(Question) Triangle Inequality Average Algorithm

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Offline mikoval

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« on: May 24, 2018, 11:52:53 PM »
I have been struggling to understand the triangle inequality average algorithm, as well as simply getting some code working that produces reasonable results.

My code is based off the following paper:
http://jussiharkonen.com/files/on_fractal_coloring_techniques%28lo-res%29.pdf

I have created a shadertoy to show my existing code:
https://www.shadertoy.com/view/Md3fDs

I have tried to follow as closely to the paper as I can however the results shown in the book:
https://imgur.com/a/boN3wSI

look nothing like mine at all.

It would be greatly appreciated if someone could give me some feedback. I have spent a bit of time trying to understand where I could be going wrong but am at a loss.

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/triangle-inequality-average-algorithm/1368/

Offline superheal

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« Reply #1 on: May 25, 2018, 04:27:33 PM »
I had to convert my code, here is a sample code, but be warned I have not tested it after the conversion.
I also included the complex.java code.

Code: [Select]
static double sum, sum2 = 0;
static double mc = 0;

public static double escapedValue(int n, Complex z, double bail, Complex p) {

double temp = p.norm_squared();
double temp2 = z.norm_squared();

temp += 0.000000001;
temp = Math.log(temp);

double f = 1 - (Math.log(bail * bail) - temp) / (Math.log(temp2) - temp);

if(n < 2)
return f;

sum = sum / n;
sum2 = sum2 / (n - 1);

return sum + (sum2 - sum) * f;
}

//n : iteration
public static Complex iteratedFunction(Complex z, Complex c, int n) {

Complex temp = z.square();
Complex temp2 = temp.plus(c);

if(n == 0) {
mc = c.norm();
}

sum2 = sum;
if(n != 0) {
double mp = temp.norm();
double m = Math.abs(mp - mc);
double M = mp + mc;

sum += (temp2.norm() - m) / (M - m);
}

return temp2;

}


public static double iterate(Complex c) {

Complex z = new Complex();
Complex p = new Complex(); //previous value
int maxIter = 300;
double bailout = 16;

for(int i = 0; i < maxIter; i++) {
if(z.norm() >= bailout) {
return escapedValue(i, z, bailout, p); // used the returned value to select a color from a palette
}

p.assign(z);
z = iteratedFunction(z, c, i);
}

return maxIter;

}

Offline Adam Majewski

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« Reply #2 on: May 25, 2018, 04:42:32 PM »

Offline 3DickUlus

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« Reply #3 on: May 26, 2018, 02:30:41 AM »
from Fragmentarium examples...
Fragmentarium is not a toy, it is a very versatile tool that can be used to make toys ;)

https://en.wikibooks.org/wiki/Fractals/fragmentarium

Offline mikoval

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« Reply #4 on: May 26, 2018, 07:16:59 AM »
Wow that you so much for the reply guys! and that fractal picture is absolutely gorgeous... wow. I actually went through your code and changed mine @superheal, however it didn't actually change the image that I was getting. Your code is nearly identical to mine. Maybe the problem is in my coloring algorithm... Can you possibly share a sample coloring algorithm that you are using with the result of the escapevalue function? I will spend a little more time going through the code though since I may have some silly mistake somewhere.

Offline mikoval

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« Reply #5 on: May 26, 2018, 07:18:05 AM »
Oh, I just noticed the fragment shader code for that image! Going to spend some time looking at it!

Offline superheal

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« Reply #6 on: May 26, 2018, 11:34:34 AM »
Here is the complete code.

Offline superheal

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« Reply #7 on: May 31, 2018, 12:01:46 AM »
I implemented the .frag code also, here is a sample image:

Offline kosalos

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« Reply #8 on: June 01, 2018, 01:14:44 AM »
I have posted a rendition for the iPad: https://github.com/Kosalos/BurningChicken

Offline mikoval

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« Reply #9 on: June 26, 2018, 08:44:25 AM »
Thank you so much for the responses guys. I spent a bit of time playing with the equations and learned a lot. Everything was really helpful. I tried making a fractal zoom using the curvature average in the end, but ran into problems with the zoom. Basically the value that you get as the result of the average sum converges slowly, so after zooming in a bit the pattern gets very small. I have been trying to find an equation to scale the values but haven't been able to find anything. Does anyone know anything about where to look? I have tried looking around but haven't seen much.

