• July 01, 2022, 01:03:11 AM

Author Topic:  Modified binary decomposition  (Read 110 times)

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Offline Adam Majewski

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Modified binary decomposition
« on: May 03, 2022, 01:17:07 PM »
Hi

Here is binary decomposition of target set  for both interior ( attraction time) and exterior ( escaping time ).

Second file is computed without modification :

Code: [Select]
unsigned char ComputeColorOfBD (complex double z)
{





  double cabs2z;


  int i; // number of iteration
  for (i = 0; i < IterMax_LSM; ++i)
    {


cabs2z = cabs2(z); // cabs2(zp-z) = cabs2(z) because zp = zcr = 0

      //
       if ( cabs2z > ER2  ||  cabs2z < AR2 )
        {
        if (creal(z) > 0)
        {  return 0;}
        else {return 255; }
     
     
      }

     

      z = f(z);

    }

  return iColorOfUnknown;


}

first  with modification :

Code: [Select]
// Modified BD
unsigned char ComputeColorOfMBD (complex double z)
{





  double cabs2z;
  double turn;

  int i; // number of iteration
  for (i = 0; i < IterMax_LSM; ++i)
    {


cabs2z = cabs2(z); // cabs2(zp-z) = cabs2(z) because zp = zcr = 0

      //
       if ( cabs2z > ER2  ) // exterior
        {
        if (creal(z) > 0)
        {  return 0;}
        else {return 255; }
     
     
      }
     
      if ( cabs2z  < AR2 ) // interior is tuned
      {
      turn = c_turn(z);
      if (turn < t0 || turn > t0+0.5)
      {  return 0;}
        else {return 255; }
     
     
      }

     

      z = f(z);

    }

  return iColorOfUnknown;


}



Compare the interior of both files :
The idea is from video: Adrien Douady _ La dynamique du lapin (1996)

Automatic translation : we  take a small sector from origins and its reciprocal images. If we rotate this small sector something special happens at some point : it aligns with the it's images and forms what are called carrots.

What do you think ?


Linkback: https://fractalforums.org/index.php?topic=4743.0

Offline Adam Majewski

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Re: Modified binary decomposition
« Reply #1 on: May 03, 2022, 02:06:40 PM »
Now with LSM . It show internal rays, and read internal angle

Offline Adam Majewski

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Re: Modified binary decomposition
« Reply #2 on: May 03, 2022, 02:38:43 PM »
only boundaries

Offline Adam Majewski

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Re: Modified binary decomposition
« Reply #3 on: May 03, 2022, 03:04:26 PM »

Offline Adam Majewski

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  • Posts: 677
Re: Modified binary decomposition
« Reply #4 on: May 04, 2022, 06:12:44 PM »
Code: [Select]
// for MBD
  t0 = ((double) numerator) / period; // Is it iternal angle from inetrnal adress  ???

so for 3/7


Offline Adam Majewski

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  • Posts: 677
Re: Modified binary decomposition
« Reply #5 on: June 12, 2022, 05:16:47 AM »
error  in the code: it should be cimag not creal ( for decomposition of target set)
Code: [Select]
// for MBD
 double t0 = 1.0 / 3.0; // period = 3

// Modified BD
unsigned char ComputeColorOfMBD (complex double z)
{

  double cabsz;
  double turn;

  int i; // number of iteration
  for (i = 0; i < IterMax_LSM; ++i)
    {


cabsz = cabs(z); // numerical speed up : cabs2(zp-z) = cabs2(z) because zp = zcr = 0

      //  if z is inside target set ( orbit trap) = exterior of circle with radius ER
       if ( cabsz > ER  ) // exterior
        {
        if (creal(z) > 0) // binary decomposition of target set
        {  return 0;}
        else {return 255; }
     
     
      }
     
      if ( cabsz  < AR ) // if z is inside target set ( orbit trap) = interior of cirlce with radius AR
      {
      turn = c_turn(z);
      if (turn < t0 || turn > t0+0.5) // modified binary decomposition of target set
      {  return 0;}
        else {return 255; }
     
     
      }

     

      z = f(z);

    }

  return iColorOfUnknown;


}


now in the exterior the radial borders of BD is at dyadic angles (shows external rays at dyadic angles).
In the interior BD also shows good approximation of Riemann mapping for each component
« Last Edit: June 12, 2022, 05:35:54 AM by Adam Majewski, Reason: descr »


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