Fractals/Computer graphic techniques/2D/plane

2D plane direct links:

• coordinate
• transformations ( maps )
• quality of the image
• graphic files
• parameter files
• Size and resolution of the image
• Scanning, sampling and decomposition

Geometry

• Euclidean
• non-Euclidean
  • hyperbolic
  • eliptic

Differences between Euclidean and non-Eucl. geometries: behavior of lines with a common perpendicular in each of the three types of geometry

Quality of image geometry:

• good ( not distorted)
• bad ( distorted)

visualisation

• grid
• level sets
  • Zebra striping in computer graphics
• checker board ( chess board)

Viewport (visible part of the plane )

A viewport is a polygon viewing region in computer graphics.

Description

Rectangle part of 2D plane can be described by :

• corners ( 4 real numbers or 2 complex numbers = 2D points)
• center and :
  • width ( 3 real numbers )
  • magnification
  • radius

"People specify fractal coordinates in many ways. Some people use the coordinates of the upper-left and lower-right visible points, specifying the coordinates as four numbers x1, y1, x2, y2. To set the same viewpoint in XaoS, set the real portion of the center to (x1+x2)/2, the imaginary part of center to (y1+y2)/2, and the radius to the greater of x2-x1 and y2-y1." ( from Xaos doc )

Corners

Standard description in Fractint, Ultra Fractal, ChaosPro and Fractal Explorer are corners. For example initial plane for Mandelbrot set is

 Corners:                X                     Y
 Top-l          -2.5000000000000000    1.5000000000000000
 Bot-r           1.5000000000000000   -1.5000000000000000
 Ctr -0.5000000000000000   0.0000000000000000  Mag 6.66666667e-01
 X-Mag-Factor     1.0000   Rotation    0.000   Skew    0.000

Display window of parameter plane has :

• a horizontal width of 4 (real)
• a vertical width (height) of 3 (imag)
• an aspect ratio (proportion) 4/3 ( also in pixels 640/480 so ther is no distorsion)
• center z=-0.5

See demo par file :

Mandel_Demo        { ; PAR for initialization of Fractint demo
  reset=1900 type=mandel corners=-2.5/1.5/-1.5/1.5 params=0/0 inside=0
  sound=no
  }

For julia set/ dynamic plane has :

Corners:                X                     Y
Top-l          -2.0000000000000000    1.5000000000000000
Bot-r           2.0000000000000000   -1.5000000000000000
Ctr  0.0000000000000000   0.0000000000000000  Mag 6.66666667e-01
X-Mag-Factor     1.0000   Rotation    0.000   Skew    0.000

Description from documentation of Fractint :

CORNERS=[xmin/xmax/ymin/ymax[/x3rd/y3rd]]

Example: corners=-0.739/-0.736/0.288/0.291

Begin with these coordinates as the range of x and y coordinates, rather than the default values of (for type=mandel) -2.0/2.0/-1.5/1.5. When you specify four values (the usual case), this defines a rectangle: x- coordinates are mapped to the screen, left to right, from xmin to xmax, y-coordinates are mapped to the screen, bottom to top, from ymin to ymax. Six parameters can be used to describe any rotated or stretched parallelogram: (xmin,ymax) are the coordinates used for the top-left corner of the screen, (xmax,ymin) for the bottom-right corner, and (x3rd,y3rd) for the bottom-left. Entering just "CORNERS=" tells Fractint to use this form (the default mode) rather than CENTER-MAG (see below) when saving parameters with the [B] command.

int SetPlaneFromCorners(double CxMin , double CxMax, double CyMin , double CyMax){
  
  
  
  Radius = fabs(CyMax-CyMin)/2.0;
  Magnification = 1.0/Radius;
  
  // http://www.purplemath.com/modules/midpoint.htm
  Center = (CxMax+CxMin)/2.0 + I*(CyMax+CyMin)/2.0;
  
    
  return 0; 
 


}

Center and ...

 "If you use the center, you can change the zoom level and the plot zooms in/out smoothly on the same center point. " Duncan C). 

magnification

Fractint uses Mag ( compare with Xmagfactor)

 CENTER-MAG=[Xctr/Yctr/Mag[/Xmagfactor/Rotation/Skew]]

This is an alternative way to enter corners as a center point and a magnification that is popular with some fractal programs and publications. Entering just "CENTER-MAG=" tells Fractint to use this form rather than CORNERS (see above) when saving parameters with the [B] command. The [TAB] status display shows the "corners" in both forms. When you specify three values (the usual case), this defines a rectangle: (Xctr, Yctr) specifies the coordinates of the center of the image.

Mag indicates the amount of magnification to use. Initial value ( no zoom ) is 6.66666667e-01.

Six parameters can be used to describe any rotated or stretched parallelogram: Xmagfactor tells how many times bigger the x- magnification is than the y-magnification,

Rotation indicates how many degrees the image has been turned.

Skew tells how many degrees the image is leaning over. Positive angles will rotate and skew the image counter-clockwise.

