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标题: 复分形难题再度征解 [打印本页]

作者: 柳烟 时间: 2011-2-20 11:22 标题: 复分形难题再度征解

搞了好几遍，尝试了各种情形，仍不能得出结果：
Carr1965 {
; Updated for UF2 by Erik Reckase, March 2000
         ; Modified Sylvie Gallet frm.
init:
  z=c=pixel
  int compt=0, int limit=round(real(p1)), float bailout=4
loop:
  IF (compt>=limit)
c = (-.743380900000982,.131850030300002)
  ELSE
c = z + c/2.125
  ENDIF
  z=z*z+c
  compt=compt+1
bailout:
  |real(z)|<=bailout
default:
  title = "Carr 1965"
  periodicity = 0
  maxiter = 500
  magn = .9
  center = (0,0)
  method = multipass
  param p1
caption = "Iter Limit"
default = (100,0)
hint = "The real part of this parameter signals a change \
         in the value of c. The imaginary part is not used."
  endparam
}
UF中效果图如下：

图片附件: Fractal2.jpg (2011-2-20 12:40, 29.52 KB) / 下载次数 1639
http://inrm3d.cn/attachment.php?aid=10235&k=cc5b6dbe738ca7f1497357cda8220729&t=1715005041&sid=13u6OU

作者: 柳烟 时间: 2011-2-20 13:01

昨天干了五六遍，失败，今天干了一上午，失败了三次，终于我已经找到解决方案了，正在扫描。
未命名2.jpg

改变P1的横坐标为2，得：

图片附件: 未命名1.jpg (2011-2-23 10:22, 29.36 KB) / 下载次数 1632
http://inrm3d.cn/attachment.php?aid=10241&k=ec2a30bf5228f470787c25dddc26ec55&t=1715005041&sid=13u6OU

图片附件: 未命名2.jpg (2011-2-23 10:22, 23.71 KB) / 下载次数 1645
http://inrm3d.cn/attachment.php?aid=10242&k=63720529eda61e26a13171be95a68e4d&t=1715005041&sid=13u6OU

作者: 榕坚 时间: 2011-2-20 20:59

这个结果很有诱惑力，可是没办法完成：
pgd_Twofold2_Julia {
  ; Two-fold rotational symmetry and complementary critical
  ; point fates make for some very intertwining (and
  ; entertaining) Julia sets.
  ; f(z) = (z^2 - 1/3)/cz^3
global:
  bool foundatt1 = false
  bool foundatt2 = true
  complex att1 = 1
  complex att2 = -1
  float b = 1/@bailout
  int j = 0
  ; Locate attractor 1 goes to, if any.
  WHILE (j < #maxit)
att1 = (sqr(att1) - 1/3)/(@seed * sqr(att1) * att1)
j = j + 1
  ENDWHILE
  ; Is it really an attractor? Or did it just wander off
  ; and die somewhere? Is the attractor self-complementary
  ; or is there a distinct mirror attractor?
  j = 0
  complex a = att1
  WHILE (j < #maxit)
att1 = (sqr(att1) - 1/3)/(@seed * sqr(att1) * att1)
IF (|att1 - a| < b)
   foundatt1 = true    ; it repeated, yay!
ENDIF
att2 = (sqr(att2) - 1/3)/(@seed * sqr(att2) * att2)
IF (|att2 - a| < b)
   foundatt2 = false    ; other cp went to same.
ENDIF
j = j + 1
  ENDWHILE
init:
  z = #pixel
  bool notdone = true
  bool landedright = false
  IF ((@what == 1) && (!foundatt1))
notdone = false
  ENDIF
  IF ((@what == 2) && (!foundatt2))
notdone = false
  ENDIF
  i = 0
loop:
  z = (sqr(z) - 1/3)/(@seed * sqr(z) * z)
  IF (|z| > @bailout)
notdone = false
landedright = (@what == 0)
  ENDIF
  IF (foundatt1)
IF (|z - att1| < b)
   notdone = false
   landedright = (@what == 1)
ENDIF
  ENDIF
  IF (foundatt2)
IF (|z - att2| < b)
   notdone = false
   landedright = (@what == 2)
ENDIF
  ENDIF
  IF (i == #maxit)
IF (@what != 3)
   notdone = false
   landedright = false
ENDIF
  ENDIF
  i = i + 1
  IF (!notdone)
IF (!landedright)
   z = (501,10)  ; Tell Smooth (Generalized) 2 to color this thing solid.
ENDIF
  ENDIF
bailout:
  notdone
default:
  title = "Twofold2 Julia"
  periodicity = 0
  maxiter = 1000
  method = onepass
  magn = 0.5
  param seed
caption = "Julia seed"
default = (0.75, 0)
  endparam
  param what
caption = "Draw what"
enum = "Main basin" "Other attractor #1" "Other attractor #2" "Trapped points"
default = 0
  endparam
  param bailout
caption = "Bailout"
hint = "Escape radius (inverse is used for finite attractors)"
default = 100000.0
min = 0
  endparam
switch:
  type = "pgd_Twofold2_Mandel"
  bailout = bailout
}

