Fractals/xaos
XaoS is an interactive fractal zoomer program written in C++ and JS. It allows the user to continuously zoom in or out of a fractal in real-time.
Here is unofficial wiki about XaoS.
Links
edit
Algorithms
editHubička algorithm
editXaoS was originally just a "poorly written" Mandelbrot viewer[1] until Jan Hubička added efficient zooming (= real time zoom[2])[3], using a technique sometimes called the XaoS algorithm or Hubička algorithm.
At the time fractal zoom movies were produced by completely recalculating each frame, even though they naturally had much of their area in common with each other. This made interactive zooming impossible without very powerful computers.[4] Furthermore, unless even more processing is used in order to do antialiasing, recalculating every frame produces a 'twinkle' effect as small bright areas hit and then disappear between pixels.[5]
Yet allowing the user to zoom, rather than jump as in Fractint, seemed like the most natural way to interact with fractals. In order to create an interactive zoom, Hubička needed to find a way to save the calculations which were already made. It would take up too much memory to save every pixel ever calculated, so the Hubička algorithm only saves the previous frame, and rather than remembering the location of each pixel it can keep them aligned in rows and columns and remember those instead.
The most difficult part of the XaoS algorithm was choosing which saved rows and columns to draw where. Doing this wrong results in distorted images, yet it must be done quickly to be useful. After several different heuristics were tried, eventually the problem was treated as an optimization problem.
The remaining rows and columns are colored in the same as the closest row/column, and are freshly calculated as the CPU gets time to do so. This is a careful balance between keeping the zoom going and increasing the level of detail. Calculating areas where the image is being zoomed to is put at a higher priority since these will be on the screen the longest and this is likely where the user is looking anyway. Zooming out, the reverse occurs and the priority is on the edges.[5]
The Hubička algorithm can also be applied to zooming in on other images where the pixels are calculated, and has been used in other software such as the rtzme complex function graphing program,[6] and other fractal zoomers.
Images
editCommand line options
editTo find it use :
xaos -help
Output is :
XaoS3.5 help text
(This help is genereated automagically. I am sorry for all inconvencies)
option string param description
-delay number
Delay screen updates (milliseconds)
-driver string
Select driver
-list List available drivers. Then exit
-config Print configuration. Then exit
-speedtest Test speed of calculation loop. Then exit
-threads number
Set number of threads (CPUs) to use
-pipe string
Accept commands from pipe (use "-" for stdin)
-maxframerate number
Maximal framerate (0 for unlimited - default)
Screen size options:
Knowledge of exact screen size makes random dot stereogram look better.
Also is used for choosing correct view area
-screenwidth f.point
exact size of screen in centimeters
-screenheight f.point
exact size of screen in centimeters
Use this option in case you use some kind of virtual screen
or something similar that confuses previous options
-pixelwidth f.point
exact size of one pixel in centimeters
-pixelheight f.point
exact size of one pixel in centimeters
-formula string
user formula
-forminit string
z0 for user formula
Animation rendering:
-render string
Render animation into seqence of .png files
-basename string
Name for .png files (XaoS will add 4 digit number and extension
-size string
widthxheight
-renderimage string
256 or truecolor
-renderframerate f.point
framerate
-antialiasing Perform antialiasing (slow, requires quite lot of memory)
-alwaysrecalc Always recalculate whole image (slowes down rendering, increases quality)
-rendervectors Render motion vectors (should be used for MPEG encoding)
-iframedist number
Recommended distance between I frames in pat file (should be used for MPEG encoding)
X11 driver options:
-display string
Select display
-size string
Select size of window (WIDTHxHEIGHT).
-sync Generate sync signals before looking for events. This
helps on old and buggy HP-UX X servers.
-shared Use shared colormap on pseudocolor display.
-usedefault Use default visual if autodetection causes troubles.
-nomitshm Disable MITSHM extension.
-fullscreen Enable fullscreen mode.
