Mandelbrot and Julian sets

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Mandelbrot and Julian sets

• Vaclav Vavra

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Definitions

• By both sets we compute the same sequence given
  by recursive formula
• zn (zn-1)2 c
• zn , zn-1, c are complex numbers(!)
• Julian set is set of z0s for which the sequence
  does not diverge
• (c is constant for a given Julian set)
• formally
• J C - z0 zn (zn-1)2 c ? 8 or
• J z0 (zn (zn-1)2 c ? 8)

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Definitions

• Mandelbrot set is set of cs for which the
  sequence does not diverge for z0 0
• formally
• J C - cz0 0, zn (zn-1)2 c ? 8 or
• J cz0 0, (zn (zn-1)2 c ? 8)

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Operations in C

• Notation
• x a b.i
• a is real part, b is imaginary part, i is
  imaginary unit, i2-1
• Operations
• y c d.i
• x y (ac) (bd).i
• x.y (ac-bd) (bcad).i
• x sqrt(a2 b2)

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Visualisation

• for a given c, z0 we compute z1, z2, z3,in a
  loop
• this way we cannot analytically determine,
  whether the sequence diverges
• However if zn gt 2, it really diverges
• if zn is still lt 2, we just stop after fixed
  number of iterations
• It works

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Julian set sudocode

• const complex c c.r,c.i
• for (every real part for z0)
  //should be between -2 and 2
• for (every imaginary part for z0)
  //should be between -2 and 2
• z z0
• for (fixed number of iterations) //120
  is ok
• z z2 c
• if (z gt 2) break // or z2 gt 4
  
• drawPixel(Re(z0), Im(z0), iterations
  needed)

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Mandelbrot set sudocode

• for (every real part for c)
  //should be between -2 and 2
• for (every imaginary part for c)
  //should be between -2 and 2
• z c //z00,
  therefore z1c
• for (fixed number of iterations) //40 is
  ok here
• z z2 c
• if (z gt 2) break // or z2 gt 4
  
• drawPixel(Re(z0), Im(z0), iterations
  needed)

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Colors

• We map iterations to colors
• Various ways how to do it
• Besides iterations maximum number of iterations
  (maxiter)
• We want to set values for r,g,b
• Maybe you want point inside the set to be black
• If (iterations maxiter) r b g 0
• Examples
• a) (r,g,b are floats from 0.0 to 1.0)
• r 1-(1-iterations/maxiter)5
• g 1-(1-iterations/maxiter)3
• b iterations/maxiter

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Colors

• b) (r,g,b are integers from 0 to 255)
• int colorTable163
• 0, 0, 0,
• 0, 0,170,
• 0,170, 0,
• 0,170,170,
• 170, 0, 0,
• 170, 0,170,
• 170, 85, 0,
• 170,170,170,
• 85, 85, 85,
• 85, 85,255,
• 85,255, 85,
• 85,255,255,
• 255, 85, 85,
• 255, 85,255,
• 255,255, 85,
• 255,255,255,

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Tips and tricks

• colors instead of mapping iterations to colors,
  we can map zn (the last one before we leave the
  loop) to colors
• So now we have
• drawPixel(Re(z0), Im(z0), Re(zn), Im(zn))
• We can use it for the inside if the set too!!
• Until now we get drawPixel(Re(z0),
  Im(z0),maxiter) there
• Mandelbrot set if we set z0 to a non-zero
  value, we get the set deformed
• z0 is then called the perturbation term
• Set the maximum number of iterations to a small
  number and we have the sets deformed, too
  (particularly useful for Mandelbrot set)

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References

• http//www.root.cz/clanky/fraktaly-v-pocitacove-gr
  afice-xiv
• http//www.cis.ksu.edu/vaclav/fractals.html