图片附件: Fractal1.jpg (2011-2-20 20:59, 52.69 KB) / 下载次数 1779
http://inrm3d.cn/attachment.php?aid=10252&k=d3a35c29a77a0f66e57623d621c4c045&t=1715005041&sid=13u6OU

作者: 柳烟 时间: 2011-2-20 21:10

这个代码读起来，太费脑细胞了，UF中有几个分形代码，要么太过漫长，要么逻辑判断太多，作起来太费劲，只好望洋兴叹了。

作者: 榕坚 时间: 2011-2-22 10:51

这个应该就是分形边界的一种构造方法,把内、外部都挖空了，要通过多次迭代才能完成。

作者: 榕坚 时间: 2011-2-22 15:58

实在挡不住诱惑，虽然伤了很多脑细胞但算是有成果：

图片附件: pgd_Twofold2_Julia.JPG (2011-2-22 15:58, 39.51 KB) / 下载次数 1569
http://inrm3d.cn/attachment.php?aid=10295&k=e7ece63305c458eb46159e32948dc851&t=1715005041&sid=13u6OU

作者: 柳烟 时间: 2011-2-22 16:03

这个代码，我可读不懂了。榕兄作得不错。

作者: 榕坚 时间: 2011-2-22 16:19

7# 柳烟

在对Z迭代之前先进行两次迭代获得一些准备数值，其它的就是按照代码依样画葫芦了。另一个它的“M"集那真是没办法了（其中有数组变量不知怎么转为几何画板能完成的），分开结构非常漂亮：
pgd_Twofold2_Mandel {
  ; Two-fold rotational symmetry and complementary critical
  ; point fates make for some very intertwining (and
  ; entertaining) Julia sets.
  ; f(z) = (z^2 - 1/3)/cz^3
init:
  z = 1
  bool notdone = true
  bool landedright = false
  int i = 0
  int j = 0
  bool odd = false
  complex z_values[@zmax]
  float b = 1/@bailout
loop:
  z = (sqr(z) - 1/3)/(#pixel * sqr(z) * z)
  IF (|z| > @bailout)
notdone = false
landedright = (@what == 0)
  ENDIF
  z_values = z
  IF (odd)
IF (|z - z_values[j]| < b)
   notdone = false
   complex w = -1
   j = 0
   bool sep = true
   WHILE ((j < #maxit) && sep)
      w = (sqr(w) - 1/3)/(#pixel * sqr(w) * w)
      IF (|w - z| < b)
      sep = false
      ENDIF
      j = j + 1
   ENDWHILE
   IF (sep)
      landedright = (@what == 2)
   ELSE
      landedright = (@what == 1)
   ENDIF
ENDIF
j = j + 1
  ENDIF
  odd = !odd
  i = i + 1
  IF (i == @zmax)
IF (@what != 3)
   notdone = false
   landedright = false
ENDIF
  ENDIF
  IF (!notdone)
IF (!landedright)
   z = (501,10)  ; Tell Smooth (Generalized) 2 to color this thing solid.
ENDIF
  ENDIF
bailout:
  notdone
default:
  title = "Twofold2 Mandelbrot"
  periodicity = 0
  maxiter = 1000
  method = onepass
  param what
caption = "Draw what"
enum = "Main basin" "One finite attractor" "Two attractors" "Trapped points"
default = 0
  endparam
  param zmax
caption = "Max iters to record"
default = 1000
  endparam
  param bailout
caption = "Bailout"
hint = "Escape radius (inverse is used for finite attractors)"
default = 100000.0
min = 0
  endparam
switch:
  type = "pgd_Twofold2_Julia"
  seed = #pixel
  bailout = bailout
}

图片附件: 捕获.JPG (2011-2-22 16:19, 94.01 KB) / 下载次数 1730
http://inrm3d.cn/attachment.php?aid=10297&k=11ff32d7c1010c2af4c35d1c795cb7bd&t=1715005041&sid=13u6OU