-windowid number
Use selected window.
-window-id number
Use selected window.
-root Use root window.
AA driver options:
-aadriver string
Select display driver used by aa-lib
-kbddriver string
Select keyboard driver used by aa-lib
-mousedriver string
Select keyboard driver used by aa-lib
-font string
Select font
-width number
Set width
-height number
Set height
-minwidth number
Set minimal allowed width
-minheight number
Set minimal allowed height
-maxwidth number
Set maximal allowed width
-maxheight number
Set maximal allowed height
-recwidth number
Set recommended width
-recheight number
Set recommended height
-normal enable usage of narmal{{typo help inline|reason=similar to harmal|date=September 2022}} characters
-nonnormal disable usage of normal characters
-dim enable usage of dim(half bright) characters
-nodim disable usage of dim(half bright) characters
-bold enable usage of bold(double bright) characters
-nobold disable usage of bold(double bright) characters
-boldfont enable usage of boldfont characters
-noboldfont disable usage of boldfont characters
-reverse enable usage of reversed characters
-noreverse disable usage of reversed characters
-all enable usage of reserved characters
-eight enable usage of non ansi characters
-extended enable usage of extended character set
-inverse enable inverse
-bright number
set bright (0-255)
-contrast number
set contrast (0-255)
-gamma f.point
set famma (0-1)
-nodither Disable dithering
-floyd_steinberg Enable floyd steinberg dithering
-error_distribution Enable error distribution dithering
-random number
Set random dithering value
-dimmul f.point
Multiply factor for dim color (5.3)
-boldmul f.point
Multiply factor for bold color (5.3)
-nomouse Disable mouse
Command line options only
-print_menus print menus specifications of all menus
-print_menu string
print menu specification
-xshl_print_menustring
print menu specification in xshl format
-xshl_print_menus print all menu specifications in xshl format
-print_dialog string
print dialog specification
File
-loadpos input_file
Load
-savepos output_file
Save
-record output_file
Record
-play input_file
Replay
-saveimg output_file
Save image
-loadexample Load random example
-savecfg Save configuration
Fractal
-usrform string
User formula
-usrformInit string
User initialization
-perturbation real_number real_number
Perturbation
-initstate Reset to defaults
-julia on|off
Julia mode
-view real_number real_number real_number real_number
View
-angle real_number
Set angle
-plane integer
Set plane
-incoloring integer
Inside coloring mode
-outcoloring integer
Outside coloring mode
-intcoloring integer
Inside truecolor coloring mode
-outtcoloring integer
Outside truecolor coloring mode
-juliaseed real_number real_number
Julia seed
Calculation
-periodicity Periodicity checking
-maxiter integer
Iterations
-bailout real_number
Bailout
-fastjulia Fast julia mode
-range integer
Solid guessing range
Filters
-edge Edge detection
-edge2 Edge detection2
-threed Pseudo 3d
-starfield Starfield
-stereogram Random dot stereogram
-interlace Interlace filter
-blur Motionblur
-emboss Emboss
-palettef Palette emulator
-anti Antialiasing
-truecolor Truecolor emulator
UI
-letterspersec integer
Letters per second
-autopilot Autopilot
-inhibittextoutput VJ mode
-recalculate Recalculate
-interrupt Interrupt
-speed real_number
Zooming speed
-fixedstep Fixed step
-nogui Disable XaoS's builtin GUI
-status Status