Parameters can be saved to parmfile called fractint.par

width

Wolf Jung uses center and width in Mandel:

/* 
   from mndlbrot.cpp  by Wolf Jung (C) 201
   These classes are part of Mandel 5.7, which is free software; you can
   redistribute and / or modify them under the terms of the GNU General
   Public License as published by the Free Software Foundation; either
   version 3, or (at your option) any later version. In short: there is
   no warranty of any kind; you must redistribute the source code as well
*/
void mndlbrot::startPlane(int sg, double &xmid, double &rewidth) const
{  if (sg > 0) 
     { xmid = -0.5; rewidth = 1.6; } // parameter plane
     else { xmid = 0; rewidth = 2.0; } // dynamic plane

width and height

to describe plane ( view ) Xaos uses:

• 4 numbers in scripts
• 3 numbers in menu

Xaos uses to call it "radius" but it is defind as : the greater of (x2-x1= width) and y2-y1=height."

radius

It is very usefull for zooming : fix center and only change radius.

Claude Heiland-Allen use center and radius for parameter plane of complex quadratic polynomial:

Radius is defined as:

• the radius of a circle that fits within the actual frame of the image
• "the difference in imaginary coordinate between the center and the top of the axis-aligned view rectangle".
• "px_radius is half the width of the image (in real coordinates)"

view="-0.75 0 1.5" # standard view of parameter plane : center_re, center_im, radius

$radius={\frac {1}{Mag}}={\frac {height}{2}}$

  // modified code using center and radius to scan the plane 
  int height = 720;
  int width = 1280;
  double dWidth;
  double dRadius = 1.5;
  double complex center= -0.75*I;
  double complex c;
  int i,j;

  double width2; // = width/2.0
  double height2; // = height/2.0
  
  width2 = width /2.0;
  height2 = height/2.0;

complex double coordinate(int i, int j, int width, int height, complex double center, double radius) {
  double x = (i - width /2.0) / (height/2.0);
  double y = (j - height/2.0) / (height/2.0);
  complex double c = center + radius * (x - I * y);
  return c;
}

for ( j = 0; j < height; ++j) {
    for ( i = 0; i < width; ++i) {
      c = coordinate(i, j, width, height, center, dRadius);
      // do smth 
       }
     }

another version

int main()
{
  int aa = 4;
  int w = 800 * aa;
  int h = 800 * aa;
  
  #pragma omp parallel for schedule(static, 1)
  for (int j = 0; j < h; ++j)
  {
    double y = (h/2 - (j + 0.5)) / (h/2) * r;
    for (int i = 0; i < w; ++i)
    {
      double x = (i + 0.5 - w/2) / (h/2) * r;
      double _Complex c = x + I * y;
      // proceed
    }
  }
  
  return 0;
}

The same in GLSL ( for Shadertoy) :

 
vec2 GiveCoordinate(vec2 center, float radius, vec2 fragCoord, vec3 iResolution)
{
 
  // from pixel to world coordinate   
  // now point (0,0) is left bottom
  // point (iResolution.x, iResolution.y) is right top   
  float x = (fragCoord.x -  iResolution.x/2.0)/ (iResolution.y/2.0);
  float y = (fragCoord.y -  iResolution.y/2.0)/ (iResolution.y/2.0);
  vec2 c = vec2(center.x + radius*x, center.y + radius*y) ;  
  return c ; 
}

Set plane corners from radius, center and aspect ratio:

 
int SetPlane(complex double center, double radius, double a_ratio){

	ZxMin = creal(center) - radius*a_ratio;	
	ZxMax = creal(center) + radius*a_ratio;	//0.75;
	ZyMin = cimag(center) - radius;	// inv
	ZyMax = cimag(center) + radius;	//0.7;
    double PixelWidth = (ZxMax-ZxMin)/iWidth;
    double PixelHeight = (ZyMax-ZyMin)/iHeight; // pixel_size = PixelWidth = PixelHeight
	return 0;

}

Orientation

There is a major incompatability between:

• the standard Math convention taught in Math class = standard coordinates (y-up)
• some CS graphics programming systems (but not all) = y-down

Check orientation of the plane by marking first quadrant of Cartesian plane :

if (x>0 && y>0) Color=MaxColor-Color;

It should be in upper right position.

OpenGL is right handed in object space and world space. But in window space (aka screen space) we are suddenly left handed.

See also

• math.stackexchange question: proper-name-for-inverted-cartesian-coordinate-system

Aspect ratio

• image
• display

image

The aspect ratio of an image is the ratio of its width to its height, and is expressed with two numbers separated by a colon, such as 16:9, sixteen-to-nine. For the x:y aspect ratio, the image is x units wide and y units high. Common aspect ratios are 1.85:1 and 2.39:1 in cinematography, 4:3 and 16:9 in television photography, and 3:2 in still photography.

Display Aspect ratio

The aspect ratio, or more precisely the Display Aspect Ratio (DAR) – the aspect ratio of the image as displayed. For TV:

• for TV DAR as traditionally 4:3 (a.k.a. Fullscreen)
• now is 16:9 (a.k.a. Widescreen) now the standard for HDTV

Optimisation

• "recursive algorithm to split the image up into blocks, and tests each block to see whether it lies inside a “black area”." by Michael Hogg
• Poincaré half-plane metric for zoom animation
• A Simple Optimization of Fractal Animation by Wei-Yin Chen, You-Sheng Yang and Kun-Mao Liang
• optimizing zoom animations by Claude Heiland-Allen
• reenigne blog : recursive subdivision
• Xaos - algorithm description
• New fast method !!!!
• Zooming by Albert Lobo Cusidó
• Rendering the Mandelbrot Set — with an example implementation in javascript By Christian Stigen Larsen
• perturbation

Patterns

• Stripe Patterns on Surfaces" given by Keenan Crane