作者: 柳烟 时间: 2011-2-22 16:34

8# 榕坚
谢谢。佩服老兄的钻研精神与思维的敏捷。UF中，每当我看到这种代码，就如珠峰挡在前面。对这种逻辑判断太多的分形，我是望而生畏。榕兄有空时，能否制成视频，详细反映出制作过程。注意有空时。另还有一个分形，是谢尔宾斯基三角形，代码也十分漫长，一时忘了在何处，我找着后发在这里。
作者: 柳烟 时间: 2011-2-23 09:15

位于gnd.ufm中，
gnd-slope-sierpinski2 {
; Based on a formula by Ramiro Perez
; Slope version by Gilles Nadeau, 2007
init:
  z1 = #pixel
  z2 = #pixel + @offset
  z3 = #pixel + flip(@offset)
  int done = 2
  float modz = 0.0
  float e1 = 0.0       ; potentials
  float e2 = 0.0
  float e3 = 0.0
  float vx = 0.0       ; normal vector
  float vy = 0.0
  float vz = 0.0
  float vd = 0.0
  float iterexp1 = 0.0
  float iterexp2 = 0.0
  float iterexp3 = 0.0

loop:

  IF ((imag(z1)>=.575*real(z1)) && (-.575*real(z1)<=imag(z1)))
z1 = 2*z1-1i
  ELSEIF (real(z1)<=0)
z1 = 2*z1+(.8660254, 0.5)
  ELSEIF (real(z1)>0)
z1 = 2*z1+(-.8660254, 0.5)
  ELSE
z1 = 2*z1
  ENDIF

  IF ((imag(z2)>=.575*real(z2)) && (-.575*real(z2)<=imag(z2)))
z2 = 2*z2-1i
  ELSEIF (real(z2)<=0)
z2 = 2*z2+(.8660254, 0.5)
  ELSEIF (real(z2)>0)
z2 = 2*z2+(-.8660254, 0.5)
  ELSE
z2 = 2*z2
  ENDIF

  IF ((imag(z3)>=.575*real(z3)) && (-.575*real(z3)<=imag(z3)))
z3 = 2*z3-1i
  ELSEIF (real(z3)<=0)
z3 = 2*z3+(.8660254, 0.5)
  ELSEIF (real(z3)>0)
z3 = 2*z3+(-.8660254, 0.5)
  ELSE
z3 = 2*z3
  ENDIF

  modz = |z1|

  iterexp1 = iterexp1 + exp(-cabs(z1))
  iterexp2 = iterexp2 + exp(-cabs(z2))
  iterexp3 = iterexp3 + exp(-cabs(z3))
  done = done + 1       ; increment iteration counter

  IF (modz > @bailout ||    \
   @everyiter ||          \
   done == #maxit + 2)    ; done, or every iteration, or last
; determine continuous iteration (height) for each point
   e1 = iterexp1 * @zscale
   e2 = iterexp2 * @zscale
   e3 = iterexp3 * @zscale

; apply transfer function
; a function is not used because these are floats
; and not all functions apply to floats
IF (@xfer == 1)          ; log
   e1 = log(e1)
   e2 = log(e2)
   e3 = log(e3)
ELSEIF (@xfer == 2)       ; sqrt
   e1 = sqrt(e1)
   e2 = sqrt(e2)
   e3 = sqrt(e3)
ELSEIF (@xfer == 3)       ; cuberoot
   e1 = (e1)^(1/3)
   e2 = (e2)^(1/3)
   e3 = (e3)^(1/3)
ELSEIF (@xfer == 4)       ; exp
   e1 = exp(e1)
   e2 = exp(e2)
   e3 = exp(e3)
ELSEIF (@xfer == 5)       ; sqr
   e1 = sqr(e1)
   e2 = sqr(e2)
   e3 = sqr(e3)
ELSEIF (@xfer == 6)       ; cube
   e1 = (e1)^3
   e2 = (e2)^3
   e3 = (e3)^3
ELSEIF (@xfer == 7)       ; sin
   e1 = sin(e1)
   e2 = sin(e2)
   e3 = sin(e3)
ELSEIF (@xfer == 8)       ; cos
   e1 = cos(e1)
   e2 = cos(e2)
   e3 = cos(e3)
ELSEIF (@xfer == 9)       ; tan
   e1 = tan(e1)
   e2 = tan(e2)
   e3 = tan(e3)
ENDIF

