inRm3D: 画板论坛 - Powered by Discuz! Board

标题: UF真是复分形的宝藏 [打印本页]

作者: 榕坚 时间: 2013-12-4 09:32 标题: UF真是复分形的宝藏

UF中的复分形还有很多没被发现，其实包括之前的DEM法也都在UF的范例中。今天看了一下还有这个东西呢：

图片附件: Fractal1.jpg (2013-12-4 09:32, 60.03 KB) / 下载次数 1287
http://inrm3d.cn/attachment.php?aid=20704&k=11d5f85a589790310d9f06952993fd91&t=1732396836&sid=nXKp6c

作者: 柳烟 时间: 2013-12-4 11:31

这种分形不知姓甚名谁。

作者: 榕坚 时间: 2013-12-4 14:27

在mt系列：
mt-latoocarfian {
; Mark Townsend, 18 August 2001
; Strange attractors from Clifford Pickover's
; book "Chaos in Wonderland" (the "De Jong"
; mutation uses similar equations from A. K.
; Dewdney's "The Armchair Universe".)
global:
  float max = 0
  int i = 0
  float xx = 0, float yy = 0
  float a = 0, float b = 0
  float c = 0, float d = 0
  int seed = @seed
  if !@rnd
a = @a
b = @b
c = @c
d = @d
  else
  ; We randomize the parameters in a loop
  ; until we find a set that has a reasonably
  ; high Lyapunov exponent.
float L = -1
while L < @lthresh
; Parameters in the range (-3 < a, b < 3)
; and (0.5 < c, d < 1.5)
   seed = random(seed)
   a = (seed / #randomrange) * 3
   seed = random(seed)
   b = (seed / #randomrange) * 3
   seed = random(seed)
   c = (seed / #randomrange) + 0.5
   seed = random(seed)
   d = (seed / #randomrange) + 0.5
   float Lsum = 0
   float lx = 0.1
   float ly = 0.1
   float xe = lx + 0.000001
   float ye = ly
   i = 0
   while i < 1000
   ; Pickover suggests 10 million iterations,
   ; but this is enough to produce decent
   ; attractors
      if @mutation == "Standard"
      xx = sin(ly * b) + c * sin(lx * b)
      yy = (sin(lx * a) + d * sin(ly * a))
      elseif  @mutation == "Alpha"
      xx = sin(ly * b) + sin(lx * b)^2 + sin(lx * b)^3
      yy = sin(lx * a) + sin(ly * a)^2 + sin(ly * c)^3
      elseif  @mutation == "Beta"
      xx = sin(ly * b) + sin(lx * b)^2
      yy = sin(lx * a) + sin(ly * a)^2
      elseif  @mutation == "Gamma"
      xx = abs(sin(ly * b)) + sin(lx * b)^2
      yy = abs(sin(lx * a)) + sin(ly * a)^2
      elseif  @mutation == "De Jong"
      xx = sin(ly * a) - cos(lx * b)
      yy = sin(lx * c) - cos(ly * d)
      endif
      float xsave = xx, float ysave = yy, lx = xe, ly = ye
      i = i + 1
      if @mutation == "Standard"
      xx = sin(ly * b) + c * sin(lx * b)
      yy = (sin(lx * a) + d * sin(ly * a))
      elseif  @mutation == "Alpha"
      xx = sin(ly * b) + sin(lx * b)^2 + sin(lx * b)^3
      yy = sin(lx * a) + sin(ly * a)^2 + sin(ly * c)^3
      elseif  @mutation == "Beta"
      xx = sin(ly * b) + sin(lx * b)^2
      yy = sin(lx * a) + sin(ly * a)^2
      elseif  @mutation == "Gamma"
      xx = abs(sin(ly * b)) + sin(lx * b)^2
      yy = abs(sin(lx * a)) + sin(ly * a)^2
      elseif  @mutation == "De Jong"
      xx = sin(ly * a) - cos(lx * b)
      yy = sin(lx * c) - cos(ly * d)
      endif