Offline FractalDave

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« Reply #10 on: August 05, 2018, 02:36:30 PM »
You can spread the pattern by using values from iterations older than the penultimate one.
UF example, you can try with smoothing parameter disabled and/or turn off the "older iteration" option (implemented in "Trap Select"), with smoothing disabled the colour rate will need increasing:
Code: [Select]
TIAolderSmoothed {
fractal:
  title="TIAolderSmoothed" width=1272 height=954 layers=1 gamma=2.5
  credits="David Makin;8/5/2018" antialiasing=yes
layer:
  caption="Background" opacity=100
mapping:
  center=0.32860675362165/0.4852981212579 magn=11745.815
formula:
  maxiter=100 filename="Standard.ufm" entry="Mandelbrot" p_start=0/0
  p_power=2/0 p_bailout=65536
inside:
  transfer=none
outside:
  density=2 transfer=linear filename="mmf5.ucl"
  entry="GenericGradientColoringRenderFalse"
  p_coloringClass="mmfnew.ulb:MMF_SmoothOrbitTraps"
  p_coloringClass.v_generic=100 p_coloringClass.v_coloring=100
  p_coloringClass.v_gradientcoloring=100
  p_coloringClass.v_mmfsmoothorbittraps=1.0
  p_coloringClass.smoothing=Divergent p_coloringClass.bailout=65536
  p_coloringClass.dauto=yes p_coloringClass.power=2/0
  p_coloringClass.fudge=1 p_coloringClass.smallbail=0.00001
  p_coloringClass.cauto=yes p_coloringClass.cpower=2/0
  p_coloringClass.fudgec=1 p_coloringClass.cfixed=no
  p_coloringClass.cfixedval=1/0 p_coloringClass.processlast=yes
  p_coloringClass.f_trapselect="dmj5.ulb:DMJ_TrapSelect"
  p_coloringClass.f_trapselect.v_generic=100
  p_coloringClass.f_trapselect.v_generator=100
  p_coloringClass.f_trapselect.v_dmj_trapselect=100
  p_coloringClass.f_trapselect.p_trapalliterations="some iterations"
  p_coloringClass.f_trapselect.p_trapstart=0
  p_coloringClass.f_trapselect.p_trapiter=1
  p_coloringClass.f_trapselect.p_trapskip=3
  p_coloringClass.f_trapselect.p_traploop=1E20
  p_coloringClass.p_trapselectthreshold=0.5
  p_coloringClass.f_traptransform="common.ulb:TrapTransform"
  p_coloringClass.f_traptransform.v_generic=100
  p_coloringClass.f_traptransform.v_transform=100
  p_coloringClass.f_traptransform.v_usertransform=100
  p_coloringClass.f_traptransform.v_traptransform=100
  p_coloringClass.f_traptransform.p_trapcenter=0/0
  p_coloringClass.f_traptransform.p_trapcentermoves=no
  p_coloringClass.f_traptransform.p_trapdrift=0/0
  p_coloringClass.f_traptransform.p_trapscale=1
  p_coloringClass.f_traptransform.p_traprotation=0
  p_coloringClass.f_traptransform.p_traprotationstep=0
  p_coloringClass.f_traptransform.p_trapskew=0
  p_coloringClass.f_traptransform.p_trapskewstep=0
  p_coloringClass.f_traptransform.p_trapaspect=1
  p_coloringClass.f_trapshape="mmfnew.ulb:MMF_TrapShapeTriangleInequal\
  ity" p_coloringClass.f_trapshape.v_generic=100
  p_coloringClass.f_trapshape.v_trapshape=101
  p_coloringClass.f_trapshape.p_usepixel=yes
  p_coloringClass.f_trapshape.p_tcentre=1/0
  p_coloringClass.f_fixzold=yes
  p_coloringClass.f_traptransfer="common.ulb:TrapTransfer"
  p_coloringClass.f_traptransfer.v_generic=100
  p_coloringClass.f_traptransfer.v_transfer=100
  p_coloringClass.f_traptransfer.v_traptransfer=100
  p_coloringClass.f_traptransfer.p_preadd1=0
  p_coloringClass.f_traptransfer.p_prescale=1
  p_coloringClass.f_traptransfer.p_preadd2=0
  p_coloringClass.f_traptransfer.p_function=linear
  p_coloringClass.f_traptransfer.p_postadd1=0
  p_coloringClass.f_traptransfer.p_postscale=1
  p_coloringClass.f_traptransfer.p_postadd2=0
  p_coloringClass.f_traptransfer.p_abs=no
  p_coloringClass.f_trapmode="mmfnew.ulb:MMF_TrapModeSmoothAverage"
  p_coloringClass.f_trapmode.v_generic=100
  p_coloringClass.f_trapmode.v_trapmode=100
  p_coloringClass.f_trapmode.v_trapmodewiththreshold=100
  p_coloringClass.f_trapmode.v_mmf_trapmodesmoothwiththreshold=100
  p_coloringClass.p_threshold=1 p_coloringClass.p_usesolid=no
  p_coloringClass.f_traptexture="dmj5.ulb:DMJ_TrapShapeFlat"
  p_coloringClass.f_traptexture.v_generic=100