-ministatus Ministatus
Misc
-playstr string
Play string
-text string
Display text
-color white|black|red
Color
-textposition left|center|right top|middle|bottom
Text position
-message string
Message
Help
-help Help
Formulae
-mandel Mandelbrot
-mandel3 Mandelbrot^3
-mandel4 Mandelbrot^4
-mandel5 Mandelbrot^5
-mandel6 Mandelbrot^6
-newton Newton
-newton4 Newton^4
-barnsley Barnsley1
-barnsley2 Barnsley2
-barnsley3 Barnsley3
-octal Octal
-phoenix Phoenix
-magnet Magnet
-magnet2 Magnet2
More formulae
-trice Triceratops
-catseye Catseye
-mbar Mandelbar
-mlambda Lambda
-manowar Manowar
-spider Spider
-sier Sierpinski
-carpet S.Carpet
-koch Koch Snowflake
-hornflake Spidron hornflake
-user User defined
Palette
-defpalette Default palette
-randompalette Random palette
-palette integer integer integer
Custom palette
-cycling Color cycling
-cyclingspeed integer
Color cycling speed
-shiftpalette integer
Shift palette
Dynamic resolution
-fastmode zero|never|animation|new|allways
Dynamic resolution mode
Rotation
-rotationspeed real_number
Rotation speed
-autorotate on|off
Automatic rotation
-fastrotate on|off
Fast rotation mode
Quit
-quit Exit now
Files
editxaf
editXaos Animnation File (*.xaf ) is a file with functions for generating sequence of images
For example :
xaos-3.5/tutorial/julia.xaf
To make xaf file use record function, syntax :
(record bool [ file ])
example :
(record #t)
or :
(record #f)
How to render image sequence from xaf file :
xaos -render [xpf_filename] -size 352x240 -antialiasing -renderframerate 24 -basename [basename]
xpf
editxpf stands for Xaos Position File. This file is human-readable, and can easily be improved by hand after saving, or used as a base for animations.[7]
See for example :
standard view (Mandelbrot set without zoom = image) :
;Position file automatically generated by XaoS 3.5 ; - a real-time interactive fractal zoomer ;Use xaos -load <filename> to display it (initstate) (defaultpalette 0) (formula 'mandel) (view -0.75 0 2.5 2.5)
when we switch to Julia mode (disabled Mandelbrot mode) :
(initstate) (defaultpalette 0) (formula 'mandel) (julia #t) (view -0.75 0 2.5 2.5)
or change c value ( = julia seed)from c=0 to c= -1
(initstate) (defaultpalette 0) (formula 'mandel) (juliaseed -1 0) (julia #t) (view -0.8434 0.07535 2.744 2.744)
or show plane value ( default = 0 is not shown ) :
(initstate) (defaultpalette 0) (formula 'mandel) (juliaseed -1 0) (julia #t) (plane 0) (view -0.9127 -0.09268 2.285 2.285)
minibrot.xpf :
(initstate) (palette 1 860134713 0) (formula 'mandel) (outcoloring 1) (view -1.9854567 -1.351727E-05 0.00029196024 0.00029196024)
where view[8] is describing viewport ( rectangle part of the plane ):
view float float float float view centerRe centerIm radius angle
List of xpf files :
- list by by Chris Gray Some Fractals Computed by XaoS with xpf files by Chris Gray
view
editto describe plane ( view ) Xaos uses:
- 4 numbers in scripts
- 3 numbers in menu
Xaos uses to call it "radius" but it is defind as : the greater of (x2-x1= width) and y2-y1=height." This is the reason why in Xaos one can set the radius to 2/zoom.
To set the same viewpoint in XaoS using menu set 3 numbers to define circle
- the real portion of the center to (x1+x2)/2
- the imaginary part of center to (y1+y2)/2
- "the radius" = the greater of x2-x1 and y2-y1.
where the coordinates of the upper-left and lower-right visible points, specifying the coordinates as four numbers x1, y1, x2, y2.
