Fractals/Iterations in the complex plane/stripeAC

introduction "Average colorings are a family of coloring functions that use the decimal part of the smooth iteration count to interpolate between average sums." Jussi Harkonen : On Fractal Coloring Techniques Here stripes are lines perpendicular to the iteration bands ( level sets of escape time). These " radial strands ... follows the external rays which are of great importance in the theoretical study of M and in holomorphic dynamics." (A Cheritat)A Cheritat wiki : Mandelbrot_set - Radial_strands " is basically a cheap way to calculate an angle. TIA is averaging the angle over all iterations to get a smooth result. So there is some initialization and some calculations per iteration to do the sum. It has to store the sum at the previous iteration before adding the next one so you can interpolate between them to get a continuous function between iteration bands (in the same way that continuous potential is usually interpolated). This interpolation is done after the iterations bail out."fractalforums : triangle-inequality-average-coloring (xenodreambuie - author of Jux) " ... it probably shows curves close to external rays, but nevertheless you do not have an explicit relation between the number returned by SAC and the true external argument." Wolf Jung name SAM = Stripe Average Method SAC = Stripe Average Coloringfractalforum : master's-thesis-on-fractal-coloring-techniques radial strands history A new coloring called the Stripe Average was introduced by Jussi Härkönen in 2007. It is based on the behavior of the Triangle Inequality Average coloring. examples Images fractalforums stripe-average-mandelbrot-2 by Zephitmaal video youtube: Mandelbrot Coloring Algorithms by sfract algorithm For every point c of rectangle from parameter plane do:fractalforums : stripe-average-coloring Calculate the orbit of the point z0. The result is series of complex numbers {z0,z1,z2,...} Apply function t to this series to get a new series of real numbers: { t0=t(z0), t1=t(z1), t2=t(z2), ...}. Take the average A ( = arithmetic mean) of this series of real numbers map real number A to a color ( coloring function) orbit Truncated orbit: Skipped orbit is a subset of truncated orbit: skip a first k+1 of items from the sequence do not use some first k+1 items to compute average addend function The addend function t maps complex number to real number is continuous on the elements of elements in skipped orbit where : s is a stripe density average full Average of full truncated orbit: compute average from all points of the truncated orbit If one compute average of all points ( from i=1 to i=n) then: stripes will be visible ( good) level sets of escape time will be also visible (bad, discontinuity ) The solution is : the interpolation skipped truncated orbit skipped Average of skipped orbit: skip first (k+1) items from computing average note that: It removes some discontinuities but not level sets of escape time interpolated Level sets of escape time are still visible ( discontinuity). One can remove them using interpolation.wikipedia : Interpolation interpolation: between 2 values using: linear function = linear interpolation splines = spline interpolation polynomial steps: Construct a smooth iteration count u = real number ( see equation 3.11 from page 21 in J Harkonen thesis) take it's fractional partwijkipedia : Fractional part d Decimal (frational) part d is: zero at boundary of previous level set converges to 1 in the neighborhood of next level set boundary "Thus it can be used as the interpolation coefficient." linear To compute linear average interpolate between two values: one obtained from the "full" series ( truncated and skipped) = Avg = one obtained from the average of all t elements except the last one = prevAvg = see equation 4.2 from page 28 in J Harkonen thesis or in GLSL code by Syntopia: float mix = frac*avg+(1.0-frac)*prevAvg; Here is