  p_coloringClass.f_traptexture.v_trapshape=101
  p_coloringClass.f_traptexture.v_dmj_trapshapeflat=100
  p_coloringClass.p_texturepositiononly=no
  p_coloringClass.p_textureposttransform=no
  p_coloringClass.p_texturestrength=1
  p_coloringClass.f_trapcoloring="mmfnew.ulb:MMF_TrapColoring"
  p_coloringClass.f_trapcoloring.v_generic=100
  p_coloringClass.f_trapcoloring.v_trapcoloring=100
  p_coloringClass.f_trapcoloring.v_mmf_trapcoloring=100
  p_coloringClass.f_trapcoloring.colourby=Distance
  p_coloringClass.f_trapcoloring.f_image="mmfnew.ulb:MMF_ImageImport"
  p_coloringClass.f_trapcoloring.f_image.v_generic=100
  p_coloringClass.f_trapcoloring.f_image.v_imagewrapper=100
  p_coloringClass.f_trapcoloring.f_image.v_imageimport=101
  p_coloringClass.f_trapcoloring.f_image.p_importedimage=""
  p_coloringClass.f_trapcoloring.f_image.p_maintainaspect=yes
  p_coloringClass.f_trapcoloring.f_image.p_transform="common.ulb:NullT\
  ransform"
  p_coloringClass.f_trapcoloring.f_image.p_transform.v_generic=100
  p_coloringClass.f_trapcoloring.f_image.p_transform.v_transform=100
  p_coloringClass.f_trapcoloring.f_image.p_transform.v_usertransform=1\
  00
  p_coloringClass.f_trapcoloring.f_image.p_transform.v_nulltransform=1\
  00 p_coloringClass.f_trapcoloring.f_image.p_tile=yes
  p_coloringClass.f_trapcoloring.f_image.p_iterate=no
  p_coloringClass.f_trapcoloring.f_image.p_transform1="common.ulb:Null\
  Transform"
  p_coloringClass.f_trapcoloring.f_image.p_transform1.v_generic=100
  p_coloringClass.f_trapcoloring.f_image.p_transform1.v_transform=100
  p_coloringClass.f_trapcoloring.f_image.p_transform1.v_usertransform=\
  100
  p_coloringClass.f_trapcoloring.f_image.p_transform1.v_nulltransform=\
  100 p_coloringClass.f_trapcoloring.f_image.p_retile=no
  p_coloringClass.f_trapcoloring.f_image.p_mirror=no
  p_coloringClass.f_trapcoloring.f_image.p_flip=no
  p_coloringClass.f_trapcoloring.f_image.p_seamless=yes
  p_coloringClass.f_trapcoloring.f_image.p_normalise=no
  p_coloringClass.f_trapcoloring.f_image.p_grey24=no
  p_coloringClass.f_trapcoloring.f_image.adjustlayer=no
  p_coloringClass.f_trapcoloring.f_image.adjusttype="Colour Parameter"
  p_coloringClass.f_trapcoloring.f_image.adjred=1
  p_coloringClass.f_trapcoloring.f_image.adjgreen=1
  p_coloringClass.f_trapcoloring.f_image.adjblue=1
  p_coloringClass.f_trapcoloring.f_image.f_adjustcolour=4294967295
  p_coloringClass.f_trapcoloring.f_image.f_adjust="mmfnew.ulb:MMF_Exte\
  ndedColorMerge"
  p_coloringClass.f_trapcoloring.f_image.f_adjust.v_generic=100
  p_coloringClass.f_trapcoloring.f_image.f_adjust.v_colormerge=100
  p_coloringClass.f_trapcoloring.f_image.f_adjust.v_defaultcolormerge=\
  100
  p_coloringClass.f_trapcoloring.f_image.f_adjust.v_mmfextendedcolorme\
  rge=1.0
  p_coloringClass.f_trapcoloring.f_image.f_adjust.p_modeselect="Origin\
  al Merge Modes"
  f_coloringClass.f_trapcoloring.f_image.f_adjust.f_mergemode=multiply
  p_coloringClass.f_trapcoloring.f_image.f_adjust.clamp=no
  p_coloringClass.f_trapcoloring.f_image.f_adjust.p_mmfmergemode=Exclu\
  sion p_coloringClass.f_trapcoloring.f_image.f_adjust.qmin=2
  p_coloringClass.f_trapcoloring.f_image.f_adjust.qmax=32
  p_coloringClass.f_trapcoloring.f_image.f_adjust.mingamma=1
  p_coloringClass.f_trapcoloring.f_image.f_adjust.maxgamma=2
  p_coloringClass.f_trapcoloring.f_image.f_adjust.redlvl=0.299
  p_coloringClass.f_trapcoloring.f_image.f_adjust.greenlvl=0.587
  p_coloringClass.f_trapcoloring.f_image.f_adjust.bluelvl=0.114
  p_coloringClass.f_trapcoloring.f_image.f_opacity=1
  p_coloringClass.f_trapcoloring.f_image.f_adjustment="mmfnew.ulb:MMF_\
  AdjustmentLayer"
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.v_generic=100
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.v_adjustmentlaye\
  r=100
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.v_mmfadjustmentl\
  ayer=1.0
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.adjusttype="Colo\
  ur Parameter"
