Fractals/Iterations in the complex plane/triangle ineq

thumb|right|TIA algorithm for coloring exterior of the set ( Mandelbrot or Julia) Names family of algorithms : Branching Average Coloring Algorithms (" because calculate an average of a function evaluated at all points in a truncated orbit ")The Coloring of Fractals by Amy Fang and Frances Ling the Triangle Inequality Average Coloring algorithm = TIA = Trippie. " linear interpolation and was designed for the Mandelbrot set. Discontinuous at some points that are mapped to zero" Curvature Average Coloring Algorithm ( developed by Damien M. Jones for Ultra Fractal in 1999, continuous at points that are mapped to zero ) Variants Fibers coloring method ( from Jux program) : " First, I prefer a running average with a weight parameter (0 - 0.999), which avoids any possible precision issues and lets you adjust detail. Second, I use sin or cos instead of the plain angle, and add a frequency parameter. And some other variations" xenodreambuie History Kerry Mitchell 1999.02.05 Damien M. Jones 2000.04.02 Explanation "The triangle inequality 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." xenodreambuie fractalforums triangle-inequality-average-coloring "creates smoothly colored ... large flame-like patterns that extend from the fractal outwards" (from UltraFractal by Frederik Slijkerman )ultrafractal help: solid-color/triangle-inequality-average Parameters escape radius : "Use very large bail-out values (1E20 = 10^20)) for good results." Ultra Fractal iteration Max = 2500 Software Ultra Fractal Fractint Fractview for android : blog - youtube - code Gnofract 4D Visions of Chaos - application for Windows ( GLSL ) - gallery Jux Examples fractals by Craig Macomber with GLSL code Code java script Mandelbrot Average Colorings using cpp + glsl GPU Fractals! By Pete Corey python mandelbrot by Karol Ciszek Ultrafractal TIA /* http://formulas.ultrafractal.com/ comment { dmj-pub.ucl 2.1 Coloring methods for Ultra Fractal 2 by Damien M. Jones April 2, 2000 For more information about this formula collection, please visit its home page: http://www.fractalus.com/ultrafractal/dmj-pub-uf.htm Many of these coloring algorithms are general-purpose tools which work with a variety of fractals. A few are more specialized. Huge portions of this compilation build upon the techniques explored by others. } */ dmj-Triangle { ; ; This coloring method implements a variation of Kerry ; Mitchell's Triangle Inequality Average coloring ; method. Because the smoothing used in this formula ; is based on the dmj-Smooth formula, which only works ; for z^n+c and derivates, the smoothing here will only ; work for those types as well. ; init: float sum = 0.0 float sum2 = 0.0 float ac = cabs(#pixel) float il = 1/log(@power) float lp = log(log(@bailout)/2.0) float az2 = 0.0 float lowbound = 0.0 float f = 0.0 BOOL first = true float ipower = 1/@apower loop: sum2 = sum IF (!first) az2 = cabs(#z - #pixel) lowbound = abs(az2 - ac) IF (@aflavor == 0) sum = sum + ((cabs(#z) - lowbound) / (az2+ac - lowbound))^@apower ELSEIF (@aflavor == 1) sum = sum + 1-(1-(cabs(#z) - lowbound) / (az2+ac - lowbound))^ipower ENDIF ELSE first = false ENDIF final: sum = sum / (#numiter) sum2 = sum2 / (#numiter-1) f = il*lp - il*log(log(cabs(#z))) #index = sum2 + (sum-sum2) * (f+1) default: title = "Triangle Inequality Average" helpfile = "dmj-pub\dmj-pub-uf-tia.htm" param apower caption = "Average Exponent" default = 1.0 hint = "This skews the values averaged by raising them to \ this power. Use 1.0 for the classic coloring." endparam param aflavor caption = "Average Flavor" default = 0 enum = "normal" "reversed" hint = "Controls