巧用对称性，一个陷阱四个球

lnszdzg

以往做相切球陷阱时，两个球得做两个陷阱，计算两次，设两个判断；4个球得做4个陷阱，计算4次，设4个判断；8个球得做8个陷阱，计算8次，设8个判断......；计算量很大，速度当然就比较慢。那么能否简化？
可以！利用对称性！
由于做分形目前基本上是用直角坐标系，这就可以利用直角坐标系的对称性简化计算了。
我们知道：点（a，b）关于x轴的对称点是(a,-b)，关于y轴的对称点是(-a,b),关于原点的对称点是(-a,-b),
这四个点可以简化成(|a|,|b|),这样四个陷阱：
#1=|(x,y)-(a,b)|,
#2=|(x,y)-(a,b)|,
#3=|(x,y)-(a,b)|,
#4=|(x,y)-(a,b)|,
可以简化为#=|(|x|,|y|)-(a,b)|,这样一个计算可以产生四个球，就是着色不好处理啊。

lnszdzg

Mathcad下的图形

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2014-7-9 10:33

程序

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2014-5-30 21:01

lnszdzg

这四个点可以简化成(|a|,|b|),这样四个陷阱：（原帖出现失误，更正再次，抱歉，请原谅）
#1=|(x,y)-(a,b)|,
#2=|(x,y)-(-a,b)|,
#3=|(x,y)-(a,-b)|,
#4=|(x,y)-(-a,-b)|,

lnszdzg

柳老师，UF中的这个在那个中啊？我想学一下着色。

lnszdzg

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2014-7-9 10:34

lnszdzg

中间好像少了一些啥，可是我加不上。

lnszdzg

谢谢柳老师，找到了。我更具Mathcad中的方法改写了一下，基本上保持了原来的结构，但只用了2个陷阱，产生了8个圆.
UF代码
8Circles {
init:
  int iter = 0
  float rangemun = 0
  float num = 4
  float Xabs = 0
  float Yabs = 0
  float Dsgd0 = 0
  float Dsgd1 = 0
  float Circle = 0
  float ZtoPsqd = 0
  float Rc = @r
  float Phi = #pi * 0.125
  float Rm = Rc * sin(Phi)
  float RcSqd = Rm^2
  bool Trapped = false
loop:
  iter = iter + 1
  if iter > @skip && !Trapped
    Xabs = abs(real(#z))
    Yabs = abs(imag(#z))
    Dsgd0 = (Xabs-Rc*sin(phi))^2 + (Yabs - Rc*cos(phi))^2
    Dsgd1 = (Xabs-Rc*sin(3*phi))^2 + (Yabs - Rc*cos(3*phi))^2
    if Dsgd0 < RcSqd
      Trapped = true
      ZtoPsqd= Dsgd0
        Circle = iter
    elseif Dsgd1 < RcSqd
      Trapped = true
      ZtoPsqd= Dsgd1
        Circle = iter
   endif
endif
final:
  if !trapped
    #solid = true
  else
    Circle = ((Circle + @off) % num)/(num-1)
    float Ratio = sqrt(ZtoPsqd/Rcsqd)
    float ColorIndex = 115 * Ratio + Circle * 140
    #index = (ColorIndex + 1) % 256 /256
  endif
default:
  title = "8 Circles"
  param r
    caption = "Circle radius"
    default = 0.4
  endparam
  param skip
    caption = "Iters to skip"
    default = 0
    hint = "Iterations to skip."
  endparam
  param off
    caption = "Hue cycle"
    default = 0
    min = 0
    max = 7
    hint = "This rotates the coloring order of the balls."
  endparam
}