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.adjred=1
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.adjgreen=1
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.adjblue=1
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjustcolour=4\
  294967295
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust="mmfnew\
  .ulb:MMF_ExtendedColorMerge"
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.v_gener\
  ic=100
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.v_color\
  merge=100
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.v_defau\
  ltcolormerge=100
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.v_mmfex\
  tendedcolormerge=1.0
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.p_modes\
  elect="Original Merge Modes"
  f_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.f_merge\
  mode=multiply
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.clamp=n\
  o
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.p_mmfme\
  rgemode=Exclusion
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.qmin=2
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.qmax=32
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.mingamm\
  a=1
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.maxgamm\
  a=2
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.redlvl=\
  0.299
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.greenlv\
  l=0.587
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_adjust.bluelvl\
  =0.114
  p_coloringClass.f_trapcoloring.f_image.f_adjustment.f_opacity=1
  p_coloringClass.f_trapcoloring.f_image.p_wrap=no
  p_coloringClass.f_trapcoloring.f_image.p_usefbm=no
  p_coloringClass.f_trapcoloring.f_image.p_fbm="mmfnew.ulb:MMF_AltfBm"
  p_coloringClass.f_trapcoloring.f_image.p_fbm.v_generic=100
  p_coloringClass.f_trapcoloring.f_image.p_fbm.v_mmfaltfbm=1.0
  p_coloringClass.f_trapcoloring.f_image.p_fbm.mode="Div 16"
  p_coloringClass.f_trapcoloring.f_image.p_fbm.seed=123456
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.mainamp=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.average=Normal
  p_coloringClass.f_trapcoloring.f_image.p_fbm.inter=Linear
  p_coloringClass.f_trapcoloring.f_image.p_fbm.dice=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.dummy=0
  p_coloringClass.f_trapcoloring.f_image.p_fbm.morph=no
  p_coloringClass.f_trapcoloring.f_image.p_fbm.exp=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.gamma=no
  p_coloringClass.f_trapcoloring.f_image.p_fbm.mid=0.5
  p_coloringClass.f_trapcoloring.f_image.p_fbm.lowp=1.5
  p_coloringClass.f_trapcoloring.f_image.p_fbm.highp=0.66666666667
  p_coloringClass.f_trapcoloring.f_image.p_fbm.levels=8
  p_coloringClass.f_trapcoloring.f_image.p_fbm.showmaps=no
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag0=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac0=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs0=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp0=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang0=0
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag1=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac1=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs1=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp1=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang1=22.5
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag2=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac2=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs2=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp2=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang2=11.25
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag3=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac3=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs3=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp3=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang3=33.75