; apply post-scale
e1 = e1 * @zscale2
e2 = e2 * @zscale2
e3 = e3 * @zscale2

   vx = e2-e1
   vy = e3-e1
   vz = -@offset
   ; normalize vector
   vd = 1/sqrt(sqr(vx)+sqr(vy)+sqr(vz))
   vx = vx*vd
   vy = vy*vd
   vz = vz*vd
   z = vx + flip(vy)       ; fudge z from vector
  ELSE             ; didn't compute z this time
z = z1          ; use primary iteration value to keep
                  ; periodicity working
  ENDIF
  IF (modz > @bailout)       ; we're done
done = 0
  ENDIF

bailout:
  (done > 0)
default:
  title = "Slope Sierpinski Triangle II"
  center = (0.0, 0.0)
  magn = 1.1538
  maxiter = 149
  method = multipass
  periodicity = 0

  float param version
default = 1.1
visible = false
  endparam

  param bailout
caption = "Bailout value"
default = 127
min = 1
  endparam

  param offset
caption = "Orbit Separation"
default = 0.01
hint = "Defines how far apart the simultaneous orbits are.  Smaller \
         distances will produce more accurate results."
  endparam
  param xfer
caption = "Height Transfer"
default = 0
enum = "linear" "log" "sqrt" "cuberoot" "exp" "sqr" "cube" \
"sin" "cos" "tan"
hint = "This function will be applied to the height value \
         before a slope is calculated."
  endparam
  param zscale
caption = "Height Pre-Scale"
default = 1.0
hint = "Specifies the ratio between height and distance.  Higher \
         values will exaggerate differences between high and low. \
   In general, you will want to use smaller numbers here."
  endparam
  param zscale2
caption = "Height Post-Scale"
default = 0.025
hint = "Specifies the ratio between height and distance; like \
         Height Pre-Scale, except that this value is applied after \
   the transfer function."
  endparam
  param everyiter
caption = "Every Iteration"
default = false
hint = "If set, the surface normal will be computed at every \
         iteration.  If you are using a coloring algorithm which \
   processes every iteration, you will need this."
  endparam
}

下载 (26.16 KB)

2011-2-23 10:20

我从面板上看了，好象那些函数开关项，一点作用都不起，榕兄再看看，是不是这样。

图片附件: Fractal2.jpg (2011-2-23 10:20, 26.16 KB) / 下载次数 1600
http://inrm3d.cn/attachment.php?aid=10322&k=dae0454263abf50a1304f965bb328412&t=1715005041&sid=13u6OU

作者: 榕坚 时间: 2011-2-23 10:22

10# 柳烟

做这个可是要有十足的耐心和充裕的时间哪。之前做过任意三角形的谢氏三角形，但不是UF的算法。等有空的时间再来试试。这种代码应该是可以用几何画板演绎的。
作者: 柳烟 时间: 2011-2-23 10:44

11# 榕坚
这个代码还容易解读，确实需要时间与足够的耐心才成。我先帖在这儿，我也觉得有点麻烦，先干到其它的，等时间合适，再来尝试尝试。我看到此图不错，萌发了制造出来的欲念。
作者: changxde 时间: 2011-2-23 11:15

下载 (16.66 KB)

2011-2-23 11:15

从UF中没看出参数的影响。

图片附件: sanj.JPG (2011-2-23 11:15, 16.66 KB) / 下载次数 1558
http://inrm3d.cn/attachment.php?aid=10328&k=269a36c4edfe3b3a5acbbe6b1073c244&t=1715005041&sid=13u6OU

作者: 榕坚 时间: 2011-2-23 21:48

10# 柳烟

好象是有影响的，这是用函数ln(x)的结果：

图片附件: 捕获.JPG (2011-2-23 21:48, 38.78 KB) / 下载次数 1611
http://inrm3d.cn/attachment.php?aid=10346&k=13dc53a4b7840194360804cdb8ee206f&t=1715005041&sid=13u6OU

作者: 柳烟 时间: 2011-2-23 22:00

14# 榕坚
榕兄速度真快，做得十分漂亮，佩服。我明天做做看，不懂再请教，这代码长，一看有些胆怯。
IF (modz > @bailout ||    \
   @everyiter ||          \
中的@everyiter是什么意思，你是如何处理的呢？问好。
作者: 榕坚 时间: 2011-2-23 22:34