In xpf file plane can be described by 4 floating point numbers :[9]
- real part of the center
- imaginary part of the center
- width in world coordinate = x2-x1 called "a real radius"
- height in world coordinte = y2-y1 called "an imaginary radius"
Syntax :
(view float float float float)
Example :
(view -0.75 0 2.5 2.5)
"People specify fractal coordinates in many ways. Some people use the coordinates of the upper-left and lower-right visible points, specifying the coordinates as four numbers x1, y1, x2, y2. To set the same viewpoint in XaoS, set the real portion of the center to (x1+x2)/2, the imaginary part of center to (y1+y2)/2, and the radius to the greater of x2-x1 and y2-y1. Other programs use a zoom factor instead of a radius. For these, you can set the radius to 2/zoom."[10]
" ... lets you specify an ellipse instead of a circle. You can specify both a real and an imaginary radius "
usrform
edit(initstate) (defaultpalette 0) (formula 'user) (usrform "Z^2+C") (usrformInit "") (view -0.4641 0.01977 2.444 2.444)
bailout
edit(bailout 6)
Bailout value or escape radius. Xaos uses here square of escape radius so if you want bailout value 2 put here 2*2 = 4.
formula
editSyntax:[11]
(formula keyword)
example
(formula mandel)
| Name | description | tutorial | User formula |
|---|---|---|---|
| mandel | Standard Mandelbrot or Julia sets. | mset.xaf | Z^2+C |
| mandel3 | Example | power.xaf | Z^3+C |
| mandel4 | Example | power.xaf | Z^4+C |
| mandel5 | Example | power.xaf | Z^5+C |
| mandel6 | Example | power.xaf | Z^6+C |
| newton | This is Newton’s approximation method for finding the roots of a polynomial. It uses the polynomial x^3=1 | newton.xaf | |
| barnsley | Example | Example | Example |
| octo | Example | Example | Example |
| phenix | Example | Example | Example |
| magnet | |||
| trice |
Changing named formule ( Menu/Fractal/Formule ) does not change user formule !!!
plane
editSyntax :
(plane integer)
Allowed values :[12]
| value | name | description |
|---|---|---|
| 0 | mu | Normal complex plane (default) |
| 1 | 1/mu | Inversion : old infinity is at 0 now and old 0 is at infinity now. |
| 2 | 1/(mu+0.25) | Similar to inversion, but also moves the center outside the Mandelbrot set, so it looks parabolic. |
| 3 | lambda | lambda plane |
| 4 | 1/lambda | Inversion of lambda plane |
| 5 | 1/(lambda-1) | Inversion with moved center |
| 6 | 1/(mu-1.40115) | Inversion with moved center |
// from xaos-3.5\src\engine\plane.c
/*
* XaoS, a fast portable realtime fractal zoomer
* Copyright (C) 1996,1997 by
*
* Jan Hubicka (hubicka@paru.cas.cz)
* Thomas Marsh (tmarsh@austin.ibm.com)
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
*/
CONST char *CONST planename[] = {
"mu",
"1/mu",
"1/(mu+0.25)",
"lambda",
"1/lambda",
"1/(lambda-1)",
"1/(mu-1.40115)",
NULL
};
// include/complex.h:57:
// #define myabs(x) ((x)>0?(x):-(x))
REGISTERS(3)
void recalculate(int plane, number_t * x1, number_t * y1)
{
number_t x = *x1, y = *y1;
switch (plane) {
case 1:
{ /* 1/mu */
number_t t;
if (myabs(x) + myabs(y) < 0.000001)
t = INT_MAX, y = INT_MAX;
else {
c_div(1, 0, x, y, t, y);
}
x = t;
}
break;
case 2:
{ /* 1/(mu + 0.25) */
number_t t;
if (myabs(x) + myabs(y) < 0.000001)
t = INT_MAX, y = INT_MAX;
else {