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag4=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac4=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs4=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp4=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang4=16.875
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag5=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac5=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs5=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp5=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang5=39.375
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag6=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac6=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs6=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp6=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang6=28.125
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag7=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac7=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs7=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp7=1
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang7=5.625
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag8=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac8=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs8=2
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp8=0.5
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang8=19.6875
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag9=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac9=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs9=2
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp9=0.5
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang9=36.5625
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag10=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac10=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs10=2
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp10=0.5
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang10=8.4375
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag11=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac11=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs11=4
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp11=0.25
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang11=25.3125
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag12=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac12=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs12=4
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp12=0.25
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang12=14.0625
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag13=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac13=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs13=8
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp13=0.125
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang13=30.9375
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag14=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac14=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs14=16
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp14=0.0625
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang14=2.8125
  p_coloringClass.f_trapcoloring.f_image.p_fbm.flag15=yes
  p_coloringClass.f_trapcoloring.f_image.p_fbm.frac15=0.7071
  p_coloringClass.f_trapcoloring.f_image.p_fbm.rs15=32
  p_coloringClass.f_trapcoloring.f_image.p_fbm.amp15=0.03125
  p_coloringClass.f_trapcoloring.f_image.p_fbm.ang15=42.1875
  p_coloringClass.f_trapcoloring.f_image.p_deflect=0
  p_coloringClass.f_trapcoloring.f_image.p_twotile=no
  p_coloringClass.f_trapcoloring.f_image.p_tileimage=""
  p_coloringClass.f_trapcoloring.f_image.p_patternx=000
  p_coloringClass.f_trapcoloring.f_image.p_patterny=0x0
  p_coloringClass.f_trapcoloring.f_image.p_xgutter=0.5
  p_coloringClass.f_trapcoloring.f_image.p_ygutter=0.5
  p_coloringClass.f_trapcoloring.f_image.p_xoff=0
  p_coloringClass.f_trapcoloring.f_image.p_yoff=0
  p_coloringClass.f_trapcoloring.f_image.p_advanced=no
