p5)
  delta = (real(p2)-imag(p2))/tiefnum
  tmp = pi/180
  rotXax = exp(flip(real(p1)*tmp)), rotYax = exp(flip(imag(p1)*tmp))
  ;
  HPixXY = rotYax
  VPixZ = real(rotXax)
  VPixXY = flip(conj(rotYax))
  NXY = VPixZ*VPixXY
  NZ = imag(conj(rotXax))
  VPixXY = -NZ*VPixXY
  ;
  tmp = NXY*imag(p2) + HPixXY*real(pixel) + VPixXY*imag(pixel)
  cx = cx0 = real(tmp), cy = cy0 = imag(tmp)
  cz = cz0 = NZ*imag(p2) + VPixZ*imag(pixel)       ;HPixZ -> 0
  tmp = NXY*delta, dcx = real(tmp), dcy = imag(tmp)
  dcz = NZ*delta
  x1 = real(p3)
  y1 = imag(p3)
  z1 = real(p4)
  j = m = i = 0:
    a = sqr(x1) + 2*y1*z1
    b = sqr(z1) + 2*x1*y1
    c = sqr(y1) + 2*x1*z1
    x1 = a + cx, y1 = b + cy, z1 = c + cz
    IF (bailout >= (sqr(x1)+sqr(y1)+sqr(z1)))
      i = i + 1
    ELSE
      i = 0
      m = m + 1
      cx = cx0 = cx0 + dcx
      cy = cy0 = cy0 + dcy
      cz = cz0 = cz0 + dcz
      x1 = real(p3), y1 = imag(p3), z1 = real(p4)
    ENDIF
    z = m - j
    j = j + 1
  tiefnum >= m && p5 >= i
  }
---------------------------- End of parameters -----------------------------
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