important fragment of GLSL code by Syntopia: // iteration for ( i = 0; i < Iterations; i++) { z = complexMul(z,z) + c; if (i>=Skip) { count++; lastAdded = 0.5+0.5*sin(StripeDensity*atan(z.y,z.x)); avg += lastAdded; } z2 = dot(z,z); if (z2> EscapeRadius2 && i>Skip) break; } prevAvg = (avg -lastAdded)/(count-1.0); avg = avg/count; frac =1.0+(log2(log(EscapeRadius2)/log(z2))); float mix = frac*avg+(1.0-frac)*prevAvg; spline Compute spline average using Catmull-Rom splines: Polynomials : then see equation 4.3 from page 28 in J Harkonen thesis Code java script by 2013 Douglas Haber dailyfractalart explorer and description of js code M set by Jeremy Ashkenas C M_LN2 is defined in math.hstackoverflow question: how-to-use-m-ln2-from-math-h ? #define M_LN2 0.69314718055994530942 /* log_e 2 */ The natural logarithm of the 2.M_LN2 /* c program: console, 1-file samm.c samm = Stripe Average Method with : - skipping first (i_skip+1) points from average - linear interpolation en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/stripeAC -------------------------------- 1. draws Mandelbrot set for complex quadratic polynomial Fc(z)=z*z +c using samm = Stripe Average Method ------------------------------- 2. technique of creating ppm file is based on the code of Claudio Rocchini http://en.wikipedia.org/wiki/Image:Color_complex_plot.jpg create 24 bit color graphic file , portable pixmap file = PPM see http://en.wikipedia.org/wiki/Portable_pixmap to see the file use external application ( graphic viewer) ----- it is example for : https://www.math.univ-toulouse.fr/~cheritat/wiki-draw/index.php/Mandelbrot_set ------------- compile : gcc samm.c -lm -Wall ./a.out -------- git -------- cd existing_folder git init git remote add origin git@gitlab.com:adammajewski/mandelbrot_wiki_ACh.git git add samm.c git commit -m "samm + linear interpolation " git push -u origin master */ #include <stdio.h> #include <math.h> #include <complex.h> // https://stackoverflow.com/questions/6418807/how-to-work-with-complex-numbers-in-c #define M_PI 3.14159265358979323846 /* pi */ /* screen ( integer) coordinate */ int iX,iY; const int iXmax = 1200; const int iYmax = 1000; /* world ( double) coordinate = parameter plane*/ double Cx,Cy; const double CxMin=-2.5; const double CxMax=1.5; const double CyMin=-5.0/3.0; const double CyMax= 5.0/3.0; /* */ double PixelWidth; //=(CxMax-CxMin)/iXmax; double PixelHeight; // =(CyMax-CyMin)/iYmax; /* color component ( R or G or B) is coded from 0 to 255 */ /* it is 24 bit color RGB file */ const int MaxColorComponentValue=255; FILE * fp; char *filename="samm_i.ppm"; // https://commons.wikimedia.org/wiki/File:Mandelbrot_set_-_Stripe_Average_Coloring.png char *comment="# ";/* comment should start with # */ static unsigned char color[3]; // 24-bit rgb color unsigned char s = 5; // stripe density const int IterationMax=1000; // N in wiki int i_skip = 1; // exclude (i_skip+1) elements from average /* bail-out value for the bailout test for exaping points radius of circle centered ad the origin, exterior of such circle is a target set */ const double EscapeRadius=10000; // = ER big !!!! double lnER; // ln(ER) double complex give_c(int iX, int iY){ double Cx,Cy; Cy=CyMin + iY*PixelHeight; if (fabs(Cy)< PixelHeight/2) Cy=0.0; /* Main antenna */ Cx=CxMin + iX*PixelWidth; return Cx+Cy*I; } // the addend function // input : complex number z // output : double number t double Give_t(double complex z){ return 0.5+0.5*sin(s*carg(z)); } /* input : - complex number - integer output = average */ double Give_Arg(double complex C , int iMax) { int i=0; // iteration double complex Z= 0.0; // initial value for iteration Z0 double A = 0.0; // A(n) double prevA = 0.0; // A(n-1) double R; // =radius = cabs(Z) double d; // smooth iteration count // iteration = computing the orbit for(i=0;i<iMax;i++) { Z=Z*Z+C; // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/qpolynomials if (i>i_skip) A += Give_t(Z); // R = cabs(Z); if(R > EscapeRadius) break; // exterior of M set prevA = A; // save value for interpolation } // for(i=0 if (i == iMax) A = -1.0; // interior else { // exterior // computing interpolated average A /= (i - i_skip) ; // A(n) prevA /= (i - i_skip - 1) ; // A(n-1) // smooth iteration count d = i + 1 + log(lnER/log(R))/M_LN2; d = d - (int)d; // only fractional part = interpolation coefficient // linear interpolation A = d*A + (1.0-d)*prevA; } return A; } /* input = complex number output - color array as Formal parameters - int = return code */ int compute_color(complex double c, unsigned char color[3]){ double arg; unsigned char b; // compute arg = Give_Arg( c, IterationMax); // if (arg < 0.0) { /* interior of Mandelbrot set = inside_color = */ color[0]=191; // color[1]=191; color[2]=191; } else // exterior of Mandelbrot set = CPM { // gray gradient b = (unsigned char) (255 - 255*arg ); color[0]= b; /* Red*/ color[1]= b; /* Green */ color[2]= b; /* Blue */ }; return 0; } void setup(){ // PixelWidth=(CxMax-CxMin)/iXmax; PixelHeight=(CyMax-CyMin)/iYmax; lnER = log(EscapeRadius); // ln(ER) /*create new file,give it a name and open it in binary mode */ fp= fopen(filename,"wb"); /* b - binary mode */ /*write ASCII header to the file*/ fprintf(fp,"P6\n %s\n %d\n %d\n %d\n",comment,iXmax,iYmax,MaxColorComponentValue); } void info(){ double distortion; // widt/height double PixelsAspectRatio = (double)iXmax/iYmax; // https://en.wikipedia.org/wiki/Aspect_ratio_(image) double WorldAspectRatio = (CxMax-CxMin)/(CyMax-CyMin); printf("PixelsAspectRatio = %.16f \n", PixelsAspectRatio ); printf("WorldAspectRatio = %.16f \n", WorldAspectRatio ); distortion = PixelsAspectRatio - WorldAspectRatio; printf("distortion = %.16f ( it should be zero !)\n", distortion ); printf("\n"); printf("bailut value = Escape Radius = %.0f \n", EscapeRadius); printf("IterationMax = %d \n", IterationMax); printf("i_skip = %d = number of skipped elements ( including t0 )= %d \n", i_skip, i_skip+1); // file printf("file %s saved. It is called .... in A Cheritat wiki\n", filename); } void close(){ fclose(fp); info(); } // ************************************* main ************************* int main() { complex double c; setup(); printf(" render = compute and write image data bytes to the file \n"); for(iY=0;iY<iYmax;iY++) for(iX=0;iX<iXmax;iX++) { // compute pixel coordinate c = give_c(iX, iY); /* compute pixel color (24 bit = 3 bytes) */ compute_color(c,color); /*write color to the file*/ fwrite(color,1,3,fp); } close(); return 0; } Ultra Fractal Ultra Fractal uf jh.ucl Stripes { ; Jussi Härkönen, 07-01-02 ; See Fibers and Things by Ron Barnett for a similar coloring ; for convergent fractals. ; ; TIA originally developed by Kerry Mitchell ; Curvature originally developed by Damien Jones ; ; 07-02-15 Added the interpolation mode parameter ; 07-03-07 Added the "None" interpolation mode ; 07-03-30 Added Skip iterations, seed (TIA only) and Mandel version (TIA only) ; parameters ; ; 16-10-30 Stripes + Perlin routines added by jam. ; ; from jh.ucl ; http://formulas.ultrafractal.com/cgi/formuladb?view;file=jh.ucl;type=.txt global: ; Precalculate the attenuation factor for attenuated sum float attenuationFactor = 1 - exp(-@attenuation) float lp = log(log(@bailout)) float twopi = 2 * #pi if (iteration > @skipIter) if (@coloring == "Stripes") ; Angle dependent component in [-1,1] float TkAng = sin(@angularFrequency * atan2(z) + @angularOffset) ; Radius-dependent component in [-1,1] float TkRad = sin(@circularFrequency * log((cabs(z))) + @circularOffset) ; The weighted sum of radius and angle dependent terms is in [-1,1] ; Scale and translate it to [0,1] Tk = 0.5 + 0.5 * ((1 - @circularWeight) * TkAng + @circularWeight * TkRad) elseif (@coloring == "Curvature") ; Skip two first iterations in order to get two nonzero previous values if (iteration > @skipIter + 2) q = (z - zPrev) / (zPrev - zPrev2) Tk = (abs(atan2(q)) / (pi)) else