# Fractals/Iterations in the complex plane/triangle ineq

algorithm for coloring exterior of the set ( Mandelbrot or Julia)

# NamesEdit

family of algorithms : Branching Average Coloring Algorithms (" because calculate an average of a function evaluated at all points in a truncated orbit ")^{[1]}

- 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 )

## VariantsEdit

- 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

# HistoryEdit

- Kerry Mitchell 1999.02.05
- Damien M. Jones 2000.04.02

# ExplanationEdit

"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 ^{[2]}

"creates smoothly colored ... large flame-like patterns that extend from the fractal outwards" (from UltraFractal by Frederik Slijkerman )^{[3]}

# ParametersEdit

- escape radius : "Use very large bail-out values (1E20 = 10^20)) for good results." Ultra Fractal
- iteration Max = 2500

# SoftwareEdit

- Ultra Fractal
- Fractint
- Fractview for android : blog - youtube - code
- Gnofract 4D
- Visions of Chaos - application for Windows ( GLSL ) - gallery
- Jux

# ExamplesEdit

- fractals by Craig Macomber with GLSL code

# CodeEdit

- java script
^{[4]} - cpp + glsl
^{[5]} - python
^{[6]}

## UltrafractalEdit

#### TIAEdit

```
/*
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
}
```

#### CAAEdit

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 = VOCEdit

```
#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);
}
```

# VideoEdit

- 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

# ReferencesEdit

- 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."