  p_coloringClass.f_trapcoloring.f_image.p_rotate=yes
  p_coloringClass.f_trapcoloring.f_image.p_random=yes
  p_coloringClass.f_trapcoloring.f_image.p_rand=yes
  p_coloringClass.f_trapcoloring.f_image.p_seedit=yes
  p_coloringClass.f_trapcoloring.f_image.p_seed=10083
  p_coloringClass.f_trapcoloring.f_image.p_gbasex=0.2
  p_coloringClass.f_trapcoloring.f_image.p_gmaxx=0.8
  p_coloringClass.f_trapcoloring.f_image.p_gbasey=0.2
  p_coloringClass.f_trapcoloring.f_image.p_gmaxy=0.8
  p_coloringClass.f_trapcoloring.f_image.p_xgutter00=0.5
  p_coloringClass.f_trapcoloring.f_image.p_xgutter01=0.5
  p_coloringClass.f_trapcoloring.f_image.p_xgutter10=0.5
  p_coloringClass.f_trapcoloring.f_image.p_xgutter11=0.5
  p_coloringClass.f_trapcoloring.f_image.p_ygutter00=0.5
  p_coloringClass.f_trapcoloring.f_image.p_ygutter01=0.5
  p_coloringClass.f_trapcoloring.f_image.p_ygutter10=0.5
  p_coloringClass.f_trapcoloring.f_image.p_ygutter11=0.5
  p_coloringClass.f_trapcoloring.f_image.p_xoff00=0
  p_coloringClass.f_trapcoloring.f_image.p_xoff01=0
  p_coloringClass.f_trapcoloring.f_image.p_xoff10=0
  p_coloringClass.f_trapcoloring.f_image.p_xoff11=0
  p_coloringClass.f_trapcoloring.f_image.p_yoff00=0
  p_coloringClass.f_trapcoloring.f_image.p_yoff01=0
  p_coloringClass.f_trapcoloring.f_image.p_yoff10=0
  p_coloringClass.f_trapcoloring.f_image.p_yoff11=0
  p_coloringClass.f_trapcoloring.f_image.p_SimplefBm=0
  p_coloringClass.f_trapcoloring.f_image.p_scale=0.5
  p_coloringClass.f_trapcoloring.f_image.p_fade=0.5
  p_coloringClass.f_trapcoloring.usetrap="Original Z value"
  p_coloringClass.f_trapcoloring.f_colortransfer="common.ulb:NullTrans\
  fer" p_coloringClass.f_trapcoloring.f_colortransfer.v_generic=100
  p_coloringClass.f_trapcoloring.f_colortransfer.v_transfer=100
  p_coloringClass.f_trapcoloring.f_colortransfer.v_nulltransfer=100
  p_coloringClass.f_trapcoloring.f_texturetransfer="common.ulb:NullTra\
  nsfer"
  p_coloringClass.f_trapcoloring.f_texturetransfer.v_generic=100
  p_coloringClass.f_trapcoloring.f_texturetransfer.v_transfer=100
  p_coloringClass.f_trapcoloring.f_texturetransfer.v_nulltransfer=100
  p_coloringClass.f_trapcoloring.f_merge="mmfnew.ulb:MMF_GradientLayer\
  " p_coloringClass.f_trapcoloring.f_merge.v_generic=100
  p_coloringClass.f_trapcoloring.f_merge.v_layer=100
  p_coloringClass.f_trapcoloring.f_merge.f_solid=4278190080
  p_coloringClass.f_trapcoloring.f_merge.f_invertmerge=no
  p_coloringClass.f_trapcoloring.f_merge.f_invertcompose=no
  p_coloringClass.f_trapcoloring.f_merge.f_opacity=1
  p_coloringClass.f_trapcoloring.f_merge.f_merge="mmfnew.ulb:MMF_Exten\
  dedColorMerge"
  p_coloringClass.f_trapcoloring.f_merge.f_merge.v_generic=100
  p_coloringClass.f_trapcoloring.f_merge.f_merge.v_colormerge=100
  p_coloringClass.f_trapcoloring.f_merge.f_merge.v_defaultcolormerge=1\
  00
  p_coloringClass.f_trapcoloring.f_merge.f_merge.v_mmfextendedcolormer\
  ge=1.0
  p_coloringClass.f_trapcoloring.f_merge.f_merge.p_modeselect="Origina\
  l Merge Modes"
  f_coloringClass.f_trapcoloring.f_merge.f_merge.f_mergemode=multiply
  p_coloringClass.f_trapcoloring.f_merge.f_merge.clamp=no
  p_coloringClass.f_trapcoloring.f_merge.f_merge.p_mmfmergemode=Exclus\
  ion p_coloringClass.f_trapcoloring.f_merge.f_merge.qmin=2
  p_coloringClass.f_trapcoloring.f_merge.f_merge.qmax=32
  p_coloringClass.f_trapcoloring.f_merge.f_merge.mingamma=1
  p_coloringClass.f_trapcoloring.f_merge.f_merge.maxgamma=2
  p_coloringClass.f_trapcoloring.f_merge.f_merge.redlvl=0.299
  p_coloringClass.f_trapcoloring.f_merge.f_merge.greenlvl=0.587
  p_coloringClass.f_trapcoloring.f_merge.f_merge.bluelvl=0.114
  p_coloringClass.f_trapcoloring.f_merge.v_gradientlayer=100
  p_coloringClass.f_trapcoloring.f_merge.f_grad="Custom Gradient"
  p_coloringClass.f_trapcoloring.f_merge.f_gradient="mmfnew.ulb:MMF_Gr\
  adient"
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.v_generic=100
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.v_mmfgradientbase=\