Tk = 0 endif else ; "TIA" ; Skip first iteration if (iteration > 1 + @skipIter) ; |z(k-1)^n| if (@usePixel) ; Use pixel value znAbs = cabs(z - #pixel) else ; Use seed specified by the user znAbs = cabs(z - @seed) endif ; Bounds: ; m(k) = ||z(k-1)|^n - |c|| float lowBound = abs(znAbs - cAbs) ; M(k) = |z(k-1)|^n + |c| float upBound = znAbs + cAbs ; T(k) Tk = (cabs(z) - lowBound) / (upBound - lowBound) endif endif ; Update previous sums sum3 = sum2 sum2 = sum1 sum1 = sum ; Calculate S(k) if (@attenuate) sum = attenuationFactor*sum + Tk else sum = sum + Tk endif endif ; ... ; #index = tempcolor default: title = "Stripes" param coloring caption = "Coloring mode" default = 0 enum = "Stripes" "TIA" "Curvature" hint = "Specifies the coloring algorithm." endparam param usePixel caption = "Mandel version" default = false visible = @coloring == "TIA" hint = "Specifies whether to use the algorithm adapted \ to Julia or Mandelbrot sets" endparam param seed caption = "Seed" default = (0.5,0.25) visible = (@coloring == "TIA") && (@usePixel == false) hint = "Seed of the Julia set." endparam param interpolation caption = "Interpolation" default = 0 enum = "Linear" "Smooth" "None" hint = "Specifies the interpolation method." endparam float param circularWeight caption = "Circle weight" default = 0.0 visible = (@coloring == "Stripes") min = 0 max = 1 hint = "Weight of circular and radial components. Weight = 0 shows only \ stripes, whereas weight = 1 shows only circles." endparam float param circularFrequency caption = "Circle frequency" default = 5.0 visible = (@coloring == "Stripes") hint = "Specifies the density of circles. Use integer values for smooth results." endparam float param circularOffset caption = "Circle offset" default = 0.0 visible = (@coloring == "Stripes") hint = "Circle offset given in radians." endparam float param angularFrequency caption = "Stripe density" default = 5.0 visible = (@coloring == "Stripes") hint = "Specifies the density of stripes. Use integer values for smooth results." endparam float param angularOffset caption = "Stripe offset" default = 0.0 visible = (@coloring == "Stripes") hint = "Stripe offset given in radians." endparam int param skipIter caption = "Skip iterations" default = 0 hint = "Excludes a given number of iterations from the end of the orbit." endparam ; Fractional iteration count parameters float param bailout caption = "Bailout" default = 1e20 min = 1 hint = "Bail-out value used by the fractal formula. Use large bailouts for \ best results." endparam ; Attenuation parameters bool param attenuate caption = "Moving average" default = false hint = "Use moving average instead of average. This may make the coloring \ structure appear clearer (especially with TIA), but may also make busy \ areas to appear flat." endparam float param attenuation caption = "Filter factor" default = 2 visible = @attenuate min = 0 hint = "Specifies the moving average strength. Values close to 0 cause the last \ terms to be weighted creating results similar to Cilia. Large values \ make the sum look more like the usual average." endparam } Fragmentarium // http://www.fractalforums.com/general-discussion/stripe-average-coloring/ // code by Syntopia #include "2D.frag" #group Mandelbrot // Number of iterations uniform int Iterations; slider[10,200,5000] uniform float R; slider[0,0,1] uniform float G; slider[0,0.4,1] uniform float B; slider[0,0.7,1] uniform bool Julia; checkbox[false] uniform float JuliaX; slider[-2,-0.6,2] uniform float JuliaY; slider[-2,1.3,2] vec2 c2 = vec2(JuliaX,JuliaY); void init() {} vec2 complexMul(vec2 a, vec2 b) { return vec2( a.x*b.x - a.y*b.y,a.x*b.y + a.y * b.x); } vec2 mapCenter = vec2(0.5,0.5); float mapRadius =0.4; uniform bool ShowMap; checkbox[true] uniform float MapZoom; slider[0.01,2.1,6] vec3 getMapColor2D(vec2 c) { vec2 p = (aaCoord-mapCenter)/(mapRadius); p*=MapZoom; p.x/=pixelSize.x/pixelSize.y; if (abs(p.x)<2.0*pixelSize.y*MapZoom) return vec3(0.0,0.0,0.0); if (abs(p.y)<2.0*pixelSize.x*MapZoom) return vec3(0.0,0.0,0.0); p +=vec2(JuliaX, JuliaY) ; vec2 z = vec2(0.0,0.0); int i = 0; for (i = 0; i < Iterations; i++) { z = complexMul(z,z) +p; if (dot(z,z)> 200.0) break; } if (i < Iterations) { float co = float( i) + 1.0 - log2(.5*log2(dot(z,z))); co = sqrt(co/256.0); return vec3( .5+.5*cos(6.2831*co),.5+.5*cos(6.2831*co),.5+.5*cos(6.2831*co) ); } else { return vec3(0.0); } } // Skip initial iterations in coloring uniform int Skip; slider[0,1,100] // Scale color function uniform float Multiplier; slider[-10,0,10] uniform float StripeDensity; slider[-10,1,10] // To test continous coloring uniform float Test; slider[0,1,1] uniform float EscapeRadius2; slider[0,1000,100000] vec3 getColor2D(vec2 c) { if (ShowMap && Julia) { vec2 w = (aaCoord-mapCenter); w.y/=(pixelSize.y/pixelSize.x); if (length(w)<mapRadius) return getMapColor2D(c); if (length(w)<mapRadius+0.01) return vec3(0.0,0.0,0.0); } vec2 z = Julia ? c : vec2(0.0,0.0); int i = 0; float count = 0.0; float avg = 0.0; // our average float lastAdded = 0.0; float z2 = 0.0; // last squared length for ( i = 0; i < Iterations; i++) { z = complexMul(z,z) + (Julia ? c2 : c); if (i>=Skip) { count++; lastAdded = 0.5+0.5*sin(StripeDensity*atan(z.y,z.x)); avg += lastAdded; } z2 = dot(z,z); if (z2> EscapeRadius2 && i>Skip) break; } float prevAvg = (avg -lastAdded)/(count-1.0); avg = avg/count; float frac =1.0+(log2(log(EscapeRadius2)/log(z2))); frac*=Test; float mix = frac*avg+(1.0-frac)*prevAvg; if (i < Iterations) { float co = mix*pow(10.0,Multiplier); co = clamp(co,0.0,10000.0); return vec3( .5+.5*cos(6.2831*co+R),.5+.5*cos(6.2831*co + G),.5+.5*cos(6.2831*co +B) ); } else { return vec3(0.0); } } #preset Default Center = -0.587525,0.297888 Zoom = 1.79585 AntiAliasScale = 1 AntiAlias = 1 Iterations = 278 R = 0 G = 0.4 B = 0.7 Julia = false JuliaX = -0.6 JuliaY = 1.3 ShowMap = true MapZoom = 2.1 Skip = 6 Multiplier = -0.1098 StripeDensity = 1.5384 Test = 1 EscapeRadius2 = 74468 #endpreset #preset Julia Center = -0.302544,-0.043626 Zoom = 4.45019 AntiAliasScale = 1 Iterations = 464 R = 0.58824 G = 0.3728 B = 0.27737 Julia = true JuliaX = -1.26472 JuliaY = -0.05884 ShowMap = false MapZoom = 1.74267 Skip = 4 Test = 1 EscapeRadius2 = 91489 Multiplier = 0.4424 StripeDensity = 2.5 AntiAlias = 2 #endpreset Processing // http://www.fractalforums.com/general-discussion/stripe-average-coloring/ // code by asimes{{typo help inline|reason=similar to amises|date=October 2022}} // result : https://commons.wikimedia.org/wiki/File:Mandelbrot_set_-_Stripe_Average_Coloring.png float xmin = -2.5; float ymin = -2.0; float wh = 4; float stripes = 5.0; int maxIterations = 1000; void setup() { size(10000, 10000, P2D); } void draw() { loadPixels(); float xmax = xmin+wh; float ymax = ymin+wh; float dx = (xmax-xmin)/width; float dy = (ymax-ymin)/height; float x = xmin; for (int i = 0; i < width; i++) { float y = ymin; for (int j = 0; j < height; j++) { float zr = x; float zi = y; float lastzr = x; float lastzi = y; float orbitCount = 0; int n = 0; while (n < maxIterations) { float zrr = zr*zr; float zii = zi*zi; float twori = 2*zr*zi; zr = zrr-zii+x; zi = twori+y; if (zrr+zii > 10000) break; orbitCount += 0.5+0.5*sin(stripes*atan2(zi, zr)); lastzr = zr; lastzi = zi; n++; } if (n == maxIterations) pixels[i+j*width] = 0; else { float lastOrbit = 0.5+0.5*sin(stripes*atan2(lastzi, lastzr)); float smallCount = orbitCount-lastOrbit; orbitCount /= n; smallCount /= n-1; float frac = -1+log(2.0*log(10000))/log(2)-log(0.5*log(lastzr*lastzr+lastzi*lastzi))/log(2); float mix = frac*orbitCount+(1-frac)*smallCount; float orbitColor = mix*255; pixels[i+j*width] = color(orbitColor); } y += dy; } x += dx; } updatePixels(); noLoop(); } void mousePressed(){ save("a10000.png"); } Programs fractview - blog - -branching-and-expsmooth, code fractalmaker deep-fractal by munrocket in java script References