# Fractals/ultrafractal

UltraFractal is a program by Frederik Slijkerman. This is unofficial wiki about it.

# licence edit

Copyright © 1997-2017 Frederik Slijkerman

# install edit

## Linux edit

use wine

# Help edit

- file types
^{[1]} - Fractal Forum
^{[2]}

# formula edit

## outside coloring edit

- Average Colorings

### TwinLamps or Standart Exp smoothing edit

TwinLamps { ; Standart Exp smoothing (invented by Ron Barnet code taken from Damien M. Jones) ; divided by fractal dimension statistics (from Kerry Mitchell). ; Tried to make universal algorithm and showing features ; of fractal and not of colour method. Goes with mandelbrots, ; patterns and julias, but not so with newton fractals. ; by Edgars Malinovskis 17.01.2012 ; Removed bug creating no colour spots. 23.02.1012 init: float sum = 0.0 float zmin=1e20 float zmax=-1 float cabsz=0.0 float lnz=0.0 loop: cabsz= cabs(#z+@posneg) lnz=exp(-cabsz) ;finding min and max z. IF (lnz < zmin) zmin = lnz ENDIF IF (lnz > zmax) zmax = lnz ENDIF ;exp smoothing sum= sum + lnz final: ;dividing exponent smooth value by fractal dimension value #index=sqrt(sum)/(1+zmax-zmin) default: title = "TwinLamps" param posneg caption = "Add to pixel value" default = (0.0, 0.0) hint = "The same as trap center.\ Adding number diverses negative Z areas\ and marks -n with dots." endparam }

### DEM edit

DistanceEstimator(OUTSIDE) { ; ; Distance-estimator coloring algorithm for Mandelbrot and ; other z^n fractal types (Phoenix, Julia). This coloring ; algorithm estimates the distance to the boundary of the ; fractal (for example the Mandelbrot set) and colors points ; accordingly. ; ; Written by Damien M. Jones ; init: complex dz = (0,0) loop: dz = @power * #z^(@power-1) * dz + 1 final: #index = (@power*log(cabs(#z)) * cabs(#z) / cabs(dz))^(1/@power) default: title = "Distance Estimator" helpfile = "Uf*.chm" helptopic = "Html/coloring/standard/distanceestimator.html" 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 usually 2." endparam }

### ExponentialSmoothing edit

ExponentialSmoothing { ; from Standard.ucl ; This coloring method provides smooth iteration ; colors for all fractal types, convergent or ; divergent (or both). It combines the two methods ; developed by Ron Barnett. It doesn't map ; precisely to iterations, but it's close. ; ; Written by Damien M. Jones ; init: float sum = 0.0 float sum2 = 0.0 complex zold = (0,0) loop: IF (@diverge) sum = sum + exp(-cabs(#z)) ENDIF IF (@converge) sum2 = sum2 + exp(-1/cabs(zold-#z)) ENDIF zold = #z final: IF (|#z - zold| < 0.5) ; convergent bailout. IF (@converge) #index = sum2 ELSE #index = 0 ENDIF ELSE ; divergent bailout. IF (@diverge) #index = sum * @divergescale ELSE #index = 0 ENDIF ENDIF default: title = "Exponential Smoothing" helpfile = "Uf*.chm" helptopic = "Html/coloring/standard/exponentialsmoothing.html" $IFDEF VER50 rating = recommended $ENDIF param diverge caption = "Color Divergent" default = FALSE hint = "If checked, points which escape to infinity will be \ colored." endparam param converge caption = "Color Convergent" default = TRUE hint = "If checked, points which collapse to one value will be \ colored." endparam param divergescale caption = "Divergent Density" default = 1.0 $IFDEF VER40 exponential = true $ENDIF hint = "Sets the divergent coloring density, relative to the \ convergent coloring. If set to 1.0, they will use \ the same color density." endparam }

### smooth edit

Smooth(OUTSIDE) { ; from Standard.ucl ; This coloring method provides smooth iteration ; colors for Mandelbrot and other z^2 formula types ; (Phoenix, Julia). Results on other types may be ; unpredictable, but might be interesting. ; ; Thanks to F. Slijkerman for some tweaks. ; Thanks to Linas Vepstas for the math. ; ; Written by Damien M. Jones ; init: complex il = 1/log(@power) ; Inverse log (power). float lp = log(log(@bailout)) ; log(log bailout). final: #index = 0.05 * real(#numiter + il*lp - il*log(log(cabs(#z)))) default: title = "Smooth (Mandelbrot)" helpfile = "Uf*.chm" helptopic = "Html/coloring/standard/smooth.html" $IFDEF VER50 rating = recommended $ENDIF 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 usually 2." endparam param bailout caption = "Bail-out value" default = 128.0 min = 1 hint = "This should be set to match the bail-out value in \ the Formula tab. This formula works best with bail-out \ values higher than 100." endparam }

### binary decomposition edit

comment { This file contains standard coloring algorithms for Ultra Fractal 3. Many of the coloring algorithms here were written by other formula authors, as noted in the comments with each formula. All formulas have been edited and simplified by Frederik Slijkerman. } BinaryDecomposition { ; from standard.ucl ; Classic binary decomposition. Can give quite abstract effects. ; Use low bail-out values in the fractal formula (if possible) for ; best effects. This coloring algorithm uses just two colors from ; the gradient: one from the left end and one from the middle. ; final: if @type == "Type 1" if real(#z) * imag(#z) >= 0 #index = 0.5 else #index = 0 endif else if atan2(#z) > 0 #index = 0.5 else #index = 0 endif endif default: title = "Binary Decomposition" helpfile = "Uf3.chm" helptopic = "Html/coloring/standard/binarydecomposition.html" param type caption = "Decomposition Type" enum = "Type 1" "Type 2" default = 0 hint = "Toggles between two types of binary decomposition. Type 2 \ reproduces the coloring used with many images in the classic \ Beauty of Fractals book." endparam }

### slope edit

// http://www.fractalforums.com/mandel-machine/maybe-perturbation-could-go-with-a-slope/ mbrot2_slope { fractal: title="mbrot2_slope" width=1024 height=400 layers=1 credits="Alef;12/4/2014;Frederik;7/23/2010" layer: caption="Background" opacity=100 mapping: center=-0.61777095064903/0.6790465094562 magn=19180.711 angle=156.5545 formula: maxiter=1000 filename="Standard.ufm" entry="SlopeMandel" p_start=0/0 p_power=2/0 p_bailout=1.0E20 p_offset=0.000000000000001 p_zmode=potential p_xfer=linear p_zscale=1.0 p_zscale2=0.005 p_everyiter=no inside: transfer=none outside: transfer=linear filename="Standard.ucl" entry="Decomposition" gradient: comments="slightly changed standart. IMHO good sky." smooth=no rotation=1 index=0 color=6303744 index=64 color=12085789 index=168 color=16777197 index=257 color=33023 index=343 color=512 opacity: smooth=no index=0 opacity=255 } mbrot3_slope { ; copyright Kerry Mitchell 15sep98 ; ; sample image to illustrate ; embossing effect fractal: title="mbrot3_slope" width=1024 height=512 layers=1 credits="Alef;12/4/2014;Kerry Mitchell;9/15/1998" layer: caption="Layer 1" opacity=100 method=linear mapping: center=-0.81810721128409/0.19884197484006 magn=4118.0018 angle=34.1619 formula: maxiter=1000 percheck=off filename="Standard.ufm" entry="SlopeMandel" p_start=0/0 p_power=2/0 p_bailout=1.0E20 p_offset=0.000000000000001 p_zmode="distance estimator" p_xfer=linear p_zscale=1.0 p_zscale2=0.005 p_everyiter=no inside: transfer=none outside: transfer=linear filename="Standard.ucl" entry="Decomposition" gradient: comments="slightly changed standart. IMHO good sky." smooth=no rotation=1 index=0 color=6303744 index=64 color=12085789 index=168 color=16777197 index=257 color=33023 index=343 color=512 opacity: smooth=no index=0 opacity=255 }

### field lines edit

"There's a later version of my formula for field lines - though somewhat extended to include smooth iteration, distance estimation and distance estimation angles in my class formulas for UF - mmf.ulb called "MMF Field Estimator" It's a UF class formula so not quite so easy to follow but I think it should be clear enough. I boiled down "corrections" to field lines that work in most places but at the moment require manual adjustment to work properly - these are the two parameters that can be modified by the user to fix errors in the field lines, unfortunately you can't always fix them everywhere in view even then but this is about as close as you can get I think. I tried including the MMF Field Lines code here but unfortunately it's too long, it's in mmf.ulb in te Ultra Fractal Formula database at: http://formulas.ultrafractal.com/ " David Makin