      float dLx = xx - xsave, float dLy = yy - ysave
      float dL2 = dLx * dLx + dLy * dLy
      float df = 1000000000000. * dL2
      float rs = 1/sqrt(df)
      xe = xsave + rs * (xx - xsave)
      ye = ysave + rs * (yy - ysave)
      xx = xsave, yy = ysave
      Lsum = Lsum + log(df)
      L = 0.721347 * lsum / i
      lx = xx
      ly = yy
   endwhile
endwhile
  endif
  int pix[#width, #height]
  if 3 * #width < 4 * #height
float scale = (#width * #magn) / 4
  else
scale = (#height * #magn) / 3
  endif
  int x = 0, int y = 0
; initialize array
  while x < #width
y = 0
while y < #height
   pix[x, y] = 0
   y = y + 1
endwhile
x = x + 1
  endwhile
  float xnew = 0
  float ynew = 0
  xx = 0.1
  yy = 0.1
  i = 0
  int iters = #width * #height * @max
  while i < iters
if @mutation == "Standard"
   xnew = sin(yy * b) + c * sin(xx * b)
   ynew = (sin(xx * a) + d * sin(yy * a))
elseif  @mutation == "Alpha"
   xnew = sin(yy * b) + sin(xx * b)^2 + sin(xx * b)^3
   ynew = sin(xx * a) + sin(yy * a)^2 + sin(yy * c)^3
elseif  @mutation == "Beta"
   xnew = sin(yy * b) + sin(xx * b)^2
   ynew = sin(xx * a) + sin(yy * a)^2
elseif  @mutation == "Gamma"
   xnew = abs(sin(yy * b)) + sin(xx * b)^2
   ynew = abs(sin(xx * a)) + sin(yy * a)^2
elseif  @mutation == "De Jong"
   xnew = sin(yy * a) - cos(xx * b)
   ynew = sin(xx * c) - cos(yy * d)
endif
xx = xnew
yy = ynew
float xc = real(#center)
float yc = imag(#center)
x = round(((xx - xc) * cos(#angle) - (yy + yc) * \
      sin(#angle)) * scale + #width / 2)
y = round(((xx - xc) * sin(#angle) + (yy + yc) * \
      cos(#angle)) * scale + #height / 2)
  ; plot the point only if inside array
if x >= 0 && x < #width && y >= 0 && y < #height
   pix[x, y] = pix[x, y] + 1
   if pix[x, y] > max
      max = pix[x, y]
   endif
endif
i = i + 1
  endwhile
final:
  #index = (log(pix[#x, #y])) / log(max)
default:
  title = "Lat鲻carfian Attractors"
  render = false
  param mutation
caption = "Mutation"
enum = "Standard" "Alpha" "Beta" "Gamma" "De Jong"
  endparam
  int param max
caption = "Sample Density"
default = 30
min = 1
  endparam
  float param a
caption = "a"
default = -0.966918
visible = !@rnd
  endparam
  param b
caption = "b"
default = 2.879879
visible = !@rnd
  endparam
  float param c
caption = "c"
default = 0.765145
visible = !@rnd && (@mutation == "Standard" || \
            @mutation == "Alpha" || @mutation == "De Jong")
  endparam
  float param d
caption = "d"
default = 0.744728
visible = !@rnd && (@mutation == "Standard" || \
            @mutation == "De Jong")
  endparam
  bool param rnd
caption = "Random Parameters"
default = false
  endparam
  float param lthresh
caption = "Lyapunov"
default = 0.3
visible = @rnd
  endparam
  int param seed
caption = "Seed"
default = 12345678
visible = @rnd
  endparam
}