  100
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.v_mmfgradient=1.0
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_transfer="common\
  .ulb:DefaultGradientTransfer"
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_transfer.v_gener\
  ic=100
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_transfer.v_trans\
  fer=100
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_transfer.v_defau\
  ltgradienttransfer=101
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_transfer.p_xfer=\
  Linear
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_transfer.p_zscal\
  e=1.0
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_transfer.p_zscal\
  e2=1.0
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_transfer.p_offse\
  t=0.0 p_coloringClass.f_trapcoloring.f_merge.f_gradient.rept=yes
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.smooth=yes
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.showpalette=no
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours="mmfn\
  ew.ulb:MMF_FourKeyColours"
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.v_gen\
  eric=100
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.v_mmf\
  keycolourbase=100
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.v_mmf\
  fourkeycolours=100
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.numco\
  ls=4
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.colou\
  r0=4278190213
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.index\
  0=0
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.colou\
  r1=4294967295
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.index\
  1=100
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.colou\
  r2=4294948096
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.index\
  2=200
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.colou\
  r3=4288413696
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.index\
  3=300
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.colou\
  r4=4294967040
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.index\
  4=50
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.colou\
  r5=4278255615
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.index\
  5=150
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.colou\
  r6=4294902015
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.index\
  6=250
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.colou\
  r7=4294967295
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.index\
  7=350
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.colou\
  r8=4294934783
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.index\
  8=75
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.colou\
  r9=4286644223
  p_coloringClass.f_trapcoloring.f_merge.f_gradient.f_keycolours.index\
  9=325 p_coloringClass.f_trapcoloring.f_merge.adjustlayer=no
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment="mmfnew.ulb:MMF_\
  AdjustmentLayer"
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.v_generic=100
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.v_adjustmentlaye\
  r=100
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.v_mmfadjustmentl\
  ayer=1.0
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.adjusttype="Colo\
  ur Parameter"
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.adjred=1
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.adjgreen=1
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.adjblue=1
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjustcolour=4\
  294967295
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust="mmfnew\
  .ulb:MMF_ExtendedColorMerge"
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.v_gener\
  ic=100
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.v_color\
  merge=100
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.v_defau\
  ltcolormerge=100
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.v_mmfex\
  tendedcolormerge=1.0