whether values are reversed before being \ raised to a power. Has no effect if Average Exponent \ is 1.0." endparam param power caption = "Exponent" default = 2.0 hint = "This should be set to match the exponent of the \ formula you are using. For Mandelbrot, this is 2." endparam param bailout caption = "Bailout" default = 1e20 min = 1 hint = "This should be set to match the bailout value in \ the Formula tab. Use a very high bailout!" endparam } // http://www.kerrymitchellart.com/tutorials/formulas2/uf2-2.htm // http://formulas.ultrafractal.com/cgi/formuladb?view;file=lkm.ufm;type=.txt comment { ; copyright Kerry Mitchell 07feb99 Triangle Inequality The triangle inequality method is based on a simple characteristic of complex numbers: the magnitude of the sum of two complex numbers, |a+b|, is strictly limited to a range determined by a and b: |a+b| >= ||a| - |b||, and |a+b| <= |a| + |b|, where |z| is the square root of the sum of the squares of the components, not the sum of the squares, as in Ultra Fractal. The extremes of this inequality are easily seen with a few examples. If a=1 and b=2, then: |a| = 1, |b| = 2; ||a| - |b|| = |1-2| = |-1| = 1; |a| + |b| = 1+2 = 3; |a+b| = |3| = 3; 1 <= 3 <= 3. The upper bound occurs when both addends have the same polar angle. The geometric interpretation of this is that the complex numbers add up, and the length of the sum is simply the sum of the individual lengths. The lower bound occurs when the polar angles of the complex numbers differ by 180 degrees; the two numbers are diametrically opposed. Then, the length of the sum is the difference of the lengths. For example, if a=3i and b=-5i, then: |a| = 3, |b| = 5; ||a| - |b|| = |3-5| = |-2| = 2; |a| + |b| = 3 + 5 = 8; |a+b| = |-2i| = 2; 2 <= 2 <= 8. In general, the length of the sum is somewhere inbetween, and can be thought of in terms of a triangle, which is how the inequality gets its name. If |a| is the length of one side of a triangle, and |b| is the length of the second side, then |a+b| is the length of the third side, and lies somewhere within the range shown above. Back to fractals. The two numbers of interest are z^n and c. Given z (the previous iterate) and c (the Mandelbrot or Julia parameter), the range for the magnitude of the new iterate can then be determined. With this range, the magnitude of the new iterate can be rescaled to a fraction between 0 and 1 inclusive: min = ||z_old| - |c||, max = |z_old| + |c|, z_new = z_old * z_old + c fraction = (|z_new| - min) / (max - min). During the iterating, these fractions are average together. After the last iteration, this average fraction is stored in the real part of z, and the actual fraction for the last iteration is stored in the imaginary part of the z. If the orbit diverges (bails out), then renormalization techniques are used to color the outside pixels smoothly without iteration bands. This works best with large bailout values. The large bailout needs to be reduced with larger exponents, to prevent color gaps due to overflow errors. For best results, use the "basic" coloring method, with "real(z)" to show the average triangulation fraction, or "imag(z)" to show the actual fraction for the last iteration. } triangle-general-julia { ; Kerry Mitchell 05feb99 ; ; z^n+c Julia set, colors by ; triangle inequality ; init: zc=#pixel c=@julparam float rc=cabs(c) float rzc=0.0 zn=(0.0,0.0) float rnz=0.0 int iter=0 bool done=false float logn=log(@nexp) float llbail=log(log(@bailout))+log(0.5) float count=0.0 float count1=0.0 float count2=0.0 float countx=0.0 float county=0.0 float fac1=0.0 float fac2=0.0 float min=0.0 float max=0.0 float k=0.0 float kfrac=0.0 loop: iter=iter+1 