作者: xuefeiyang 时间: 2013-12-4 20:01

3# 榕坚

你们用的这些在哪里有？能给我传一份吗？邮箱422161240@qq.com

作者: 榕坚 时间: 2013-12-4 20:34

4# xuefeiyang

你有安装UF程序吧。所有的范例都在里面的。

作者: xuefeiyang 时间: 2013-12-4 20:37

我安装了UF程序，但找不到你所说的那个系列：

图片附件: New.jpg (2014-7-9 11:59, 18.25 KB) / 下载次数 1296
http://inrm3d.cn/attachment.php?aid=20710&k=d569fefc6ad4fd46fc7b3c04c8839a83&t=1732396836&sid=nXKp6c

作者: 榕坚 时间: 2013-12-4 21:04

6# xuefeiyang

不是在formula面版中找，在outside的面版中的mt系列中。它是一个分形外部着色程序，象那希尔伯特曲线一样。

作者: 榕坚 时间: 2013-12-5 08:20

代码太吓人了：
KleinianLimitSets {
; modified june 2009, original June 2006 Ronald Barnett
;
; Based upon the depth-first tree search algorithm in the book \
; Indra's Pearls by Mumford, Series and Wright. It is considerably faster \
; than most other algorithms in generating the limit set.
; modified Feb 2010 to allow mapping transforms to work.
;
global:
  import "common.ulb"
  import "reb.ulb"

  Circle C = new Circle(0,0)
  Circle cir[4]
  Mobius gen[4]
  MobiusArray word = new MobiusArray(@level)
  complex cs[4]
  float rad[4]
  int tag[]
  setlength(tag,@level)

  float magn = #magn
  float scrsize = #width*#height
  float epsilon = @epsilon

  ; this uses a 640x640 screen with magn = 1 as the reference
  if @screen
epsilon = epsilon*409600/scrsize/magn
  endif

  int w = 0
  int h = 0
  float Inc = @percentInc*0.01
  float wd = Inc*#width
  float ht = Inc*#height
  int wd2 = round(wd*0.5)
  int ht2 = round(ht*0.5)
  int ilev = 0
  complex ta = @ta
  complex tb = @tb
  complex taj = @taj
  complex tbj = @tbj
  if !@expert && @typeb == "Double Cusps"
tb = (2,0)
if @tb2 == "1 100 Cusp"
   ta = (1.99901430461837,-3.94468287279129e-005)
elseif @tb2 == "1 99 Cusp"
   ta = (1.99899431720688,-4.06536410107979e-005)
elseif @tb2 == "1 98 Cusp"
   ta = (1.9989737159102,-4.19101798220381e-005)
elseif @tb2 == "1 97 Cusp"
   ta = (1.99895247534081,-4.32190326141746e-005)
elseif @tb2 == "1 96 Cusp"
   ta = (1.99893056878569,-4.45829500728911e-005)
elseif @tb2 == "1 95 Cusp"
   ta = (1.99890796812253,-4.60048582566762e-005)
elseif @tb2 == "1 94 Cusp"
   ta = (1.99888464372953,-4.74878717791897e-005)
elseif @tb2 == "1 93 Cusp"
   ta = (1.99886056438849,-4.90353081361803e-005)
elseif @tb2 == "1 92 Cusp"
   ta = (1.99883569718054, -5.06507032899056e-005 )
elseif @tb2 == "1 91 Cusp"
   ta = (1.99881000737372, -5.23378286366251e-005 )
elseif @tb2 == "1 90 Cusp"
   ta = (1.99878345830198, -5.41007094967586e-005 )
elseif @tb2 == "1 89 Cusp"
   ta = (1.99875601123446, -5.59436452831306e-005 )
elseif @tb2 == "1 88 Cusp"
   ta = (1.99872762523451, -5.78712315206112e-005 )
elseif @tb2 == "1 87 Cusp"
   ta = (1.99869825700728, -5.98883839104726e-005 )
elseif @tb2 == "1 86 Cusp"
   ta = (1.99866786073491, -6.20003646555514e-005 )
elseif @tb2 == "1 85 Cusp"
   ta = (1.99863638789812, -6.42128112880125e-005 )
elseif @tb2 == "1 84 Cusp"
   ta = (1.99860378708289, -6.6531768270374e-005 )

……

图片附件: New.gif (2014-7-9 12:00, 16.48 KB) / 下载次数 1172
http://inrm3d.cn/attachment.php?aid=20716&k=1b772259a84f8dcaa7755d9c4d19b90d&t=1732396836&sid=nXKp6c