MMF3-FieldLines { ; ; This version of calculating field lines will work reasonably well with both ; Mandelbrot and Julia Sets for divergent fractals with a divergence that is ; close to being a positive integer >=2 but can also produce quite interesting ; results with fractals that do not fit that criteria - just not rendering the ; field lines correctly in other cases :-) ; ; Thanks to Chris Hayton for showing me how to fix the "off-by-one" problems. ; http://www.fractalforums.com/programming/smooth-external-angle-of-mandelbrot-set/ ; David Makin ; Here's my code from mmf3.ucl for Ultra Fractal: ; global: float twopi = 2.0*#pi float dtwopi = 0.5/#pi float dp = 1.0/@power float m = @power*dtwopi float dp2 = 1.0/(2.0+@power) ; Changed from originally trying 1/power init: complex zv[#maxiter] int i = 0 float a = 0.0 float b = 0.0 float c = 0.0 float s = 0.0 if @julia zv[i] = #z i = i + 1 endif loop: zv[i] = #z i = i + 1 final: if (s = atan2(#z))<0 s = s + twopi endif while i > @skipiter i = i - 1 if @version==0 b = (@power*atan2(zv[i]))%twopi else b = atan2(zv[i]^@power) endif if b < 0.0 b = b + twopi endif c = c + b - s if c >= #pi s = s + twopi if @fixit || @accident c = c - twopi endif elseif c < -#pi s = s - twopi if @fixit || @accident c = c + twopi endif endif if @fixit c = dp2*c elseif !@accident c = 0.0 endif if (a = atan2(zv[i])) < 0.0 a = a + twopi endif s = dp*(s + twopi*floor(a*m)) endwhile s = s*dtwopi if @fixskip while s<0.0 s = s + 1.0 endwhile while s>=1.0 s = s - 1.0 endwhile endif #index = s default: title = "MMF3 Field Lines (use high bailouts)" heading caption = "Information" ; text = "This colouring is a little touchy around certain values, if you \ ; get obvious lines or dots that shouldn't be there but the 'Degree \ ; of Divergence' is set properly for the main formula and you are \ ; using a high bailout of say 1e20 or higher then try \ ; adjusting the location very slightly. For example if vertical \ ; lines appear on an unrotated fractal then try changing the x \ ; (real) location very slightly, if horizontal lines appear then try \ ; changing the y (imag) location very slightly." endheading heading endheading param version caption = "Version" enum = "Original" "Alternative" default = 1 hint = "Allows you to modify one of the calculations in the formula that \ is particularly sensitive to small errors. Ideally both methods \ should produce identical results but they don't, so if you get \ visible errors using one method then try the other (assuming you \ have the divergence set correctly and are using a high bailout)." endparam param power caption = "Degree of divergence" default = 2.0 hint = "This is an estimate of the divergence of the main formula, e.g. 2 \ for z^2+c, 3 for z^3+c etc." endparam param julia caption = "Julia ?" default = false hint = "Enable for correct rendering of Julia Sets." endparam param skipiter caption = "'Start' iteration" default = 0 min = 0 hint = "You can use this to attempt to get better detail at higher \ iteration depths, note that it only works as intended with 'Color \ Density' as a whole integer >=1." endparam param fixskip caption = "Fix 'Start' iteration" default = false hint = "Fixes a problem when the start iteration is non-zero and maybe in \ other cases - specifically if you get solid areas of palette \ colour zero when the Transfer Function is Linear this may fix the \ problem." endparam param fixit caption = "Fix Field Lines" default = false hint = "Enable this to make the formula much more accurate at rendering \ the field lines correctly. It's a fudge produced after trial and \ error but is quite effective at correcting some areas." endparam param accident caption = "Happy Accident" default = false hint = "You may find the colouring useful when this is enabled. This \ is available as I found the results when initially adding the \ 'Fix it' option quite interesting." visible = !@fixit endparam }

