Re: [Fractint] At it again...

22 May 2002

      From: "Morgan L. Owens"
Sent: Monday, May 20, 2002 9:38 AM

[Way too much interesting stuff (Circles formulas) snipped...]
...
...and while we're on the subject of Fractint formulae, can someone explain 
why it is that it's still possible after all this time for BoF to still 
contain things that haven't been reproduced in Fractint yet? You'd have 
thought that the two-dimensional Newton systems described therein would've 
been:
twodnewton{
h = real(p1)
m = imag(p1)
z=pixel:
oz=z
x=real(z),y=imag(z)
f=2*x-y-m*(x-x*x)
g=2*y-x-m*(y-y*y)
fx=2-m*(1-2*x)
gy=2-m*(1-2*y)
delta  =(-g-f*gy)/(fx*gy-1)
epsilon=(-f-g*fx)/(fx*gy-1)
nx=x+h*delta
ny=y+h*epsilon
z=nx+flip(ny)
|oz-z|>.001
}
(E.g. h=0.5, m=2)
Arrgh!
Another chance to worldwide fame missed! ;-)

As it happens, this - and the Voltera-Lottka system - where
among the first formulas I tried when I first met Fractint in
the German language edition of "Fractal Creations" back in '92.
("What? There are really fractals they *haven't* implemented?")

Well, publish or perish... 8-/

(BTW, a Newton formula of a system of 2 real equations - not
this one - was once published by Sylvie Gallet.)

Concerning your "Circles" formulas:
Some time ago I stumbled over "Wada Basins in Chaotic Scattering" and the following websites:

http://webs1152.im1.net/~dsweet/Spheres/

...and on Paul Bourke's site:

http://astronomy.swin.edu.au/~pbourke/fractals/wada/

Regards,
Gerald

P.S.
So here ("Who needs POV-Ray?") are the respective Fractint-
formulas:

------------------------- WADA.FRM ----------------------------
TetraSpheres {;Four mirrored spheres at the corners of a
              ;tetrahedron, each touching the other three.
 ;( Viewing direction is positive z )
 ;colouring: number of iterations
 ;periodicity=no
 ;p1r: z-depth of viewing ray's start (try -1.5 or 0.5)
 ;
 ;Spheres' centers
 xy1 = (-1,-0.577350269189625)              ;(-1, -1/sqrt(3))
 xy2 = (1,-0.577350269189625)               ;( 1, -1/sqrt(3))
 xy3 = (0,1.15470053837925)                 ;( 0,  2/sqrt(3))
 ;xy4 = (0,0)
 z123 = -0.408248290463863                  ;(-1/sqrt(6), 0)
 z4 = 1.22474487139159                      ;( sqrt(3/2), 0)
 pxy = pixel, pz = real(p1)
 vxy = 0, vz = 1:
   isect = 0, dist = 999999
   dz = z123 - pz, vzdz = vz*dz, dzmod = |dz|
   dxy = xy1 - pxy
   tmp1 = real(vxy)*real(dxy) + imag(vxy)*imag(dxy) + vzdz
   IF (tmp1 > 0)
     tmp2 = |dxy| + dzmod - |tmp1|
     IF (tmp2 < 1)
       dist = tmp1 - sqrt(1-tmp2)
       xy0 = xy1, z0 = z123
       isect = 1
     ENDIF
   ENDIF
   dxy = xy2 - pxy
   tmp1 = real(vxy)*real(dxy) + imag(vxy)*imag(dxy) + vzdz
   IF (tmp1 > 0)
     tmp2 = |dxy| + dzmod - |tmp1|
     IF (tmp2 < 1)
       tmp1 = tmp1 - sqrt(1-tmp2)
       IF (tmp1 < dist)
         dist = tmp1
         xy0 = xy2, z0 = z123
         isect = 1
       ENDIF
     ENDIF
   ENDIF
   dxy = xy3 - pxy
   tmp1 = real(vxy)*real(dxy) + imag(vxy)*imag(dxy) + vzdz
   IF (tmp1 > 0)
     tmp2 = |dxy| + dzmod - |tmp1|
     IF (tmp2 < 1)
       tmp1 = tmp1 - sqrt(1-tmp2)
       IF (tmp1 < dist)
         dist = tmp1
         xy0 = xy3, z0 = z123
         isect = 1
       ENDIF
     ENDIF
   ENDIF
   dxy = -pxy, dz = z4 - pz
   tmp1 = real(vxy)*real(dxy) + imag(vxy)*imag(dxy) + vz*dz
   IF (tmp1 > 0)
     tmp2 = |dxy| + |dz| - |tmp1|
     IF (tmp2 < 1)
       tmp1 = tmp1 - sqrt(1-tmp2)
       IF (tmp1 < dist)
         dist = tmp1
         xy0 = 0, z0 = z4
         isect = 1
       ENDIF
     ENDIF
   ENDIF
   IF (isect)
     pxy = pxy + dist*vxy, pz = pz + dist*vz
     nxy = pxy - xy0, nz = pz - z0
     tmp1 = sqrt(|nxy|+|nz|), nxy = nxy/tmp1, nz = nz/tmp1
     tmp1 = (real(vxy)*real(nxy)+imag(vxy)*imag(nxy)+vz*nz)*2
     vxy = vxy - tmp1*nxy, vz = vz - tmp1*nz
     ;tmp1 = sqrt(|vxy|+|vz|), vxy = vxy/tmp1, vz = vz/tmp1
   ENDIF
 isect == 1
 }