作者: 榕坚 时间: 2013-12-5 08:29

这个应该可以用几何画板来做一下：
ESAFractalTree {
;
; version 1.0
; By Etienne Saint-Amant, 2002/07/02
;
; Go to http://ESAfractal.com for examples and tutorial
;
; Based on FractalTree
; By Samuel Monnier, 11.1.00
; Modified with his kind permission
;
init:
  z = 0
  float arg = 0
  int i = 0
  float j = 0
  float d = 0
  float dist = 1e5
  float x = 0
  float y = 0
  float imagCof = @imaginaryCoefficient
  if @mode == 0
z = #pixel
  endif

  float dr = pi/@rotincrCoefficient*@rotincr/@altitudeTree

loop:
  if @mode == 1
i = i + 1
if i == @itertr
   z = #z
endif
  endif
final:
  ; Performs translation, rotation
  ; and magnification
  z = z - @center
  z = z*exp(-1i*@rot*pi/@rotationchange)
  z = z/@size
  i = 0

  while i < @niter
i = i + 1

imagCof = imagCof + @alterimaginary

x = abs(real(z))
y = imag(z)
z = x + imagCof * 1i * y
arg = -atan2(z) + pi/@weirdness
; Estimate distance
if y >= -1 && y <= 0
   d = x^@power
elseif y < -1
   d = cabs(z + imagCof * 1i)^@power
elseif y > 0
   d = cabs(z)^@power
endif

if d < dist
   dist = d
endif

; Computes the rotation to apply
j = @order-1
rotincr = j*dr
while j > 0
   j = j - 2
   if arg < (j+1)*dr
      rotincr = j*dr
   endif
endwhile

; Performs transformation
z = z*exp(imagCof*1i*rotincr)
z = z*@magnincr
z = z - imagCof*1i

  endwhile

  #index = dist

default:
  title = "ESA Fractal Tree"
  param mode
caption = "Mode"
default = 0
enum = "Pixel" "Distance Estimator"
  endparam
  param power
caption = "Dist. Est. Power"
default = 1.0
  endparam
  param order
caption = "Tree Order"
default = 4.0
  endparam
  param rotincr
caption = "Angle Between Branches"
default = 40.0
  endparam
  param magnincr
caption = "Tree Magnification Increment"
default = 1.9
  endparam
  param center
caption = "Center"
default = (0,0)
  endparam
  param rot
caption = "Rotation"
default = 0.0
  endparam
  param size
caption = "Size"
default = 1.0
  endparam
  param niter
caption = "Number of Iteration"
default = 10
  endparam
  param itertr
caption = "Iteration to observe"
default = 1
  endparam

  ;parameter added by ESA
  param imaginaryCoefficient
caption = "Imaginary Coefficient"
default = 0.75
hint = "Imaginary Coefficient to change the shape of the branches (default .75)"
  endparam

  ;parameter added by ESA
  param alterimaginary
caption = "Alter Imaginary Coefficient"
default = 0.1
hint = "Alter the Imaginary Coefficient for each iteration (default 0.1)"
  endparam

  ;parameter added by ESA
  param altitudeTree
caption = "Altitude Tree"
default = 2.0
hint = "Create a tree developing more vertically (default 2.0)"
  endparam

  ;parameter added by ESA
  param weirdness
caption = "Weirdness"
default = 2.2
hint = "Weirdness (default 2.2)"
endparam

  ;parameter added by ESA
  param rotincrCoefficient
caption = "Rotation Incrementation"
default = 120
hint = "Rotation Incrementation Coefficient (default 120)"
endparam

  ;parameter added by ESA
  param rotationchange
caption = "Rotation Change"
default = 50
hint = "Rotation Change (default 50)"
endparam
}

图片附件: New.jpg (2014-7-9 12:01, 21.28 KB) / 下载次数 1185
http://inrm3d.cn/attachment.php?aid=20717&k=fac73ca5fb4641ba638b42d75c8ac985&t=1732396836&sid=nXKp6c

作者: xuefeiyang 时间: 2013-12-5 09:31

2# 柳烟

flame!

作者: 榕坚 时间: 2014-4-27 21:22

今天无意间看到UF中的一个范例，想半天不知道是怎么能把小M给变到那些位置上去：

图片附件: Recurring Dream.jpg (2014-4-27 21:22, 26.89 KB) / 下载次数 1871
http://inrm3d.cn/attachment.php?aid=21455&k=f8afeb81fdb2988ea7a28350bafed747&t=1732396836&sid=nXKp6c