### multiwave edit

Gradient file by Pauldebrot ^{[3]}

MandelMultiwave { global: color c2x0 = hsl(0,0.5,0.5 + @lmag/2) color c2x1 = hsl(0,0.5,0.5) color c2x2 = hsl(0,0.5,0.5 - @lmag/2) color c2x3 = hsl(0,0.5,0.5) color c3x0 = hsl(0,0.5,0.5 + @lmag2/2) color c3x1 = hsl(0,0.5,0.5) color c3x2 = hsl(0,0.5,0.5 - @lmag2/2) color c3x3 = hsl(0,0.5,0.5) color c4x0 = hsl(0,0.5,0.5 + @lmag3/2) color c4x1 = hsl(0,0.5,0.5) color c4x2 = hsl(0,0.5,0.5 - @lmag3/2) color c4x3 = hsl(0,0.5,0.5) color c5x0 = hsl(0,0.5,0.5) color c5x1 = hsl(0,0.5,0.5 - @lmag4/2) color c5x2 = hsl(0,0.5,0.5) color c5x3 = hsl(0,0.5,0.5 + @lmag4/2) init: complex il = 1/log(@power) ; Inverse log (power). float lp = log(log(@bailout)) ; log(log bailout). final: float i = real(#numiter + il*lp - il*log(log(cabs(#z)))) + @displacement IF (i < 1) i = 1 ENDIF i = (i - 1)*@rescale + 1 float ix = (i - 1)/@hfreq2 ix = ix - trunc(ix) color c1a color c1b color c1c color c1d color c2a color c2b color c2c color c2d color c3a color c3b color c3c color c3d color c4a color c4b color c4c color c4d IF (ix < 1/5) ix = ix*5 c1a = @c1e c1b = @c1a c1c = @c1b c1d = @c1c c2a = @c2e c2b = @c2a c2c = @c2b c2d = @c2c c3a = @c3e c3b = @c3a c3c = @c3b c3d = @c3c c4a = @c4e c4b = @c4a c4c = @c4b c4d = @c4c ELSEIF (ix < 2/5) ix = (ix - 1/5)*5 c1a = @c1a c1b = @c1b c1c = @c1c c1d = @c1d c2a = @c2a c2b = @c2b c2c = @c2c c2d = @c2d c3a = @c3a c3b = @c3b c3c = @c3c c3d = @c3d c4a = @c4a c4b = @c4b c4c = @c4c c4d = @c4d ELSEIF (ix < 3/5) ix = (ix - 2/5)*5 c1a = @c1b c1b = @c1c c1c = @c1d c1d = @c1e c2a = @c2b c2b = @c2c c2c = @c2d c2d = @c2e c3a = @c3b c3b = @c3c c3c = @c3d c3d = @c3e c4a = @c4b c4b = @c4c c4c = @c4d c4d = @c4e ELSEIF (ix < 4/5) ix = (ix - 3/5)*5 c1a = @c1c c1b = @c1d c1c = @c1e c1d = @c1a c2a = @c2c c2b = @c2d c2c = @c2e c2d = @c2a c3a = @c3c c3b = @c3d c3c = @c3e c3d = @c3a c4a = @c4c c4b = @c4d c4c = @c4e c4d = @c4a ELSE ix = (ix - 4/5)*5 c1a = @c1d c1b = @c1e c1c = @c1a c1d = @c1b c2a = @c2d c2b = @c2e c2c = @c2a c2d = @c2b c3a = @c3d c3b = @c3e c3c = @c3a c3d = @c3b c4a = @c4d c4b = @c4e c4c = @c4a c4d = @c4b ENDIF float rp0 = red(c1b) float gp0 = green(c1b) float bp0 = blue(c1b) float rm0 = (red(c1c) - red(c1a))/2 float gm0 = (green(c1c) - green(c1a))/2 float bm0 = (blue(c1c) - blue(c1a))/2 float rp1 = red(c1c) float gp1 = green(c1c) float bp1 = blue(c1c) float rm1 = (red(c1d) - red(c1b))/2 float gm1 = (green(c1d) - green(c1b))/2 float bm1 = (blue(c1d) - blue(c1b))/2 float ixt2 = ix^2 float ixt3 = ix^3 float ixa = 2*ixt3 - 3*ixt2 + 1 float ixb = ixt3 - 2*ixt2 + ix float ixc = -2*ixt3 + 3*ixt2 float ixd = ixt3 - ixt2 float rrr = ixa*rp0 + ixb*rm0 + ixc*rp1 + ixd*rm1 float ggg = ixa*gp0 + ixb*gm0 + ixc*gp1 + ixd*gm1 float bbb = ixa*bp0 + ixb*bm0 + ixc*bp1 + ixd*bm1 IF (rrr < 0.0) rrr = 0.0 ELSEIF (rrr > 1.0) rrr = 1.0 ENDIF IF (ggg < 0.0) ggg = 0.0 ELSEIF (ggg > 1.0) ggg = 1.0 ENDIF IF (bbb < 0.0) bbb = 0.0 ELSEIF (bbb > 1.0) bbb = 1.0 ENDIF color c1 = rgb(rrr,ggg,bbb) rp0 = red(c2b) gp0 = green(c2b) bp0 = blue(c2b) rm0 = (red(c2c) - red(c2a))/2 gm0 = (green(c2c) - green(c2a))/2 bm0 = (blue(c2c) - blue(c2a))/2 rp1 = red(c2c) gp1 = green(c2c) bp1 = blue(c2c) rm1 = (red(c2d) - red(c2b))/2 gm1 = (green(c2d) - green(c2b))/2 bm1 = (blue(c2d) - blue(c2b))/2 ixt2 = ix^2 ixt3 = ix^3 ixa = 2*ixt3 - 3*ixt2 + 1 ixb = ixt3 - 2*ixt2 + ix ixc = -2*ixt3 + 3*ixt2 ixd = ixt3 - ixt2 rrr = ixa*rp0 + ixb*rm0 + ixc*rp1 + ixd*rm1 ggg = ixa*gp0 + ixb*gm0 + ixc*gp1 + ixd*gm1 bbb = ixa*bp0 + ixb*bm0 + ixc*bp1 + ixd*bm1 IF (rrr < 0.0) rrr = 0.0 ELSEIF (rrr > 1.0) rrr = 1.0 ENDIF IF (ggg < 0.0) ggg = 0.0 ELSEIF (ggg > 1.0) ggg = 1.0 ENDIF IF (bbb < 0.0) bbb = 0.0 ELSEIF (bbb > 1.0) bbb = 1.0 ENDIF color c2 = rgb(rrr,ggg,bbb) rp0 = red(c3b) gp0 = green(c3b) bp0 = blue(c3b) rm0 = (red(c3c) - red(c3a))/2 gm0 = (green(c3c) - green(c3a))/2 bm0 = (blue(c3c) - blue(c3a))/2 rp1 = red(c3c) gp1 = green(c3c) bp1 = blue(c3c) rm1 = (red(c3d) - red(c3b))/2 gm1 = (green(c3d) - green(c3b))/2 bm1 = (blue(c3d) - blue(c3b))/2 ixt2 = ix^2 ixt3 = ix^3 ixa = 2*ixt3 - 3*ixt2 + 1 ixb = ixt3 - 2*ixt2 + ix ixc = -2*ixt3 + 3*ixt2 ixd = ixt3 - ixt2 rrr = ixa*rp0 + ixb*rm0 + ixc*rp1 + ixd*rm1 ggg = ixa*gp0 + ixb*gm0 + ixc*gp1 + ixd*gm1 bbb = ixa*bp0 + ixb*bm0 + ixc*bp1 + ixd*bm1 IF (rrr < 0.0) rrr = 0.0 ELSEIF (rrr > 1.0) rrr = 1.0 ENDIF IF (ggg < 0.0) ggg = 0.0 ELSEIF (ggg > 1.0) ggg = 1.0 ENDIF IF (bbb < 0.0) bbb = 0.0 ELSEIF (bbb > 1.0) bbb = 1.0 ENDIF color c3 = rgb(rrr,ggg,bbb) rp0 = red(c4b) gp0 = green(c4b) bp0 = blue(c4b) rm0 = (red(c4c) - red(c4a))/2 gm0 = (green(c4c) - green(c4a))/2 bm0 = (blue(c4c) - blue(c4a))/2 rp1 = red(c4c) gp1 = green(c4c) bp1 = blue(c4c) rm1 = (red(c4d) - red(c4b))/2 gm1 = (green(c4d) - green(c4b))/2 bm1 = (blue(c4d) - blue(c4b))/2 ixt2 = ix^2 ixt3 = ix^3 ixa = 2*ixt3 - 3*ixt2 + 1 ixb = ixt3 - 2*ixt2 + ix ixc = -2*ixt3 + 3*ixt2 ixd = ixt3 - ixt2 rrr = ixa*rp0 + ixb*rm0 + ixc*rp1 + ixd*rm1 ggg = ixa*gp0 + ixb*gm0 + ixc*gp1 + ixd*gm1 bbb = ixa*bp0 + ixb*bm0 + ixc*bp1 + ixd*bm1 IF (rrr < 0.0) rrr = 0.0 ELSEIF (rrr > 1.0) rrr = 1.0 ENDIF IF (ggg < 0.0) ggg = 0.0 ELSEIF (ggg > 1.0) ggg = 1.0 ENDIF IF (bbb < 0.0) bbb = 0.0 ELSEIF (bbb > 1.0) bbb = 1.0 ENDIF color c4 = rgb(rrr,ggg,bbb) ix = (i - 1)/@hfreq1 ix = ix - trunc(ix) color ca color cb color cc color cd IF (ix < 1/4) ix = ix*4 ca = c4 cb = c1 cc = c2 cd = c3 ELSEIF (ix < 1/2) ix = (ix - 1/4)*4 ca = c1 cb = c2 cc = c3 cd = c4 ELSEIF (ix < 3/4) ix = (ix - 1/2)*4 ca = c2 cb = c3 cc = c4 cd = c1 ELSE ix = (ix - 3/4)*4 ca = c3 cb = c4 cc = c1 cd = c2 ENDIF rp0 = red(cb) gp0 = green(cb) bp0 = blue(cb) rm0 = (red(cc) - red(ca))/2 gm0 = (green(cc) - green(ca))/2 bm0 = (blue(cc) - blue(ca))/2 rp1 = red(cc) gp1 = green(cc) bp1 = blue(cc) rm1 = (red(cd) - red(cb))/2 gm1 = (green(cd) - green(cb))/2 bm1 = (blue(cd) - blue(cb))/2 ixt2 = ix^2 ixt3 = ix^3 ixa = 2*ixt3 - 3*ixt2 + 1 ixb = ixt3 - 2*ixt2 + ix ixc = -2*ixt3 + 3*ixt2 ixd = ixt3 - ixt2 rrr = ixa*rp0 + ixb*rm0 + ixc*rp1 + ixd*rm1 ggg = ixa*gp0 + ixb*gm0 + ixc*gp1 + ixd*gm1 bbb = ixa*bp0 + ixb*bm0 + ixc*bp1 + ixd*bm1 IF (rrr < 0.0) rrr = 0.0 ELSEIF (rrr > 1.0) rrr = 1.0 ENDIF IF (ggg < 0.0) ggg = 0.0 ELSEIF (ggg > 1.0) ggg = 1.0 ENDIF IF (bbb < 0.0) bbb = 0.0 ELSEIF (bbb > 1.0) bbb = 1.0 ENDIF c2 = rgb(rrr,ggg,bbb) ix = (i - 1)/@sfreq ix = ix - trunc(ix) IF (ix < 1/3) ix = ix*3 ca = @cz cb = @cx cc = @cy cd = @cz ELSEIF (ix < 2/3) ix = (ix - 1/3)*3 ca = @cx cb = @cy cc = @cz cd = @cx ELSE ix = (ix - 2/3)*3 ca = @cy cb = @cz cc = @cx cd = @cy ENDIF rp0 = red(cb) gp0 = green(cb) bp0 = blue(cb) rm0 = (red(cc) - red(ca))/2 gm0 = (green(cc) - green(ca))/2 bm0 = (blue(cc) - blue(ca))/2 rp1 = red(cc) gp1 = green(cc) bp1 = blue(cc) rm1 = (red(cd) - red(cb))/2 gm1 = (green(cd) - green(cb))/2 bm1 = (blue(cd) - blue(cb))/2 ixt2 = ix^2 ixt3 = ix^3 ixa = 2*ixt3 - 3*ixt2 + 1 ixb = ixt3 - 2*ixt2 + ix ixc = -2*ixt3 + 3*ixt2 ixd = ixt3 - ixt2 rrr = ixa*rp0 + ixb*rm0 + ixc*rp1 + ixd*rm1 ggg = ixa*gp0 + ixb*gm0 + ixc*gp1 + ixd*gm1 bbb = ixa*bp0 + ixb*bm0 + ixc*bp1 + ixd*bm1 IF (rrr < 0.0) rrr = 0.0 ELSEIF (rrr > 1.0) rrr = 1.0 ENDIF IF (ggg < 0.0) ggg = 0.0 ELSEIF (ggg > 1.0) ggg = 1.0 ENDIF IF (bbb < 0.0) bbb = 0.0 ELSEIF (bbb > 1.0) bbb = 1.0 ENDIF c3 = rgb(rrr,ggg,bbb) float hh = hue(c2) float ss = sat(c2) float ll = lum(c2) IF (ll != 1.0 && lum(c3) != 1.0) ll = 1 - 1/((1/(1 - ll) - 1)*(1/(1 - lum(c3)) - 1) + 1) ENDIF IF (@smode == 0) ss = ss*(1 - sat(c3)*(((1 - cos((hh - hue(c3))*#pi/3))/2)^@spow)) ELSE hh = (hh + hue(c3))%6 IF (ss != 1.0 && sat(c3) != 1.0) ss = 1 - 1/((1/(1 - ss) - 1)*(1/(1 - sat(c3)) - 1) + 1) ENDIF ENDIF c1 = hsl(hh,ss,ll) ix = (i - 1)/@lfreq ix = ix - trunc(ix) IF (ix < 1/4) ix = ix*4 ca = c2x3 cb = c2x0 cc = c2x1 cd = c2x2 ELSEIF (ix < 1/2) ix = (ix - 1/4)*4 ca = c2x0 cb = c2x1 cc = c2x2 cd = c2x3 ELSEIF (ix < 3/4) ix = (ix - 1/2)*4 ca = c2x1 cb = c2x2 cc = c2x3 cd = c2x0 ELSE ix = (ix - 3/4)*4 ca = c2x2 cb = c2x3 cc = c2x0 cd = c2x1 ENDIF rp0 = red(cb) gp0 = green(cb) bp0 = blue(cb) rm0 = (red(cc) - red(ca))/2 gm0 = (green(cc) - green(ca))/2 bm0 = (blue(cc) - blue(ca))/2 rp1 = red(cc) gp1 = green(cc) bp1 = blue(cc) rm1 = (red(cd) - red(cb))/2 gm1 = (green(cd) - green(cb))/2 bm1 = (blue(cd) - blue(cb))/2 ixt2 = ix^2 ixt3 = ix^3 ixa = 2*ixt3 - 3*ixt2 + 1 ixb = ixt3 - 2*ixt2 + ix ixc = -2*ixt3 + 3*ixt2 ixd = ixt3 - ixt2 rrr = ixa*rp0 + ixb*rm0 + ixc*rp1 + ixd*rm1 ggg = ixa*gp0 + ixb*gm0 + ixc*gp1 + ixd*gm1 bbb = ixa*bp0 + ixb*bm0 + ixc*bp1 + ixd*bm1 IF (rrr < 0.0) rrr = 0.0 ELSEIF (rrr > 1.0) rrr = 1.0 ENDIF IF (ggg < 0.0) ggg = 0.0 ELSEIF (ggg > 1.0) ggg = 1.0 ENDIF IF (bbb < 0.0) bbb = 0.0 ELSEIF (bbb > 1.0) bbb = 1.0 ENDIF c2 = rgb(rrr,ggg,bbb) ix = (i - 1)/@lfreq2 ix = ix - trunc(ix) IF (ix < 1/4) ix = ix*4 ca = c3x3 cb = c3x0 cc = c3x1 cd = c3x2 ELSEIF (ix < 1/2) ix = (ix - 1/4)*4 ca = c3x0 cb = c3x1 cc = c3x2 cd = c3x3 ELSEIF (ix < 3/4) ix = (ix - 1/2)*4 ca = c3x1 cb = c3x2 cc = c3x3 cd = c3x0 ELSE ix = (ix - 3/4)*4 ca = c3x2 cb = c3x3 cc = c3x0 cd = c3x1 ENDIF rp0 = red(cb) gp0 = green(cb) bp0 = blue(cb) rm0 = (red(cc) - red(ca))/2 gm0 = (green(cc) - green(ca))/2 bm0 = (blue(cc) - blue(ca))/2 rp1 = red(cc) gp1 = green(cc) bp1 = blue(cc) rm1 = (red(cd) - red(cb))/2 gm1 = (green(cd) - green(cb))/2 bm1 = (blue(cd) - blue(cb))/2 ixt2 = ix^2 ixt3 = ix^3 ixa = 2*ixt3 - 3*ixt2 + 1 ixb = ixt3 - 2*ixt2 + ix ixc = -2*ixt3 + 3*ixt2 ixd = ixt3 - ixt2 rrr = ixa*rp0 + ixb*rm0 + ixc*rp1 + ixd*rm1 ggg = ixa*gp0 + ixb*gm0 + ixc*gp1 + ixd*gm1 bbb = ixa*bp0 + ixb*bm0 + ixc*bp1 + ixd*bm1 IF (rrr < 0.0) rrr = 0.0 ELSEIF (rrr > 1.0) rrr = 1.0 ENDIF IF (ggg < 0.0) ggg = 0.0 ELSEIF (ggg > 1.0) ggg = 1.0 ENDIF IF (bbb < 0.0) bbb = 0.0 ELSEIF (bbb > 1.0) bbb = 1.0 ENDIF c3 = rgb(rrr,ggg,bbb) ix = (i - 1)/@lfreq3 ix = ix - trunc(ix) IF (ix < 1/4) ix = ix*4 ca = c4x3 cb = c4x0 cc = c4x1 cd = c4x2 ELSEIF (ix < 1/2) ix = (ix - 1/4)*4 ca = c4x0 cb = c4x1 cc = c4x2 cd = c4x3 ELSEIF (ix < 3/4) ix = (ix - 1/2)*4 ca = c4x1 cb = c4x2 cc = c4x3 cd = c4x0 ELSE ix = (ix - 3/4)*4 ca = c4x2 cb = c4x3 cc = c4x0 cd = c4x1 ENDIF rp0 = red(cb) gp0 = green(cb) bp0 = blue(cb) rm0 = (red(cc) - red(ca))/2 gm0 = (green(cc) - green(ca))/2 bm0 = (blue(cc) - blue(ca))/2 rp1 = red(cc) gp1 = green(cc) bp1 = blue(cc) rm1 = (red(cd) - red(cb))/2 gm1 = (green(cd) - green(cb))/2 bm1 = (blue(cd) - blue(cb))/2 ixt2 = ix^2 ixt3 = ix^3 ixa = 2*ixt3 - 3*ixt2 + 1 ixb = ixt3 - 2*ixt2 + ix ixc = -2*ixt3 + 3*ixt2 ixd = ixt3 - ixt2 rrr = ixa*rp0 + ixb*rm0 + ixc*rp1 + ixd*rm1 ggg = ixa*gp0 + ixb*gm0 + ixc*gp1 + ixd*gm1 bbb = ixa*bp0 + ixb*bm0 + ixc*bp1 + ixd*bm1 IF (rrr < 0.0) rrr = 0.0 ELSEIF (rrr > 1.0) rrr = 1.0 ENDIF IF (ggg < 0.0) ggg = 0.0 ELSEIF (ggg > 1.0) ggg = 1.0 ENDIF IF (bbb < 0.0) bbb = 0.0 ELSEIF (bbb > 1.0) bbb = 1.0 ENDIF c4 = rgb(rrr,ggg,bbb) ix = (i - 1)/@lfreq4 ix = ix - trunc(ix) IF (ix < 1/4) ix = ix*4 ca = c5x3 cb = c5x0 cc = c5x1 cd = c5x2 ELSEIF (ix < 1/2) ix = (ix - 1/4)*4 ca = c5x0 cb = c5x1 cc = c5x2 cd = c5x3 ELSEIF (ix < 3/4) ix = (ix - 1/2)*4 ca = c5x1 cb = c5x2 cc = c5x3 cd = c5x0 ELSE ix = (ix - 3/4)*4 ca = c5x2 cb = c5x3 cc = c5x0 cd = c5x1 ENDIF rp0 = red(cb) gp0 = green(cb) bp0 = blue(cb) rm0 = (red(cc) - red(ca))/2 gm0 = (green(cc) - green(ca))/2 bm0 = (blue(cc) - blue(ca))/2 rp1 = red(cc) gp1 = green(cc) bp1 = blue(cc) rm1 = (red(cd) - red(cb))/2 gm1 = (green(cd) - green(cb))/2 bm1 = (blue(cd) - blue(cb))/2 ixt2 = ix^2 ixt3 = ix^3 ixa = 2*ixt3 - 3*ixt2 + 1 ixb = ixt3 - 2*ixt2 + ix ixc = -2*ixt3 + 3*ixt2 ixd = ixt3 - ixt2 rrr = ixa*rp0 + ixb*rm0 + ixc*rp1 + ixd*rm1 ggg = ixa*gp0 + ixb*gm0 + ixc*gp1 + ixd*gm1 bbb = ixa*bp0 + ixb*bm0 + ixc*bp1 + ixd*bm1 IF (rrr < 0.0) rrr = 0.0 ELSEIF (rrr > 1.0) rrr = 1.0 ENDIF IF (ggg < 0.0) ggg = 0.0 ELSEIF (ggg > 1.0) ggg = 1.0 ENDIF IF (bbb < 0.0) bbb = 0.0 ELSEIF (bbb > 1.0) bbb = 1.0 ENDIF color c5 = rgb(rrr,ggg,bbb) float mn = @fitminit float mx = @fitmaxit ix = i IF (@transfer == 1) ix = ix^(1/@transpower) mn = mn^(1/@transpower) mx = mx^(1/@transpower) ELSEIF (@transfer == 2) ix = log(ix) mn = log(mn) mx = log(mx) ENDIF IF (@fit) IF (ix < mn) ix = 0 ELSE ix = (ix - mn)/(mx - mn) ENDIF ix = ix * @fittimes ELSE ix = 0.05*ix ENDIF color c6 = gradient(ix) IF (@mmode == 1) hh = (hue(c1)+hue(c6))%6 float sc1 = sat(c1) float sc6 = sat(c6) IF (sc1 > 0.99) sc1 = 0.99 ENDIF IF (sc6 > 0.99) sc6 = 0.99 ENDIF sc1 = 1/(1 - sc1) - 1 sc6 = 1/(1 - sc6) - 1 ss = sc1*sc6 ss = 1 - 1/(ss + 1) float ll1 = lum(c1) float ll2 = lum(c2) float ll3 = lum(c3) float ll4 = lum(c4) float ll5 = lum(c5) float ll6 = lum(c6) IF (ll1 == 1.0 || ll2 == 1.0 || ll3 == 1.0 || ll4 == 1.0 || ll5 == 1.0 || ll6 == 1.0) ll = 1.0 ELSE ll1 = 1/(1 - ll1) - 1 ll2 = 1/(1 - ll2) - 1 ll3 = 1/(1 - ll3) - 1 ll4 = 1/(1 - ll4) - 1 ll5 = 1/(1 - ll5) - 1 ll6 = 1/(1 - ll6) - 1 ll = ll1*ll2*ll3*ll4*ll5*ll6 ll = 1 - 1/(ll + 1) ENDIF IF (ss < 0) ss = 0 ELSEIF (ss > 1) ss = 1 ENDIF IF (ll < 0) ll = 0 ELSEIF (ll > 1) ll = 1 ENDIF #color = hsl(hh,ss,ll) ELSE hh = hue(c1) ss = sat(c1) float ll1 = lum(c1) float ll2 = lum(c2) float ll3 = lum(c3) float ll4 = lum(c4) float ll5 = lum(c5) IF (ll1 == 1.0 || ll2 == 1.0 || ll3 == 1.0 || ll4 == 1.0 || ll5 == 1.0) ll = 1.0 ELSE ll1 = 1/(1 - ll1) - 1 ll2 = 1/(1 - ll2) - 1 ll3 = 1/(1 - ll3) - 1 ll4 = 1/(1 - ll4) - 1 ll5 = 1/(1 - ll5) - 1 ll = ll1*ll2*ll3*ll4*ll5 ll = 1 - 1/(ll + 1) ENDIF IF (ll < 0) ll = 0 ELSEIF (ll > 1) ll = 1 ENDIF IF (@mmode == 0) c1 = hsl(hh,ss,ll) float rr6 = red(c6) float gg6 = green(c6) float bb6 = blue(c6) IF (rr6 > 0.99) rr6 = 0.99 ENDIF IF (gg6 > 0.99) gg6 = 0.99 ENDIF IF (bb6 > 0.99) bb6 = 0.99 ENDIF float rr1 = red(c1) float gg1 = green(c1) float bb1 = blue(c1) IF (rr1 > 0.99) rr1 = 0.99 ENDIF IF (gg1 > 0.99) gg1 = 0.99 ENDIF IF (bb1 > 0.99) bb1 = 0.99 ENDIF rr6 = 1/(1 - rr6) - 1 gg6 = 1/(1 - gg6) - 1 bb6 = 1/(1 - bb6) - 1 rr1 = 1/(1 - rr1) - 1 gg1 = 1/(1 - gg1) - 1 bb1 = 1/(1 - bb1) - 1 float rrr = 1 - 1/(rr6*rr1 + 1) float ggg = 1 - 1/(gg6*gg1 + 1) float bbb = 1 - 1/(bb6*bb1 + 1) #color = rgb(rrr,ggg,bbb) ELSE float aa = alpha(c5) float omaa = 1 - aa IF (@mmode == 2) c1 = hsl(hh,ss,ll) float rrr = red(c6)*aa + red(c1)*omaa float ggg = green(c6)*aa + green(c1)*omaa float bbb = blue(c6)*aa + blue(c1)*omaa #color = rgb(rrr,ggg,bbb) ELSE hh = (hue(c6)+hh)%6 ss = sat(c6)*aa + ss*omaa ll = lum(c6)*aa + ll*omaa #color = hsl(hh,ss,ll) ENDIF ENDIF ENDIF default: title = "Mandelbrot Multiwave Coloring" 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. \ Only needed when coloring divergent points." endparam color param c1a caption = "Hue 1A" default = rgb(130/255,91/255,40/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c2a caption = "Hue 2A" default = rgb(186/255,153/255,102/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c3a caption = "Hue 3A" default = rgb(248/255,114/255,12/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c4a caption = "Hue 4A" default = rgb(74/255,0,0) hint = "Altering these hues will alter the way the gradient is modified." endparam param hfreq1 caption = "Short hue shift period (iters)" default = 530.0 hint = "Changes the period of the short hue modification cycle. Early in the long cycle it will cycle among the four colors above." endparam color param c1b caption = "Hue 1B" default = rgb(77/255,49/255,19/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c2b caption = "Hue 2B" default = rgb(195/255,179/255,131/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c3b caption = "Hue 3B" default = rgb(231/255,227/255,23/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c4b caption = "Hue 4B" default = rgb(240/255,164/255,0) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c1c caption = "Hue 1C" default = rgb(0,72/255,16/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c2c caption = "Hue 2C" default = rgb(133/255,146/255,128/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c3c caption = "Hue 3C" default = rgb(179/255,220/255,72/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c4c caption = "Hue 4C" default = rgb(243/255,224/255,83/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c1d caption = "Hue 1D" default = rgb(24/255,23/255,103/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c2d caption = "Hue 2D" default = rgb(144/255,143/255,163/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c3d caption = "Hue 3D" default = rgb(64/255,153/255,192/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c4d caption = "Hue 4D" default = rgb(31/255,173/255,131/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c1e caption = "Hue 1E" default = rgb(120/255,22/255,22/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c2e caption = "Hue 2E" default = rgb(177/255,129/255,130/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c3e caption = "Hue 3E" default = rgb(189/255,65/255,68/255) hint = "Altering these hues will alter the way the gradient is modified." endparam color param c4e caption = "Hue 4E" default = rgb(33/255,31/255,79/255) hint = "Altering these hues will alter the way the gradient is modified." endparam param hfreq2 caption = "Long hue shift period (iters)" default = 5147.0 hint = "Changes the period of the long hue modification cycle." endparam color param cx caption = "Superslow bias color 1" default = rgb(0.5,0.5,0.5) hint = "Altering these hues will alter the way the gradient is modified." endparam color param cy caption = "Superslow bias color 2" default = hsl(0,0.5,0.7) hint = "Altering these hues will alter the way the gradient is modified." endparam color param cz caption = "Superslow bias color 3" default = hsl(5,1.0,0.3) hint = "Altering these hues will alter the way the gradient is modified." endparam param sfreq caption = "Superslow bias period (iters)" default = 82117.0 hint = "Changes the period of the superslow bias cycle." endparam param smode caption = "Superslow bias mode" enum = "saturation bias" "hsl bias" default = 1 hint = "Luminances adjust luminance of colors above; in saturation bias mode hue/saturation adjusts saturation. Hues matching this hue are unchanged, hues opposite have their saturation reduced the higher this color's saturation. The effect is to suppress opposite hues; set this to saturated red to suppress cyan for example. Neutral gray has no effect. In hsl bias mode hsl addition is used instead." endparam param spow caption = "Superslow bias sensitivity" default = 4.0 min = 0.01 hint = "Higher values narrow the range of hues desaturated; lower ones widen the range. 2 and 0.5 have opposite effects." endparam param lfreq caption = "First luminance shift period (iters)" default = 17.0 hint = "Changes the period of the first luminance modification cycle. The four colors of the short cycle will change gradually to the second set of four, then the third, then the fourth, before returning." endparam param lmag caption = "First luminance shift amplitude" default = 0.2 min = 0.0 max = 1.0 hint = "Changes the amplitude of the first luminance modification cycle." endparam param lfreq2 caption = "Second luminance shift period (iters)" default = 94.0 hint = "Changes the period of the second luminance modification cycle." endparam param lmag2 caption = "Second luminance shift amplitude" default = 0.4 min = 0.0 max = 1.0 hint = "Changes the amplitude of the second luminance modification cycle." endparam param lfreq3 caption = "Third luminance shift period (iters)" default = 2544.0 hint = "Changes the period of the third luminance modification cycle." endparam param lmag3 caption = "Third luminance shift amplitude" default = 0.6 min = 0.0 max = 1.0 hint = "Changes the amplitude of the third luminance modification cycle." endparam param lfreq4 caption = "Fourth luminance shift period (iters)" default = 18544.0 hint = "Changes the period of the fourth luminance modification cycle." endparam param lmag4 caption = "Fourth luminance shift amplitude" default = 0.6 min = 0.0 max = 1.0 hint = "Changes the amplitude of the fourth luminance modification cycle." endparam param mmode caption = "Primary gradient merge mode" enum = "rgb bias" "hsl bias" "rgb blend" "hsl blend" default = 1 hint = "Affects how the gradient is applied to the color ripples; the blend options use the gradient's alpha" endparam param fit caption = "Fit Gradient to Range" default = true hint = "Check this to spread the gradient out over the range of iteration values." endparam param fittimes caption = "Number of repetitions" default = 1.0 min = 1.0 hint = "Repeats gradient the specified number of times over the range of iteration values." endparam param fitminit caption = "Start iteration" default = 1.0 min = 1.0 hint = "Gradient begins at this iteration number. It is best if it's approximately the lowest \ actual number of iterations in the image. You can find the exact number by looking at \ Statistics after generating the image once." endparam param fitmaxit caption = "End iteration" default = 1000.0 min = 1.0 hint = "Gradient fitting is based on this range of iterations. Can be profitably made lower than \ maxiter -- try reducing it by factors of 10 until the gradient doesn't fit well, then raise \ it by a factor of 10 once." endparam param transfer caption = "Super transfer function" enum = "Linear" "Power" "Log" default = 2 hint = "Linear distributes gradient evenly over iterations. \ Power weights gradient towards lower iterations for powers > 1. \ Log weights gradient towards lower iterations." endparam param transpower caption = "Transfer power" default = 3.0 hint = "Larger values weight gradient more towards low iterations. \ 3.0 with a regular transfer function of Linear and a super transfer \ function of Linear with a regular transfer function of CubeRoot \ produce the same results." visible = (@transfer == 1) endparam param bailout caption = "Bail-out value" default = 100000.0 hint = "Larger gives smoother coloring, up to a point." min = 1 endparam param displacement caption = "Displacement" default = 0.0 hint = "Skips the first N iterations of the color gradient, effectively shifting the whole color scheme to lower iterations as a block. Use this for testing." min = 0.0 endparam param rescale caption = "Rescaling" default = 1.0 hint = "Compresses the entire color gradient by this factor, after application of displacement. Use this for testing." min = 1.0 endparam }