15# 柳烟

面板中的开关项，值为0或1.
作者: 柳烟 时间: 2011-2-24 09:48

init:
  z1 = #pixel
  z2 = #pixel + @offset
  z3 = #pixel + flip(@offset)
IF ((imag(z1)>=.575*real(z1)) && (-.575*real(z1)<=imag(z1)))
z1 = 2*z1-1i
  ELSEIF (real(z1)<=0)
z1 = 2*z1+(.8660254, 0.5)
  ELSEIF (real(z1)>0)
z1 = 2*z1+(-.8660254, 0.5)
  ELSE
z1 = 2*z1
  ENDIF
我用画板造此分形时，我在理解这段代码时，卡住了！这段代码的意思是：满足不同的条件时，Z1的算法不同，由红色部分的 Z1，判断当((imag(z1)>=.575*real(z1)) && (-.575*real(z1)<=imag(z1)))成立时，按前面红色部分的z1计算 z1 = 2*z1-1i。当此条件不成立，而按红色部分的z1判断real(z1)<=0成立时，由红色部分的z1按算式z1 = 2*z1+(.8660254, 0.5)算出z1；当这二个条件均不成立，但real(z1)>0成立，则按 z1 = 2*z1+(-.8660254, 0.5)仍由红色的z1代入此式算z1.
这样问题出现了，ELSE下的算式，意思是上面三种情况均不满足，按z1 = 2*z1算z1，但当同时否定上面三种情况时，似又出现三种情况之一，这矛盾，如何解决呢？或者是我理解有误？
作者: 榕坚 时间: 2011-2-24 16:18

其实起作用的是：
IF ((imag(z1)>=.575*real(z1)) && (-.575*real(z1)<=imag(z1)))
z1 = 2*z1-1i
  ELSEIF (real(z1)<=0)
z1 = 2*z1+(.8660254, 0.5)
  ELSE
z1 = 2*z1+(-.8660254, 0.5)
ENDIF
从代码看不会执行:Z1=2*Z1 ，我就把它删了。
作者: 柳烟 时间: 2011-2-24 18:04

18# 榕坚
谢过榕兄指点，我正在画板中准备各数据，这段：
IF (modz > @bailout ||    \
   @everyiter ||          \
   done == #maxit + 2)    ; done, or every iteration, or last
; determine continuous iteration (height) for each point
   e1 = iterexp1 * @zscale
   e2 = iterexp2 * @zscale
   e3 = iterexp3 * @zscale
中的条件：modz > @bailout ||    \
   @everyiter ||          \
   done == #maxit + 2如何处理？之前的数据已经准备好了，到这里，又进行不下去了。
作者: 榕坚 时间: 2011-2-24 21:41

if1:0.5*(1+sgn(sgn(modZ-@bailout)-0.5))
if2: @everyiter
if3:1-abs(sgn(done-(n+2)))
IF=sgn(if1+if2+if3)
n为迭代次数，@everyiter的值预先赋以0或1。
作者: 柳烟 时间: 2011-2-25 11:41

完成后，扫不出图，检查了几遍，找不到问题在那里。我想可能是逻辑判断出错，迭代问题，也可能是其它问题，我发文件在这，榕兄抽空看看，帮找找原因。谢谢。
谢氏三角形尝试.gsp (43.22 KB)
下载次数: 2362

2011-2-25 11:41

附件: 谢氏三角形尝试.gsp (2011-2-25 11:41, 43.22 KB) / 下载次数 2362
http://inrm3d.cn/attachment.php?aid=10373&k=bf881a22703814b3854e30204908288c&t=1715005041&sid=13u6OU
作者: 榕坚 时间: 2011-2-26 19:50

21# 柳烟

已经帮你改好了，顺便扫描一图：

图片附件: 捕获.JPG (2011-2-26 19:50, 32.73 KB) / 下载次数 1483
http://inrm3d.cn/attachment.php?aid=10401&k=31142c996dc663eab83c296fdecd962c&t=1715005041&sid=13u6OU

附件: 谢氏三角形尝试.gsp (2011-2-26 19:50, 45.41 KB) / 下载次数 1918
http://inrm3d.cn/attachment.php?aid=10402&k=8a836ef82145275a1ad61ea9c933cbde&t=1715005041&sid=13u6OU
作者: 柳烟 时间: 2011-2-27 11:09

22# 榕坚
感谢榕兄帮忙找错误原因，并修改正确。下来我对照你改的文件，再结合代码，找问题之所在。这个题目解决后，UF中的sg.ufm系列里的最后几个分形的代码，与此分形代码类似，解决起来应该说就容易了。问好。
作者: 柳烟 时间: 2011-2-28 09:41

22# 榕坚
这两天，我在看你修改后的文件，有二问请教：
1、修改后的文件扫出的图片，从右斜向左有一朦胧的斜线，不知原因为何。
2、该分形的迭代是否有点多，从t4、t5、……、t11，还有e1、e2、e3等，都要参与迭代呢？好象是这样。
作者: 榕坚 时间: 2011-2-28 11:11