作者: changxde 时间: 2014-4-27 22:12

要是每个圆饼上都有一个小M就更好了。
这是UF中的那个？

作者: 榕坚 时间: 2014-4-27 22:21

12# changxde

不要打开主程序，直接从public文件夹中打开lkm.upr，有个recurring dream就是。

作者: xiaongxp 时间: 2014-4-29 20:10

11# 榕坚
这是用叠加六个相似变换和一个翻折变换的办法扫出的图形，足以乱真。

图片附件: M Set.jpg (2014-4-29 23:00, 19.37 KB) / 下载次数 1952
http://inrm3d.cn/attachment.php?aid=21476&k=60091fa53c015bcf41ad17341dc5c3f4&t=1732396836&sid=nXKp6c

作者: xiaongxp 时间: 2014-4-30 13:08

图片附件: M Set 2.jpg (2014-4-30 23:55, 27.15 KB) / 下载次数 2021
http://inrm3d.cn/attachment.php?aid=21480&k=0a0ea4081b5471223c316ff56335e048&t=1732396836&sid=nXKp6c

作者: 榕坚 时间: 2014-4-30 13:39

这个图也在这里：

图片附件: Multi-Embossed Julia, Smallest Mag.jpg (2014-4-30 13:39, 24.51 KB) / 下载次数 1857
http://inrm3d.cn/attachment.php?aid=21482&k=b59e876741620aaf738a110a01d308be&t=1732396836&sid=nXKp6c

作者: changxde 时间: 2014-4-30 16:54

此UF图是增加映射Circle Zoom得到的，比1#多了两个映射共128个映射。

图片附件: New.jpg (2014-7-9 12:01, 22.94 KB) / 下载次数 2020
http://inrm3d.cn/attachment.php?aid=21483&k=af298c39ea091e4f1ed8888bf9be36d9&t=1732396836&sid=nXKp6c

作者: 榕坚 时间: 2014-4-30 19:42

17# changxde
哪些中心点的坐标，缩放比，旋转角是怎么算出来的？UF是一个个做变换，能否一次性解决？

作者: changxde 时间: 2014-4-30 20:02

我加上去的两个是试着来的，

作者: xiaongxp 时间: 2014-4-30 20:04

17# changxde
哪些中心点的坐标，缩放比，旋转角是怎么算出来的？UF是一个个做变换，能否一次性解决？
榕坚发表于 2014-4-30 19:42

在画板中用几何法比用代数公式法作变换要方便得多，而且扫描速度会更快。榕老师试试我这工具：

附件: 相似变换(局部缩放同步扫描).gsp (2014-4-30 20:25, 4.06 KB) / 下载次数 2330
http://inrm3d.cn/attachment.php?aid=21486&k=139bcd8f5fb141661c5af5ed1375f392&t=1732396836&sid=nXKp6c

作者: xiaongxp 时间: 2014-5-1 19:04

再加两个变换，行列数可设

M Set 3.gsp (20.9 KB)

图片附件: New.jpg (2014-7-9 12:02, 28.77 KB) / 下载次数 1741
http://inrm3d.cn/attachment.php?aid=21500&k=f9ab6440cbeb7c4f3d76d0ea50158ebc&t=1732396836&sid=nXKp6c

附件: M Set 3.gsp (2014-5-1 19:04, 20.9 KB) / 下载次数 3015
http://inrm3d.cn/attachment.php?aid=21501&k=7dacf5f8d7a4effd6bcff38699b60522&t=1732396836&sid=nXKp6c

作者: lnszdzg 时间: 2014-5-3 13:07

Carr1984

图片附件: New.jpg (2014-7-9 12:02, 15.15 KB) / 下载次数 1665
http://inrm3d.cn/attachment.php?aid=21520&k=2f94f435b82732bcabd1508833e4c799&t=1732396836&sid=nXKp6c

作者: lnszdzg 时间: 2014-5-3 13:11

Carr2003

图片附件: New.jpg (2014-7-9 12:03, 11.89 KB) / 下载次数 1656
http://inrm3d.cn/attachment.php?aid=21521&k=eae520012e3b00f0012eab18417097fb&t=1732396836&sid=nXKp6c

作者: lnszdzg 时间: 2014-5-3 13:14

Carr1972

图片附件: New.jpg (2014-7-9 12:03, 13.64 KB) / 下载次数 1666
http://inrm3d.cn/attachment.php?aid=21522&k=dbb7b70cc81d6a3d2e24513c40745324&t=1732396836&sid=nXKp6c

欢迎光临 inRm3D: 画板论坛 (http://inrm3d.cn/)