# transformations (*.uxf) edit

## equirectangular edit

equirectangular project by pauldelbrot^{[4]}

float lon = real(#pixel) float lat = imag(#pixel) IF ((|lon| > 4) || (|lat| > 1)) #solid = true ELSE lon = lon * 0.5 * #pi lat = (lat * 0.25 + 0.25) * #pi float r = tan(lat) #pixel = r*(cos(lon) + (0,1)*sin(lon)) ENDIF

Description

- Outside the rectangle will be black (or whatever the transform solid color is set to).
- The rectangle will be in -2 to 2 horizontally and -1 to 1 vertically, so with magnification 1 and center 0 and a 2:1 image aspect ratio it will exactly fill the image rect.
- Inside the rectangle will be an equirectangular projection of the Riemann sphere.
- The #pixel= assignment actually does the transform. The RHS of that assignment is just a final polar-to-cartesian conversion.
- The (0,1) constant is i, the square root of minus one.

## exponential edit

ExponentialMap { ; https://fractalforums.org/ultrafractal/59/exponential-map-transform/1697 ; Vertical exponential transform. ; Make narrow high window (say 100X800), zoom, put point of interest at bottom. ; Turn on. Tweak with the 2 controls, and select appropriate width (depends on image, ; narrower for deeper zooms). ; global: if (4 * #height < 3 * #width) pixeldim = 3/#magn/#height else pixeldim = 4/#magn/#width endif w = #width * pixeldim h = #height * pixeldim cc = #center c0 = cc - 1i/2 * h +0i*w -w/4 * @sh b = w/h*log(#magn) *@b a = 1i*w/b transform: if @ison c = #pixel dc = a*(exp(-1i*b/w*(c-c0))) #pixel = c0+dc endif default: title = "Exponential map" param ison caption = "On" default = false endparam float param b caption = "vert. control" default = 1.2 endparam float param sh caption = "hor. shift" default = 0 endparam }

# images edit

# files edit

- ufr : Fractal Files *.ufr
- upr : Displays parameter files (*.upr) : additional parameters specific to the selected fractal formula
- par : Older Fractint parameter files (*.par)
- ugr : Displays gradient files (*.ugr)
- ual : Older gradient files (*.ual)
- map : Fractint palette files (*.map)
- uxf : Displays transformation files (*.uxf)
- ufm : Displays fractal formula files (*.ufm). Fractal formula files contain multiple fractal formulas. Older Fractint formula files (*.frm) are also shown. See Fractal formulas.
- ucl : Displays coloring algorithm files (*.ucl). See also writing coloring algorithm
- ulb : Displays plug-in library files (*.ulb)

## upr edit

troubledTree4 by pauldelbrotfractalforums.org: ultrafractal histogram-de-coloring

a very dense location in the Mandelbrot set, which I have used as a test for various DE methods in the past. This method has given the overall best result here, with one caveat: the spiral centers are too dark, losing detail of the very center of each one, likely because the sampling grid misses points this close and they are not included in the histogram. With the default settings, this image is computed at 200 megapixels in a day and a half on decent modern hardware; with a higher sampling density it would be significantly slower still.

troubledTree4 { fractal: title="troubled tree 4" width=960 height=540 layers=1 credits="Owner;9/9/2023" antialiasing=yes layer: caption="Background" opacity=100 method=linear transparent=yes mapping: center=-1.1454771035128796953843/0.26992867352137579904915 magn=1.1827E13 formula: maxiter=1000000 percheck=off filename="Standard.ufm" entry="FastMandel" p_start=0/0 p_bailout=4.0 inside: transfer=none outside: transfer=linear filename="pgd_DE_histogram.ucl" entry="pgd_MandelbrotDEH" p_seed=-1/0 p_mand=yes p_swidth=100 p_sheight=100 p_finalheight=1080 gradient: smooth=yes index=0 color=8716288 index=100 color=16121855 index=200 color=46591 index=300 color=156 opacity: smooth=no index=0 opacity=255 }

Julia set Dragon for Deh Test ( Distance Estimation Histogram)

juliaDragonDehTest { fractal: title="julia dragon deh test" width=960 height=540 layers=1 credits="Owner;9/10/2023" antialiasing=yes layer: caption="Background" opacity=100 method=linear transparent=yes mapping: center=0/0 magn=1.6666667 formula: maxiter=1000000 percheck=off filename="Standard.ufm" entry="Julia" p_seed=-0.747/0.08 p_power=2/0 p_bailout=1.0E70 inside: transfer=none outside: transfer=linear filename="pgd_DE_histogram.ucl" entry="pgd_MandelbrotDEH" p_seed=-0.747/0.08 p_mand=no p_swidth=100 p_sheight=100 p_finalheight=1080 gradient: smooth=yes index=0 color=8716288 index=100 color=16121855 index=200 color=46591 index=300 color=156 opacity: smooth=no index=0 opacity=255 }