24# 柳烟

1、不止右边有那条斜线，左边也有。那是着色的问题UF中也有（因为你选的着色类型是BASIC所以看不见）。几何画板也可以调成没有那条线。附图：
2、这个分形的迭代原象比较多，我是按代码要求给的。至于是否可以删减可以尝试。

图片附件: Fractal1.jpg (2011-2-28 11:11, 35.98 KB) / 下载次数 1474
http://inrm3d.cn/attachment.php?aid=10435&k=eb0ba42f2cff21a3f7de5d708bb2a1c0&t=1715005041&sid=13u6OU

图片附件: Fractal1.jpg (2011-2-28 11:12, 49.27 KB) / 下载次数 1489
http://inrm3d.cn/attachment.php?aid=10436&k=8783f6cc9d808b61e09f1019fb4525e2&t=1715005041&sid=13u6OU

图片附件: gnd-slope-sierpinski2.JPG (2011-2-28 11:16, 46.88 KB) / 下载次数 1468
http://inrm3d.cn/attachment.php?aid=10437&k=478190f0a00ab6397968aacc0571e430&t=1715005041&sid=13u6OU

作者: 榕坚 时间: 2011-2-28 11:23

面板中的函数改变好象只是逃逸区的形状有些变化，这是选tan函数的：

图片附件: 捕获.JPG (2011-2-28 11:23, 64.98 KB) / 下载次数 1646
http://inrm3d.cn/attachment.php?aid=10438&k=8e21a6d6ee3ce4d6420dafe4c4c04f8e&t=1715005041&sid=13u6OU

作者: 柳烟 时间: 2011-2-28 11:49

原来如此，明白了，谢谢。
作者: 柳烟 时间: 2011-2-28 22:08

位于tfa.ufm系列中的:mandala Epsilon Julia
mandalaEpsilonJulia {
; Toby Marshall, Jan 2002
init:
  z = #pixel
  complex fudge = @fudge*(0.0,1.0)

  loop:
  z = fn1(fn2(z^@e1-@budge)*fn4(imag(@seed^@e3-fudge))^@fn5(real(@seed^@e4-@kludge)))^fn3(@pow/((@seed^@e2)-@nudge))
  bailout:
  |z| < @bailout
  switch:
  type = "mandalaepsilon"
  bailout = bailout
  fn1 = @fn1
  fn2 = @fn2
  fn3 = @fn3
  fn4 = @fn4
  fn5 = @fn5
  budge = @budge
  nudge = @nudge
  kludge = @kludge
  fudge = @fudge
  e1 = @e1
  e2 = @e2
  e3 = @e3
  e4 = @e4
  pow = @pow

  default:
  title = "mandala epsilon julia"
  periodicity = 0
  maxiter = 500
  param bailout
  caption = "bailout"
  default = 7.0
  endparam
  param seed
  caption = "seed"
  default = (0.95, -0.332)
  endparam
  param e1
  caption = "exponent1"
  default = (3.0,0.0)
  hint = "The real component determines the number \
of arms in the figure. The imaginary component \
changes the scale of the elements moving around the figure."
  endparam
  param e2
  caption = "exponent2"
  default = (1.0,0.0)
  endparam
  param e3
  caption = "exponent3"
  default = (1.0,0.0)
  endparam
  param e4
  caption = "exponent4"
  default = (1.0,0.0)
  endparam
  param budge
  caption = "budge"
  default = (0.0,0.0)
  endparam
  param fudge
  caption = "fudge"
  default = 0.0
  endparam
  param kludge
  caption = "kludge"
  default = 0.0
  endparam
  param nudge
  caption = "nudge"
  default = (0.0,0.0)
  endparam
  param pow
  caption = "power"
  default = (1.0,0.0)
  endparam

  func fn1
  caption = "function 1"
  default = flip ()
  endfunc
  func fn2
  caption = "function 2"
  default = conj ()
  endfunc
func fn3
  caption = "function 3"
  default = log ()
  endfunc
  func fn4
  caption = "function 4"
  default = ident ()
  endfunc
  func fn5
  caption = "function 5"
  default = ident ()
  endfunc
}

下载 (69.79 KB)

2011-3-1 08:52

图片附件: Fractal4.jpg (2011-3-1 08:52, 69.79 KB) / 下载次数 1445
http://inrm3d.cn/attachment.php?aid=10445&k=2dcf399bb6fe426c3b72e160481d3832&t=1715005041&sid=13u6OU