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.p_modes\
  elect="Original Merge Modes"
  f_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.f_merge\
  mode=multiply
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.clamp=n\
  o
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.p_mmfme\
  rgemode=Exclusion
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.qmin=2
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.qmax=32
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.mingamm\
  a=1
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.maxgamm\
  a=2
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.redlvl=\
  0.299
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.greenlv\
  l=0.587
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_adjust.bluelvl\
  =0.114
  p_coloringClass.f_trapcoloring.f_merge.f_adjustment.f_opacity=1
  p_coloringClass.f_trapcoloring.f_merge.v_mmfgradientlayer=1.0
  p_coloringClass.f_trapcoloring.adjustlayer=no
  p_coloringClass.f_trapcoloring.adjusttype="Colour Parameter"
  p_coloringClass.f_trapcoloring.adjred=1
  p_coloringClass.f_trapcoloring.adjgreen=1
  p_coloringClass.f_trapcoloring.adjblue=1
  p_coloringClass.f_trapcoloring.f_adjustcolour=4294967295
  p_coloringClass.f_trapcoloring.f_adjust="mmfnew.ulb:MMF_ExtendedColo\
  rMerge" p_coloringClass.f_trapcoloring.f_adjust.v_generic=100
  p_coloringClass.f_trapcoloring.f_adjust.v_colormerge=100
  p_coloringClass.f_trapcoloring.f_adjust.v_defaultcolormerge=100
  p_coloringClass.f_trapcoloring.f_adjust.v_mmfextendedcolormerge=1.0
  p_coloringClass.f_trapcoloring.f_adjust.p_modeselect="Original Merg\
  e Modes"
  f_coloringClass.f_trapcoloring.f_adjust.f_mergemode=multiply
  p_coloringClass.f_trapcoloring.f_adjust.clamp=no
  p_coloringClass.f_trapcoloring.f_adjust.p_mmfmergemode=Exclusion
  p_coloringClass.f_trapcoloring.f_adjust.qmin=2
  p_coloringClass.f_trapcoloring.f_adjust.qmax=32
  p_coloringClass.f_trapcoloring.f_adjust.mingamma=1
  p_coloringClass.f_trapcoloring.f_adjust.maxgamma=2
  p_coloringClass.f_trapcoloring.f_adjust.redlvl=0.299
  p_coloringClass.f_trapcoloring.f_adjust.greenlvl=0.587
  p_coloringClass.f_trapcoloring.f_adjust.bluelvl=0.114
  p_coloringClass.f_trapcoloring.f_opacity=1
  p_coloringClass.f_finaltransfer="mmfnew.ulb:MMF_TrapTransfer"
  p_coloringClass.f_finaltransfer.v_generic=100
  p_coloringClass.f_finaltransfer.v_transfer=100
  p_coloringClass.f_finaltransfer.v_mmf_traptransfer=100
  p_coloringClass.f_finaltransfer.p_offset0=0
  p_coloringClass.f_finaltransfer.p_zscale0=1
  p_coloringClass.f_finaltransfer.p_offset1=0
  p_coloringClass.f_finaltransfer.p_forcepos=no
  p_coloringClass.f_finaltransfer.p_xfer=Linear
  p_coloringClass.f_finaltransfer.p_iatpower=2
  p_coloringClass.f_finaltransfer.p_power=4
  p_coloringClass.f_finaltransfer.p_offset2=0
  p_coloringClass.f_finaltransfer.p_zscale1=1
  p_coloringClass.f_finaltransfer.p_offset3=0
  p_coloringClass.f_additer=yes
  p_coloringClass.f_itertransfer="mmfnew.ulb:MMF_TrapTransfer"
  p_coloringClass.f_itertransfer.v_generic=100
  p_coloringClass.f_itertransfer.v_transfer=100
  p_coloringClass.f_itertransfer.v_mmf_traptransfer=100
  p_coloringClass.f_itertransfer.p_offset0=0
  p_coloringClass.f_itertransfer.p_zscale0=1
  p_coloringClass.f_itertransfer.p_offset1=0
  p_coloringClass.f_itertransfer.p_forcepos=no
  p_coloringClass.f_itertransfer.p_xfer=Linear
  p_coloringClass.f_itertransfer.p_iatpower=2
  p_coloringClass.f_itertransfer.p_power=4
  p_coloringClass.f_itertransfer.p_offset2=0
  p_coloringClass.f_itertransfer.p_zscale1=1
  p_coloringClass.f_itertransfer.p_offset3=0
gradient:
  smooth=yes rotation=-138 index=120 color=145 index=180 color=46591
  index=-181 color=8024433 index=-38 color=16121855
opacity:
  smooth=no index=0 opacity=255
}
The meaning and purpose of life is to give life purpose and meaning.


xx
Triangle Inequality Average using older values with iteration smoothing

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