Lichtenberg upr with dense fragment of parameter plane by pauldelbrot^{[5]}

Lichtenberg { fractal: title="Lichtenberg" width=960 height=540 layers=1 credits="Owner;7/1/2020" antialiasing=yes layer: caption="Background" opacity=100 method=linear mapping: center=-1.1477908048866525/0.2752899772141804 magn=17309286 formula: maxiter=10000000 filename="Standard.ufm" entry="FastMandel" p_start=0/0 p_bailout=4.0 inside: transfer=none outside: transfer=linear filename="pgd.ucl" entry="pgd_MSetBoundary4" p_bwidth=1.0 p_pow=0.5 p_power=2/0 p_mandel=yes p_convergent=no p_preit=100 p_bailout=100000.0 gradient: comments="Use with Lighting coloring algorithm." smooth=yes index=0 color=16777215 index=399 color=0 opacity: smooth=no index=0 opacity=255 } pgd.ucl:pgd_MSetBoundary4 { ; A superior distance estimator boundary render that colors ; the boundary using a gradient. Works well with the ; "lighting" gradient. Points not near the boundary get ; solid colored; pixels that contain M-set boundary get ; gradient colored. The gradient can roughly be considered ; to approximate the fractal dimension of the boundary, with ; earlier colors indicating D near 1 and later ones ; indicating D close to 2. ; ; It should also work for Julia sets. ; ; Non-z^n + c fractals will not generally give accurate ; results. ; ; Tip: Turn off "repeat gradient" when coloring non-convergent ; points (so, on the Outside tab). ; ; Uses a different algorithm to map distances to grey shades ; than the first two Mandelbrot Boundaries. ; init: complex der = 1 complex dc = 0 complex dzz = 0 complex dcz = 0 int i = 0 complex t = 0 bool done = false bool edge = false float pixsize = 4*@bwidth/(#magn*#width) loop: IF (@convergent && !done) IF (i == @preit) t = #z ELSEIF (i > @preit) IF (@power != 2) dzz = @power*(sqr(der)*(@power - 1)*(#z)^(@power - 2) + dzz*(#z)^(@power - 1)) dcz = @power*(der*dc*(@power - 1)*(#z)^(@power - 2) + dcz*(#z)^(@power - 1)) der = @power*der*(#z)^(@power - 1) dc = @power*dc*(#z)^(@power - 1) ; Generalized from below to z^n + c. ELSE dzz = 2*(sqr(der) + #z*dzz) dcz = 2*(der*dc + #z*dcz) der = 2*der*#z dc = 2*dc*#z + 1 ; From TSOFI D.2 "Finding Disks in the Interior of M". ENDIF IF (|#z - t| < (1/@bailout)) done = true IF (|der| >= 1) edge = true ; Misiurewicz point or component boundary ENDIF ENDIF ENDIF i = i + 1 ELSEIF (!done) IF (@power != 2) der = @power*der*(#z)^(@power - 1) ; Generalized from below to z^n + c. ELSE der = 2*der*#z ; From TSOFI D.1 "Bounding the Distance to M" ENDIF IF (@mandel) der = der + 1 ENDIF ENDIF final: float dist = 0 IF (@convergent) complex cc = 1 - der complex dd = dc*conj(cc)/|cc| complex ee = dcz + dzz*dd dist = 0.25*(1.0 - |der|)/cabs(ee) ELSE float d = cabs(#z) dist = real(d*log(d)*0.5/cabs(der)) ENDIF IF (edge) dist = 1.0 ENDIF IF ((dist < pixsize) && (done || (!@convergent))) ; Pixel contains boundary points. dist = dist / pixsize IF (@pow < 0) dist = 1 - dist ^ (1/@pow) ELSE dist = dist ^ (1/@pow) ENDIF #index = 4*atan(log(1/dist))/#pi - 1 ELSEIF (@convergent && (!done)) ; Probably right on the edge. #index = 1.0 ELSE ; Pixel does not contain boundary points. #solid = true ENDIF default: title = "Mandelbrot Boundary 4" heading caption = "Coloring settings" endheading param bwidth caption = "Border width (pix)" default = 1.0 min = 0.0 hint = "Border width in pixels. You may want to reduce \ this for antialiased renders." endparam param pow caption = "Curve Power" default = 1.0 hint = "Adjusting this affects how fast the gradient changes in denser \ parts of the m-set boundary. Try adjusting it if filaments are \ too dark or seahorse centers are washed out." endparam heading caption = "Formula settings" endheading param power caption = "Exponent" default = (2,0) hint = "This should match the power used in the Mandelbrot \ or Julia formula parameters." endparam param mandel caption = "Mandelbrot" default = true hint = "Uncheck this for Julia fractals." endparam param convergent caption = "Convergent" default = false visible = (@mandel) hint = "Check this to color Mandelbrot interior points \ close to the boundary." endparam heading caption = "Cycle-detection tuning" visible = (@mandel && @convergent) endheading param preit caption = "Preiterations" default = 100 min = 0 visible = (@mandel && @convergent) hint = "Iterations to calculate before looking for a cycle \ when estimating interior distances. Larger values \ make cycle detection more accurate." endparam param bailout caption = "Bailout" default = 100000.0 min = 0.0 hint = "Larger values make cycle detection more accurate, \ but more preiterations are needed." visible = (@mandel && @convergent) endparam }

## ucl edit

Coloring algorithms:

- are stored in coloring algorithm files (*.ucl). Each file can contain multiple coloring algorithms.
- define how fractals are colored. The fractal formula creates the basic shape of the fractal, and coloring algorithms provide ways to color that shape. This gives you the flexibility to freely combine coloring algorithms with any fractal formula.

Examples:

Mu-Ency(OUTSIDE) { ; ; Hybrid direct colouring scheme combining dwell, binary decomposition and distance estimator. ; ; Written by Aleph0, based on an algorithm designed by Robert Munafo and documented here: ; ; http://mrob.com/pub/muency/color.html ; ; See also his examples of hybrid colouring schemes: ; ; http://mrob.com/pub/muency/demdwellhybrid.html ; http://mrob.com/pub/muency/binarydecomposition.html ; ; The above pages are part of Robert's Mu-Ency Encyclopedia of the Mandelbrot Set web site: ; ; http://mrob.com/pub/muency.html ; ; This is a good colouring scheme to use when exploring and studying the structure of the Mandelbrot Set at deep zoom levels, as the distance ; estimator scales automatically to reflect the zoom level and the colouring effects are subtle enough to not detract from the filament detail. ; Conventional colouring approaches that use dwell and a wide colour pallette can be problematic at deep zoom levels, as they introduce excessive ; noise that tends to obscure the structure of the set's connecting filaments. ; ; I have followed Robert's general design, but implemented the following user-configurable settings for individual elements of the scheme. The ; default settings match the style used for most of the images hosted at the Mu-Ency web site. ; ; - Binary decomposition (checker-board) effect can be switched on or off (default = on). ; - Dwell band lightening can selected for either odd or even numbered stripes (default = odd). ; - Rainbow effect can be switched on or off (default = off). ; - Distance estimator filament widths can be scaled (multiplier values 0.01-100, default = 1). ; - Dwell colour slope is adjustable (log base selectable 2.0-1000000.0, default 80). ; - Dwell colour pallette starting point can be rotated (range 0.0-6.0, default 0). ; ; NOTE: Set Inside Solid Colour = White to match the style used for most of the images hosted at the Mu-Ency web site. ; global: $define DEBUG float pixel_spacing = 4 / (#magn * #width) ; Unzoomed Mandelbrot set width = 4.0. init: float angle = 0 complex dz = (0,0) float distance_estimate = 0 float dscale = 0 int dwell = 0 float D = 0 ; float finalang = 0 float finalrad = 0 float hue = 0 float P = 0 float radius = 0 float saturation = 0 float value = 0 float z = 0 loop: dz = @power * #z^(@power-1) * dz + 1 final: z = cabs(#z) ; Reordered relative to Robert's description, for substitution into D formula, hence avoiding repetition of cabs calculation. D = #numiter + log(log(z)/log(2))/log(2) - log(log(128)/log(2))/log(2) dwell = floor(D) finalrad = D - dwell ; finalang = atan(imag(#z)/real(#z)) distance_estimate = (log(z*z) * z)/cabs(dz) dscale = log(distance_estimate/(10 * @DE_width * pixel_spacing))/log(2) ; Convert scaled distance estimate to value (luminance in HSL colour space) from 0.0 to 1.0 in 8 bands. if dscale > 0 value = 1.0 elseif dscale > -8 value = (8 + dscale)/8 else value = 0 endif ; Scale value downwards, as 1.0 results in colour white. Spec may be wrong - washes out colours with that range. value = value*0.65 ; Apply logarithmic scaling to dwell. P = log(dwell)/log(@C_slope) ; Mu-Ency algorithm specifies 100000, but that makes colours change too slowly. ; P = dwell ; Map scaled dwell to an angle and radius on the colour wheel. if P < 0.5 P = 1.0 - 1.5*P angle = 1 - P radius = sqrt(P) else P = 1.5*P - 0.5 angle = P radius = sqrt(P) endif ; Make every even or odd numbered stripe a bit lighter, dependent on setting of stripe_phase parameter. if (dwell + @stripe_phase) % 2 == 1 value = 0.85 * value radius = 0.667 * radius endif ; Break the stripes into squares to make the external angles evident. In combination with the stripes, produces a checkerboard appearance. ; if finalang > #pi ; Mu-Ency algorithm specifies > pi, but that's impossible. However, page http://mrob.com/pub/muency/binarydecomposition.html ; specifies testing for positive imaginary component of final iteration. if @binary == 1 && imag(#z) < 0 angle = angle + 0.1 ; Mu-Ency algorithm specifies 0.02, but that's not generally enough. endif ; Add rainbow-like gradient if @rainbow == 1 angle = angle + 0.2 * finalrad ; Mu-Ency algorithm specifies 0.0001, but effect is then invisible. endif ; Set up the hue. ; hue = angle * 10.0 ; I think multiplier 10 in spec may be wrong; band colours change too rapidly? hue = angle hue = (hue - floor(hue))*6 ; Hue is in range 0-6 for Ultra Fractal, scale up from 0-1. hue = (hue + @C_rotation) % 6 ; Hue starts at red in UF, rotate by 3 to start at cyan. ; Set up the saturation. ; saturation = radius - floor(radius) ; This wraps to zero saturation (flat grey) when radius reaches a new integer! saturation = 0.7 + 0.3 * (radius - floor(radius)) ; This wraps to 0.7 saturation. ; if real(#screenpixel) == 0 && imag(#screenpixel) == 0 ; int i = #numiter ; print("-----") ; print("screenpixel = ", #screenpixel, " ... finaliter = ", i) ; print("#z = ", #z, " ... z = ", z) ; print("D = ", D, " ... dwell = ", dwell, " ... finalrad = ", finalrad) ; print("dscale = ", dscale, " ... value = ", value) ; print("P = ", P, " ... angle = ", angle, " ... radius = ", radius) ; print("hue = ", hue, " ... saturation = ", saturation) ; endif ; Set pixel colour using HSL colour model (hue, saturation, luminance). ; #color = hsl(hue, 1.0, 1.0) #color = hsl(hue, saturation, value) default: title = "Mu-Ency" float 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 usually 2." endparam int param binary caption = "Binary decomposition" enum = "Off" "On" default = 1 hint = "Set on to enable binary decomposition, off to disable." endparam int param stripe_phase caption = "Dwell stripe phase" enum = "Even" "Odd" default = 1 hint = "Selects whether to lighten odd or even dwell stripes." endparam int param rainbow caption = "Rainbow effect" enum = "Off" "On" default = 0 hint = "Set on to enable rainbow effect, off to disable." endparam float param DE_width caption = "DE width" default = 1.0 min = 0.01 Max = 100 hint = "Width multiplier for filaments rendered by distance estimator." endparam float param C_slope caption = "Colour slope." default = 80.0 min = 2.0 Max = 1000000.0 hint = "Rate at which dwell colouring changes; lower values = faster changes, higher = slower." endparam float param C_rotation caption = "Colour rotation." default = 5.0 min = 0.0 Max = 6.0 hint = "Rotates starting point of dwell colour pallette; adjust to obtain desired colours." endparam } Monochrome DE(OUTSIDE) { ; ; Monochrome distance estimator colouring scheme with configurable colours for outside set and near to set. Inside set colour can also ; be configured using Ultra Fractal's Inside layer properties. ; ; Written by Aleph0, based on a distance estimator algorithm designed by Robert Munafo and documented here: ; ; http://mrob.com/pub/muency/color.html ; ; The above page is part of Robert's Mu-Ency Encyclopedia of the Mandelbrot Set web site: ; ; http://mrob.com/pub/muency.html ; ; This is a good colouring scheme to use when exploring and studying the structure of the Mandelbrot Set at deep zoom levels, as the distance ; estimator scales automatically to reflect the zoom level. Conventional colouring approaches that use dwell and a wide colour pallette can be ; problematic at deep zoom levels, as they introduce excessive noise that tends to obscure the structure of the set's connecting filaments. ; ; The following user-configurable settings are available: ; ; - Near colour can be selected (selects colour of points near the set, i.e. the filaments, colour darkens progressively to black based on ; proximity to the set, default = black giving a grey scale transition to black). ; - Blend outside and near colours. On = blend, off = use near colour directly. Both effects can be useful, dependent on the colour ; combinations selected. ; - Distance estimator filament widths can be scaled (multiplier values 0.01-100, default = 1). ; ; NOTE: Use Solid Colour setting in the Inside and Outside layer property pages to set the fill colour for inside and outside the set, ; respectively. Select same colour for inside set and near set for monochrome renderings, e.g. to . ; global: $define DEBUG float pixel_spacing = 4 / (#magn * #width) ; Unzoomed Mandelbrot set width = 4.0. init: complex dz = (0,0) float distance_estimate = 0 float dscale = 0 float value = 0 float z = 0 loop: dz = @power * #z^(@power-1) * dz + 1 final: ; #solid = false z = cabs(#z) distance_estimate = (log(z*z) * z)/cabs(dz) dscale = log(distance_estimate/(10 * @DE_width * pixel_spacing))/log(2) ; Convert scaled distance estimate to value (outside/DE blend if blend on, luminance if blend off) from 0.0 to 1.0 in 8 bands. if dscale > 0 #color = hsl(hue(@outside_colour), sat(@outside_colour), lum(@outside_colour)) else if @DE_smooth == 0 if dscale > -1 #color = hsl(hue(@outside_colour), sat(@outside_colour), lum(@outside_colour)) else #color = hsl(hue(@DE_colour), sat(@DE_colour), lum(@DE_colour)) endif else if dscale > -8 value = (8 + dscale)/8 else value = 0 endif value = value ^ @DE_slope ; print("dscale = ", dscale, "... value = ", value) #color = blend(@DE_colour, @outside_colour, value) endif endif ; if real(#screenpixel) == 0 && imag(#screenpixel) == 0 ; int i = #numiter ; print("-----") ; print("screenpixel = ", #screenpixel, " ... finaliter = ", i) ; print("#z = ", #z, " ... z = ", z) ; print("dscale = ", dscale, " ... value = ", value) ; endif default: title = "Monochrome DE" float 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 usually 2." endparam color param outside_colour caption = "Outside colour" default = rgb(255,255,255) hint = "Colour to be used for points outside set." endparam color param DE_colour caption = "DE colour" default = rgb(0,0,0) hint = "Colour to be used for distance estimator filaments." endparam int param DE_smooth caption = "DE smoothing" enum = "Off" "On" default = 1 hint = "On = use filament smoothing, off = no filament smoothing." endparam float param DE_width caption = "DE width" default = 1.0 min = 0.00001 Max = 100 hint = "Width multiplier for filaments rendered by distance estimator." endparam float param DE_slope caption = "DE slope" default = 1.0 min = 0.01 Max = 100.0 hint = "Slope for transition to DE colour: 1.0 = linear from threshold to set; \ < 1.0 = slow change near threshold accelerating to rapid change near set; \ > 1.0 = rapid change at threshold decelerating to slow change near set." endparam } DistanceEstimatorScaled(OUTSIDE) { ; ; Distance-estimator coloring algorithm for Mandelbrot and ; other z^n fractal types (Phoenix, Julia). This coloring ; algorithm estimates the distance to the boundary of the ; fractal (for example the Mandelbrot set) and colors points ; accordingly. ; ; Written by Damien M. Jones. ; Modified by Aleph0 to scale the distance estimate with zoom. ; global: $define DEBUG float pixel_spacing = 4 / (#magn * #width) ; Unzoomed Mandelbrot set width = 4.0. init: complex dz = (0,0) loop: dz = @power * #z^(@power-1) * dz + 1 final: #index = (@power*log(cabs(#z)) * (cabs(#z)) / (cabs(dz) * 100000 * @DE_width * pixel_spacing))^(1/@power) default: title = "Distance Estimator Scaled" helpfile = "Uf*.chm" helptopic = "Html/coloring/standard/distanceestimator.html" 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 usually 2." endparam float param DE_width caption = "DE width" default = 1.0 min = 0.00001 Max = 10000000 hint = "Width multiplier for filaments rendered by distance estimator." endparam }

### uxf edit

Transformations globally transform and warp the shape of a fractal. You can combine various transformations to create complex effects.