作者: 柳烟 时间: 2011-2-28 22:15

z = fn1(fn2(z^@e1-@budge)*fn4(imag(@seed^@e3-fudge))^@fn5(real(@seed^@e4-@kludge)))^fn3(@pow/((@seed^@e2)-@nudge))
此分形看起来不错的，按此算式，应该先算红色部分的绿色部分次方，但我算出后，无意义，下面不知如何办了。
作者: 榕坚 时间: 2011-3-1 08:25

29# 柳烟

变通一下：
作者: 榕坚 时间: 2011-3-1 08:26

传图片卡住了：

图片附件: mandalaEpsilonJulia.JPG (2011-3-1 08:53, 46.45 KB) / 下载次数 2592
http://inrm3d.cn/attachment.php?aid=10453&k=9f614ecbad89ab03a8171b27f884fdff&t=1715005041&sid=13u6OU

作者: 柳烟 时间: 2011-3-1 08:41

30# 榕坚
如何变通？
作者: inRm 时间: 2011-3-1 08:54

31# 榕坚
这个别致，很强的立体感和金属感。
作者: 榕坚 时间: 2011-3-1 09:11

32# 柳烟

这是指数幂运算的不尽人意问题了，因为底数为负值时几何画板认定为无意义的。只要把底数改为复数（如（-2）^3改为real(（-2，0）^3)）就不会出现不存在了。
作者: 柳烟 时间: 2011-3-1 09:23

它后面的mandalaGammaJulia，将seed实部改为0.93，扫出的图不错，放**点，居然又发现直线分形。

下载 (41.03 KB)

2011-3-1 09:33

下载 (29.41 KB)

2011-3-1 17:50

下载 (9.65 KB)

2011-3-1 17:50

图片附件: 未命名.jpg (2011-3-1 09:33, 41.03 KB) / 下载次数 2358
http://inrm3d.cn/attachment.php?aid=10456&k=ebeb67defb093f54fdc58ee59e9c42dd&t=1715005041&sid=13u6OU

图片附件: 未命名.jpg (2011-3-1 17:50, 29.41 KB) / 下载次数 2397
http://inrm3d.cn/attachment.php?aid=10460&k=6e88065e6d4bb6eabc73b2df07e4bce1&t=1715005041&sid=13u6OU

图片附件: 未命名(1).jpg (2011-3-1 17:50, 9.65 KB) / 下载次数 2359
http://inrm3d.cn/attachment.php?aid=10461&k=d040b6d88654be65c56a8fb5eeeccb9e&t=1715005041&sid=13u6OU

作者: 柳烟 时间: 2011-3-1 09:27

34# 榕坚
此变通太神妙了！
作者: 榕坚 时间: 2011-3-1 11:27

下载 (45.52 KB)

2011-3-1 17:51

图片附件: mandalaGammaJulia-2.JPG (2011-3-1 17:51, 45.52 KB) / 下载次数 2624
http://inrm3d.cn/attachment.php?aid=10458&k=c84d01ac00332ad3b27f196949b44c24&t=1715005041&sid=13u6OU

作者: 榕坚 时间: 2011-3-1 11:42

35# 柳烟

改一下确实好看多了：

图片附件: mandalaGammaJulia-1.JPG (2011-3-1 11:42, 50.68 KB) / 下载次数 2839
http://inrm3d.cn/attachment.php?aid=10459&k=2d349e3e6f18741a82270f71cdc24fcb&t=1715005041&sid=13u6OU

作者: 柳烟 时间: 2011-3-1 16:14

38# 榕坚
是的。这类分形的美有其特色，玩起来有趣。
作者: 柳烟 时间: 2011-3-3 10:00

位于sam.ufm系列中的科赫雪花曲线，我看了，扫描法出来的图形，美。我正在造第一个开关项，觉得这开关项可能简单点，单就这开关项的程序还没完全懂，我感觉到此分形能用几何画板做。
KochCurve {
; By Samuel Monnier, 2.1.00
init:
  z = #pixel
  zz = 0
  ; Twist
  z = 1/sqrt(@r)*real(z) + 1i*sqrt(@r)*imag(z)
  x = real(z)
  y = imag(z)
  sq3 = sqrt(3)
  bool bail2 = false
  bool bail = false
  i = 0
loop:
  i = i + 1
  ;----------------
  ; First Iteration
  ;----------------
  if i == 2

; Calculate the argument of z
arg = atan2(z)