TetraSpheresBas {;Four mirrored spheres at the corners of a
                 ;tetrahedron, each touching the other three.
 ;( Viewing direction is positive z )
 ;colouring: on which side of the tetrahedron the ray escapes
 ;outside=summ, periodicity=no
 ;p1r: z-depth of viewing ray's start (try -1.5 or 0.5)
 ;
 ;Spheres' centers
 xy1 = (-1,-0.577350269189625)                ;(-1, -1/sqrt(3))
 xy2 = (1,-0.577350269189625)                 ;( 1, -1/sqrt(3))
 xy3 = (0,1.15470053837925)                   ;( 0,  2/sqrt(3))
 ;xy4 = (0,0)
 z123 = -0.408248290463863                    ;(-1/sqrt(6), 0)
 z4 = 1.22474487139159                        ;( sqrt(3/2), 0)
 ;Unit vectors towards tetrahedron's sides
 e1xy = (0.816496580927725,0.816496580927725) ;(sqrt(2/3),sqrt(2/3))
 e2xy = (-0.816496580927725,0.816496580927725);(-sqrt(2/3), sqrt(2/3))
 e3xy = (0,-0.942809041582067)                ;(0,-2*sqrt(2)/3)
 ;e4xy = (0,0)
 ;e123z = (1/3, 0)
 ;e4z = (-1, 0)
 pxy = pixel, pz = real(p1)
 vxy = 0, vz = j = 1:
   isect = 0, dist = 999999
   dz = z123 - pz, vzdz = vz*dz, dzmod = |dz|
   dxy = xy1 - pxy
   tmp1 = real(vxy)*real(dxy) + imag(vxy)*imag(dxy) + vzdz
   IF (tmp1 > 0)
     tmp2 = |dxy| + dzmod - |tmp1|
     IF (tmp2 < 1)
       dist = tmp1 - sqrt(1-tmp2)
       xy0 = xy1, z0 = z123
       isect = 1
     ENDIF
   ENDIF
   dxy = xy2 - pxy
   tmp1 = real(vxy)*real(dxy) + imag(vxy)*imag(dxy) + vzdz
   IF (tmp1 > 0)
     tmp2 = |dxy| + dzmod - |tmp1|
     IF (tmp2 < 1)
       tmp1 = tmp1 - sqrt(1-tmp2)
       IF (tmp1 < dist)
         dist = tmp1
         xy0 = xy2, z0 = z123
         isect = 1
       ENDIF
     ENDIF
   ENDIF
   dxy = xy3 - pxy
   tmp1 = real(vxy)*real(dxy) + imag(vxy)*imag(dxy) + vzdz
   IF (tmp1 > 0)
     tmp2 = |dxy| + dzmod - |tmp1|
     IF (tmp2 < 1)
       tmp1 = tmp1 - sqrt(1-tmp2)
       IF (tmp1 < dist)
         dist = tmp1
         xy0 = xy3, z0 = z123
         isect = 1
       ENDIF
     ENDIF
   ENDIF
   dxy = -pxy, dz = z4 - pz
   tmp1 = real(vxy)*real(dxy) + imag(vxy)*imag(dxy) + vz*dz
   IF (tmp1 > 0)
     tmp2 = |dxy| + |dz| - |tmp1|
     IF (tmp2 < 1)
       tmp1 = tmp1 - sqrt(1-tmp2)
       IF (tmp1 < dist)
         dist = tmp1
         xy0 = 0, z0 = z4
         isect = 1
       ENDIF
     ENDIF
   ENDIF
   IF (isect)
     pxy = pxy + dist*vxy, pz = pz + dist*vz
     nxy = pxy - xy0, nz = pz - z0
     tmp1 = sqrt(|nxy|+|nz|), nxy = nxy/tmp1, nz = nz/tmp1
     tmp1 = (real(vxy)*real(nxy)+imag(vxy)*imag(nxy)+vz*nz)*2
     vxy = vxy - tmp1*nxy, vz = vz - tmp1*nz
     ;tmp1 = sqrt(|vxy|+|vz|), vxy = vxy/tmp1, vz = vz/tmp1
   ELSE
     col = 5
     dist = 999999
     pz3 = pz/3, vz3 = vz/3        ;pz * e123z, vz * e123z
     tmp1 = real(vxy)*real(e1xy) + imag(vxy)*imag(e1xy) + vz3
     IF (tmp1 > 0)
       tmp2 = real(pxy)*real(e1xy) + imag(pxy)*imag(e1xy) + pz3
       dist = (1.5-tmp2)/tmp1
       col = 1
     ENDIF
     tmp1 = real(vxy)*real(e2xy) + imag(vxy)*imag(e2xy) + vz3
     IF (tmp1 > 0)
       tmp2 = real(pxy)*real(e2xy) + imag(pxy)*imag(e2xy) + pz3
       tmp1 = (1.5-tmp2)/tmp1
       IF (tmp1 <  dist)
         dist = tmp1
         col = 2
       ENDIF
     ENDIF
     tmp1 = real(vxy)*real(e3xy) + imag(vxy)*imag(e3xy) + vz3
     IF (tmp1 > 0)
       tmp2 = real(pxy)*real(e3xy) + imag(pxy)*imag(e3xy) + pz3
       tmp1 = (1.5-tmp2)/tmp1
       IF (tmp1 <  dist)
         dist = tmp1
         col = 3
       ENDIF
     ENDIF
     IF (vz < 0)
       IF ((-1.5-pz)/vz <  dist)
         col = 4
       ENDIF
     ENDIF
     z = col - j
   ENDIF
   j = j + 1
 isect == 1
 }

---------------------------------------------------------------

Re: [Fractint] At it again...

Gerald K. Dobiasovsky