c_div(1, 0, x, y, t, y);
}
x = t;
x += 0.25;
}
break;
case 3: /* lambda */
{
number_t tr, ti, mr, mi;
mr = x, mi = y;
c_pow2(x, y, tr, ti);
c_div(tr, ti, 4, 0, x, y);
c_div(mr, mi, 2, 0, tr, ti);
c_sub(tr, ti, x, y, mr, mi);
x = mr, y = mi;
}
break;
case 4: /* 1/lambda */
{
number_t tr, ti, mr, mi;
c_div(1, 0, x, y, tr, y);
x = tr;
mr = x, mi = y;
c_pow2(x, y, tr, ti);
c_div(tr, ti, 4, 0, x, y);
c_div(mr, mi, 2, 0, tr, ti);
c_sub(tr, ti, x, y, mr, mi);
x = mr, y = mi;
}
break;
case 5: /* 1/(lambda-1) */
{
number_t tr, ti, mr, mi;
c_div(1, 0, x, y, tr, y);
x = tr + 1;
mr = x, mi = y;
c_pow2(x, y, tr, ti);
c_div(tr, ti, 4, 0, x, y);
c_div(mr, mi, 2, 0, tr, ti);
c_sub(tr, ti, x, y, mr, mi);
x = mr, y = mi;
}
break;
case 6:
{ /* 1/(mu + 0.25) */
number_t t;
if (myabs(x) + myabs(y) < 0.000001)
t = INT_MAX, y = INT_MAX;
else {
c_div(1, 0, x, y, t, y);
}
x = t;
x -= 1.40115;
}
break;
default:
break;
}
*x1 = x;
*y1 = y;
}
perturbation
editIt changes the point at which orbits start. Traditionally zero is used.[13]
(perturbation complex)
doc
editinfo xaosdev.info
How to render big images ?
editIt is not simple.[14] One can do it using :
- using animation thru menu ( "saving your fractal to a .xpf file (File->Save), then rendering it using the Misc->Render Animation function. "[15]
- using command line options[16]
XaoS -render fractal.xpf -size 18000x12000
How to change the resolution
edit"XaoS usually starts in low resolution (320x200 or so) to make calculations faster. In case you have fast computer or you need to save bigger .gif images you may change resolution. This should be done by pressing '=' in full screen drivers or simply by resizing XaoS window." [17]
Formula
edit- use xpf file
- use Main Menu/ Fractal/user formula
- Adding Built-in Formulas[18]
- edit the source code in src/engine/formulas.c
formulas.c
edit#define VARIABLES register number_t n,sqrr,sqri,zre1,zim1;
#define INIT sqri=zim*zim,n=zre,zre=pre,pre=n,n=zim,zim=pim,pim=n,n=(number_t)1;
#define BTEST greater_then_1Em6(n)
#define FORMULA \
zre1 = zre; \
zim1 = zim; \
n = zim * zim; \
sqri = zre * zre; \
sqrr = sqri - n; \
sqri = n + sqri; \
n = 0.3333333333 / ((sqri * sqri)); \
zim = (0.66666666) * zim - (zre + zre) * zim * n + pim; \
zre = (0.66666666) * zre + (sqrr) * n + pre; \
zre1 -= zre; \
zim1 -= zim; \
n = zre1 * zre1 + zim1 * zim1;
#define CALC newton_calc
#include "docalc.c"
After modifying the code you must recompile XaoS to take any effect.[19]
References
edit- ↑ XaoS man page
- ↑ Real-Time Zooming Math Engine by Zoltán Kovács
- ↑ xaosdev.info : algorithm
- ↑ CS and Dance (PDF), archived from the original on 2005-12-20
- ↑ a b Hubička, Jan (1997), XaoS Algorithms, archived from the original on 2014-03-28.
- ↑ Visualizations on the Complex Plane, archived from the original on 2006-10-17
- ↑ Xaos- format description
- ↑ Xaos - view
- ↑ Xaos - plane description
- ↑ Xaos - view
- ↑ Xaos doc - formula
- ↑ GNU XaoS » Documentation » XaoS » Plane
- ↑ Xaos doc - perturbation
- ↑ Can't (easily) render high-resolution static images #53
- ↑ xaos-users group : resolution
- ↑ xaos-users group : Higher resolution on Mac OSX
- ↑ Xaos tutorial
- ↑ Xaos doc : Adding Built-in Formulas by Arpad Fekete
- ↑ xaos-users group : Newton fractals