if(@nexp==2.0) zn=sqr(zc) rnz=|zc| else zn=zc^@nexp rnz=cabs(zn) endif zc=zn+c min=cabs(rnz-rc) max=rnz+rc rzc=cabs(zc) county=(rzc-min)/(max-min) count=count+county countx=count/iter z=countx+flip(county) if(|zc|>@bailout) done=true count1=count/iter fac1=log(log(rzc))-llbail iter=iter+1 if(@nexp==2.0) zn=sqr(zc) rnz=|zc| else zn=zc^@nexp rnz=cabs(zn) endif min=cabs(rnz-rc) max=rnz+rc zc=zn+c rzc=cabs(zc) county=(rzc-min)/(max-min) count=count+county count2=count/iter fac2=log(log(rzc))-llbail-logn k=exp((fac1+fac2)/2.0) kfrac=(k-1.0)/(@nexp-1.0) countx=kfrac*count1+(1.0-kfrac)*count2 z=countx+flip(county) endif bailout: done==false default: title="Triangle General Julia" maxiter=100 periodicity=0 center=(0.0,0.0) magn=1.0 angle=0 param julparam caption="Julia parameter" default=(1.0,0.0) endparam param bailout caption="bailout value" default=1000000.0 min=0.0 endparam param nexp caption="exponent" default=2.0 hint="z exponent, > 1.0" min=1.0 endparam switch: type="triangle-general-mandelbrot" bailout=bailout nexp=nexp } triangle-general-mandelbrot { ; Kerry Mitchell 05feb99 ; ; z^n+c Mandelbrot set, colors by ; triangle inequality ; init: c=#pixel zc=@manparam+c float rc=cabs(c) float rzc=0.0 zn=(0.0,0.0) float rnz=0.0 int iter=0 bool done=false float logn=log(@nexp) float llbail=log(log(@bailout))+log(0.5) float count=0.5 float count1=0.0 float count2=0.0 float countx=0.0 float county=0.0 float fac1=0.0 float fac2=0.0 float min=0.0 float max=0.0 float k=0.0 float kfrac=0.0 loop: iter=iter+1 if(@nexp==2.0) zn=sqr(zc) rnz=|zc| else zn=zc^@nexp rnz=cabs(zn) endif zc=zn+c min=cabs(rnz-rc) max=rnz+rc rzc=cabs(zc) county=(rzc-min)/(max-min) count=count+county countx=count/iter z=countx+flip(county) if(|zc|>@bailout) done=true count1=count/iter fac1=log(log(rzc))-llbail iter=iter+1 if(@nexp==2.0) zn=sqr(zc) rnz=|zc| else zn=zc^@nexp rnz=cabs(zn) endif min=cabs(rnz-rc) max=rnz+rc zc=zn+c rzc=cabs(zc) county=(rzc-min)/(max-min) count=count+county count2=count/iter fac2=log(log(rzc))-llbail-logn k=exp((fac1+fac2)/2.0) kfrac=(k-1.0)/(@nexp-1.0) countx=kfrac*count1+(1.0-kfrac)*count2 z=countx+flip(county) endif bailout: done==false default: title="Triangle General Mandelbrot" maxiter=100 periodicity=0 center=(0.0,0.0) magn=1.0 angle=0 param manparam caption="Mandelbrot start" default=(0.0,0.0) hint="use (0,0) for standard Mandelbrot" endparam param bailout caption="bailout value" default=1000000.0 min=0.0 endparam param nexp caption="exponent" default=2.0 hint="z exponent, > 1.0" min=1.0 endparam switch: type="triangle-general-julia" julparam=pixel bailout=bailout nexp=nexp } CAA from dmj.ucl: comment { dmj-pub.ucl 2.1 Coloring methods for Ultra Fractal 2 by Damien M. Jones April 2, 2000 For more information about this formula collection, please visit its home page: http://www.fractalus.com/ultrafractal/dmj-pub-uf.htm Many of these coloring algorithms are general-purpose tools which work with a variety of fractals. A few are more specialized. Huge portions of this compilation build upon the techniques explored by others. } dmj-Curvature { ; ; This formula averages curvature over the course ; of all iterations by averaging the difference in ; absolute value of angles between orbit steps. The ; technique was suggested by Prof. Javier Barrallo. ; ; Note that the results are often very similar to ; Kerry Mitchell's Triangle Inequality Average method, ; but are slightly more angular. ; init: complex zold = (0,0) complex zold2 = (0,0) float a = 0.0 float a2 = 0.0 int i = 0 loop: a2 = a IF (i >= 2 && @aflavor != 2) ; zold and zold2 are valid a = a + abs(atan2((#z-zold)/(zold-zold2))) ; update average ENDIF i = i + 1 ; count the