; Inside approximation
if @region == 0 || @region == 2

   ; Draw the Star of David
   if (y + 1/sq3 > 0) && (sq3*x - y + 2/sq3 > 0) \
      && (sq3*x + y - 2/sq3 < 0)
      bail = true
   endif
   if (-y + 1/sq3 > 0) && (sq3*x + y + 2/sq3 > 0) \
      && (sq3*x - y - 2/sq3 < 0)
      bail = true
   endif

   ; Deal the case where z needs a rotation
   if (arg > 5/6*pi) || (arg < -pi/2)
      z = z*exp(1i*4/3*pi)
   endif
   if (arg < pi/6) && (arg > -pi/2)
      z = z*exp(1i*2/3*pi)
   endif
   z = z - 1i*1/sq3
   if @region == 2
      zz = z
      z = #pixel
   endif
endif

; Outside approximation
if @region == 1 || @region == 2

   ; Draw the hexagon
   if |x| > 1 || x/sq3 + y - 2*sq3/3 > 0 \
      || x/sq3 + y + 2*sq3/3 < 0 || x/sq3 \
      - y + 2*sq3/3 < 0 || x/sq3 - y - \
      2*sq3/3 > 0
      bail = true
   endif

   ; Deal the case where z needs a rotation
   if (abs(arg) < pi/6)
      z = z*exp(-1i*pi/2)
   elseif (arg > pi/6) && (arg < pi/2)
      z = z*exp(-1i*5*pi/6)
   elseif (arg > pi/2) && (arg < 5*pi/6)
      z = z*exp(1i*5*pi/6)
   elseif (abs(arg) > 5*pi/6)
      z = z*exp(1i*pi/2)
   elseif (arg < -pi/6) && (arg > -pi/2)
      z = z*exp(-1i*pi/6)
   elseif (arg < -pi/2) && (arg > -5*pi/6)
      z = z*exp(1i*pi/6)
   endif
   z = z + 1i
   z = sq3*z

endif

  ;----------------
  ;Other Iterations
  ;----------------
  elseif i > 2
; "Undo" the offset parameter's
; action
z = z + @offset
z = 1/sqrt(@r)*real(z) + 1i*sqrt(@r)*imag(z)
if @region == 2
   oldz = z
   z = zz
   zz = oldz
endif
; Magnify z so that a simple formula
; allows to compute...
z = 3*z
x = real(z)
y = imag(z)
; ...the new triangle on the segment
if (y > 0) && (sq3*x - y + sq3 > 0) \
   && (sq3*x + y - sq3 < 0)
   bail2 = true
endif
; Restore the original value of z
z = z/3

x = real(z)
y = imag(z)
; Transforms the left segment
if x < -1/3
   z = 3*z + 2
; Transforms the right segment
elseif x > 1/3
   z = 3*z - 2
else
   ; Transforms the middle-left segment
   if x < 0
      z = z + 1/3
      z = z*exp(-1i*pi/3)
      z = 3*z - 1
   ; Transforms the middle-right segment
   else
      z = z - 1/3
      z = z*exp(1i*pi/3)
      z = 3*z + 1
   endif
endif

  endif

  ; A parameter to fool some coloring
  ; algorythms like final decomposition...
  if i >= 2
z = z - @offset
  endif

bailout:
  bail == false && bail2 == false

default:
  title = "Koch Curve"
  helpfile = "sam-help/kochcurves.htm"
  helptopic = "kcurve"
  magn = 1.5
  center = (0.0002,0)
  maxiter = 50
  param region
caption = "Trapped Region"
default = 0
enum = "Inside" "Outside" "Both"
  endparam
  param offset
caption = "Offset"
default = (0,0)
  endparam
  param r
caption = "Twist Parameter"
default = (1,0)
  endparam
}
作者: 柳烟 时间: 2011-3-3 10:04

下载 (27.32 KB)

2011-3-3 17:52

图片附件: Fractal1.jpg (2011-3-3 17:52, 27.32 KB) / 下载次数 622
http://inrm3d.cn/attachment.php?aid=10521&k=38d43a8a09b839ee7950ee675018683a&t=1715005041&sid=13u6OU

作者: 榕坚 时间: 2011-3-3 15:16

40# 柳烟

这个才真是繁杂的判断哪，难！
作者: 柳烟 时间: 2011-3-3 16:18

42# 榕坚
判断是有点多，我看到代码有点长，判断又多，吓了一跳。看了几遍，感觉只造第一个开关项，可除去代码中的好几段，也许能成，我算了一部分数据了，晚上再看看，能否整个样子，不过，实在没把握。
作者: 柳烟 时间: 2011-3-3 21:54

太困难了，已决定放弃了。

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