UF效果

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2014-7-9 10:34

lnszdzg

可以仿照原来的弄。
8Circles2 {
init:
  int iter = 0
  float rangemun = 0
  float x = 0
  float y = 0
  float num = 8
  float Xabs = 0
  float Yabs = 0
  float Dsgd0 = 0
  float Dsgd1 = 0
  float Circle = 0
  float ZtoPsqd = 0
  float Rc = @r
  float Phi = #pi * 0.125
  float Rm = Rc * sin(Phi)
  float RcSqd = Rm^2
  bool Trapped = false
loop:
  iter = iter + 1
  x = real(#z)
  y = imag(#z)
  if iter > @skip && !Trapped
    Xabs = abs(real(#z))
    Yabs = abs(imag(#z))
    Dsgd0 = (Xabs-Rc*sin(phi))^2 + (Yabs - Rc*cos(phi))^2
    Dsgd1 = (Xabs-Rc*sin(3*phi))^2 + (Yabs - Rc*cos(3*phi))^2
   if Dsgd0 < RcSqd
      Trapped = true
      ZtoPsqd= Dsgd0
     if Y >  0&& x > 0
        Circle = 1
      elseif y < 0 && x > 0
        Circle = 0
      elseif y < 0 && x < 0
        Circle = 5
      else
        Circle = 4
      endif
   elseif Dsgd1 < RcSqd
      Trapped = true
      ZtoPsqd= Dsgd1
      if Y >  0&& x > 0
        Circle = 2
      elseif y < 0 && x > 0
        Circle = 3
      elseif y < 0 && x < 0
        Circle = 6
      else
        Circle = 7
      endif
   endif
endif
final:
  if !trapped
    #solid = true
  else
    Circle = ((Circle + @off) % num)/(num-1)
    float Ratio = sqrt(ZtoPsqd/Rcsqd)
    float ColorIndex = 120 * Ratio + Circle * 100
    #index = (ColorIndex + 1) % 256 /256
  endif
default:
  title = "8 Circles2"
  param r
    caption = "Circle radius"
    default = 0.4
  endparam
  param skip
    caption = "Iters to skip"
    default = 0
    hint = "Iterations to skip."
  endparam
  param off
    caption = "Hue cycle"
    default = 0
    min = 0
    max = 7
    hint = "This rotates the coloring order of the balls."
  endparam
}

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2014-7-9 10:35

lnszdzg

将上文件稍作修改，可以得到：

代码：
8Circles2 {
init:
  int iter = 0
  float rangemun = 0
  float x = 0
  float y = 0
  float num = 10
  float Xabs = 0
  float Yabs = 0
  float Dsgd0 = 0
  float Dsgd1 = 0
  float Dsgd2 = 0
  float Ratio = 0
  float Circle = 0
  float ZtoPsqd = 0
  float Rc = @r
  float Phi = #pi /12
  float Rm = Rc * sin(Phi)
  float RcSqd = Rm^2
  bool Trapped = false
loop:
  iter = iter + 1
  x = real(#z)
  y = imag(#z)
  if iter > @skip && !Trapped
    Xabs = abs(real(#z))
    Yabs = abs(imag(#z))
    Dsgd0 = (Xabs-Rc*sin(phi))^2 + (Yabs - Rc*cos(phi))^2
    Dsgd1 = (Xabs-Rc*sin(5*phi))^2 + (Yabs - Rc*cos(5*phi))^2
    Dsgd2 = abs(abs(x)-abs(y))
   if Dsgd0 < RcSqd
      Trapped = true
      ZtoPsqd= Dsgd0
     if Y >  0&& x > 0
        Circle = 1
      elseif y < 0 && x > 0
        Circle = 0
      elseif y < 0 && x < 0
        Circle = 5
      else
        Circle = 4
      endif
      Ratio = sqrt(ZtoPsqd/Rcsqd)
   elseif Dsgd1 < RcSqd
      Trapped = true
      ZtoPsqd= Dsgd1
      if Y >  0&& x > 0
        Circle = 2
      elseif y < 0 && x > 0
        Circle = 3
      elseif y < 0 && x < 0
        Circle = 6
      else
        Circle = 7
         endif
         Ratio = sqrt(ZtoPsqd/Rcsqd)
   elseif Dsgd2 < 4*RcSqd
      trapped = true
      ZtoPsqd = 4*Dsgd2
      if x > 0
        Circle = 9
      else
        Circle = 8
     endif
     Ratio = sqrt(0.15*ZtoPsqd/Rcsqd)
   endif
endif
final:
  if !trapped
    #solid = true
  else
    Circle = ((Circle + @off) % num)/(num-0)
    float ColorIndex = 120 * Ratio + Circle * 100
    #index = (ColorIndex + 1) % 256 /256
  endif
default:
  title = "8 Circles2"
  param r
    caption = "Circle radius"
    default = 0.4
  endparam
  param skip
    caption = "Iters to skip"
    default = 0
    hint = "Iterations to skip."
  endparam
  param off
    caption = "Hue cycle"
    default = 0
    min = 0
    max = 7
    hint = "This rotates the coloring order of the balls."
  endparam
}

lnszdzg

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