iteration zold2 = zold ; save the orbit values zold = #z final: IF (@aflavor == 0) float il = 1/log(@power) float lp = log(log(@bailout)/2.0) float f = il*lp - il*log(log(cabs(#z))) a = a / i a2 = a2 / (i-1) #index = (a2 + (a-a2) * (f+1)) / #pi ELSEIF (@aflavor == 1) #index = (a/i) / #pi + 1 ELSE a = atan2((#z-zold)/(zold-zold2)) ; update average #index = a / #pi + 1 ENDIF Visions of Chaos = VOC #version 400 // http://www.fractalforums.com/programming/triangle-inequality-average-coloring/ // code by Softology uniform vec2 resolution; uniform vec3 palette[256]; uniform double xmin; uniform double xmax; uniform double ymin; uniform double ymax; uniform double bailout; uniform int maxiters; double bailout_squared=double(bailout*bailout); double magnitude,r1,r2,g1,g2,b1,b2,tweenval; double realiters; vec4 finalcol,col; int superx,supery; double stepx=(xmax-xmin)/resolution.x; double stepy=(ymax-ymin)/resolution.y; int colval,colval1,colval2; dvec2 z,c; //triangle inequality average coloring double sum,sum2,ac,il,lp,az2,lowbound,f,index,tr,ti; int mandelbrotPower; double rval,gval,bval,rval1,gval1,bval1,rval2,gval2,bval2; void main(void) { sum=0; sum2=0; ac=0; il=0; lp=0; mandelbrotPower=2; finalcol=vec4(0,0,0,0); c.x = xmin+gl_FragCoord.x/resolution.x*(xmax-xmin); c.y = ymin+gl_FragCoord.y/resolution.y*(ymax-ymin); int i; z = dvec2(0.0,0.0); //triangle inequality average coloring sum = 0; sum2 = 0; ac = sqrt(c.x * c.x + c.y * c.y); il = 1.0 / log(mandelbrotPower); lp = log(float(log(float(bailout)) / mandelbrotPower)); az2 = 0.0; lowbound = 0.0; f = 0.0; index = 0.0; for(i=0; i<maxiters; i++) { //START OF FRACTAL FORMULA double x = (z.x * z.x - z.y * z.y) + c.x; double y = (z.y * z.x + z.x * z.y) + c.y; //END OF FRACTAL FORMULA magnitude=(x * x + y * y); if(magnitude>bailout_squared) break; z.x = x; z.y = y; //tia sum2=sum; if ((i!=0)&&(i!=maxiters-1)) { tr=z.x-c.x; ti=z.y-c.y; az2=sqrt(tr * tr + ti * ti); lowbound=abs(az2 - ac); sum+=((sqrt(z.x * z.x + z.y * z.y)-lowbound)/(az2+ac-lowbound)); } } if (i==maxiters) { col=vec4(0.0,0.0,0.0,1.0); } else { //triangle inequality average sum=sum/i; sum2=sum2/(i-1.0); f=il*lp - il*log(log(float(length(z)))); index=sum2+(sum-sum2)*(f+1.0); realiters=255*index; colval1= int(mod(realiters,255)); colval2=int(mod((colval1+1),255)); tweenval=fract(realiters); if (colval1<0) { colval1=colval1+255; } if (colval2<0) { colval2=colval2+255; } rval1 =palette[colval1].r; gval1 =palette[colval1].g; bval1 =palette[colval1].b; rval2 =palette[colval2].r; gval2 =palette[colval2].g; bval2 =palette[colval2].b; rval =rval1 +((rval2 - rval1)*tweenval); gval =gval1 +((gval2 - gval1)*tweenval); bval =bval1 +((bval2 - bval1)*tweenval); col=vec4(rval,gval,bval,1.0); } gl_FragColor = vec4(col); } Video Mandelbrot Coloring Algorithms by sfract Julia fractal animation by Jérémy Belin youtube: Mandelbrot Fractal Zoom: Bach, Fractals, and The Art of Fugue (HD) by Harlan Brothers Mandlebrot zoom Medium by Mike Koval -"it actually is the Curvature Average Coloring Algorithm, not the raw triangle inequality. " Triangle Inequality Average - Mandelbrot Fractal Mini - Zoom (8k 60fps) by Maths Town The Last Colour - mini-Mandelbrot Fractal Zoom by Maths Town References wikipedia : triangle inequality On Smooth Fractal Coloring Techniques master thesis by Jussi Haerkoenen /formulas.ultrafractal : lkm.ufm fractalforums : triangle-inequality-average-algorithm field lines in fractalshades and the code : " an orbit-average of the angle", "The key is to define a continuous mapping between points that escape just before and just after iteration n, using the fractional part of the continuous iteration number." References