Fractals/mandelbrot-graphics

Parts Library and c programs for CPU-based visualisation of the Mandelbrot set by Claude Heiland-Allenmandelbrot-graphics - CPU-based visualisation of the Mandelbrot set by Claude Heiland-Allen mandelbrot-prelude library for Haskell (low resolution image in terminal with block graphics characters) Install Dependencies pkg-config math gmp mpfr mpc pari ghci cairo ( and pixman ) mandelbrot-numerics mandelbrot-symbolics openmp shared libraries ldd m-render linux-vdso.so.1 => (0x00007ffcae4e7000) libmandelbrot-graphics.so => /home/a/opt/lib/libmandelbrot-graphics.so (0x00007fb8f9a12000) libcairo.so.2 => /usr/lib/x86_64-linux-gnu/libcairo.so.2 (0x00007fb8f96df000) libmandelbrot-numerics.so => /home/a/opt/lib/libmandelbrot-numerics.so (0x00007fb8f94cf000) libpthread.so.0 => /lib/x86_64-linux-gnu/libpthread.so.0 (0x00007fb8f92b2000) libc.so.6 => /lib/x86_64-linux-gnu/libc.so.6 (0x00007fb8f8ee9000) libm.so.6 => /lib/x86_64-linux-gnu/libm.so.6 (0x00007fb8f8bdf000) libgomp.so.1 => /usr/lib/x86_64-linux-gnu/libgomp.so.1 (0x00007fb8f89bd000) libpixman-1.so.0 => /usr/lib/x86_64-linux-gnu/libpixman-1.so.0 (0x00007fb8f8715000) libfontconfig.so.1 => /usr/lib/x86_64-linux-gnu/libfontconfig.so.1 (0x00007fb8f84d1000) libfreetype.so.6 => /usr/lib/x86_64-linux-gnu/libfreetype.so.6 (0x00007fb8f8227000) libpng12.so.0 => /lib/x86_64-linux-gnu/libpng12.so.0 (0x00007fb8f8002000) libxcb-shm.so.0 => /usr/lib/x86_64-linux-gnu/libxcb-shm.so.0 (0x00007fb8f7dfd000) libxcb-render.so.0 => /usr/lib/x86_64-linux-gnu/libxcb-render.so.0 (0x00007fb8f7bf3000) libxcb.so.1 => /usr/lib/x86_64-linux-gnu/libxcb.so.1 (0x00007fb8f79d1000) libXrender.so.1 => /usr/lib/x86_64-linux-gnu/libXrender.so.1 (0x00007fb8f77c6000) libX11.so.6 => /usr/lib/x86_64-linux-gnu/libX11.so.6 (0x00007fb8f748c000) libXext.so.6 => /usr/lib/x86_64-linux-gnu/libXext.so.6 (0x00007fb8f727a000) libz.so.1 => /lib/x86_64-linux-gnu/libz.so.1 (0x00007fb8f705f000) librt.so.1 => /lib/x86_64-linux-gnu/librt.so.1 (0x00007fb8f6e57000) libmpc.so.3 => /usr/local/lib/libmpc.so.3 (0x00007fb8f6c3e000) libmpfr.so.4 => /usr/local/lib/libmpfr.so.4 (0x00007fb8f69db000) libgmp.so.10 => /usr/local/lib/libgmp.so.10 (0x00007fb8f6764000) /lib64/ld-linux-x86-64.so.2 (0x0000564eca780000) libdl.so.2 => /lib/x86_64-linux-gnu/libdl.so.2 (0x00007fb8f6560000) libexpat.so.1 => /lib/x86_64-linux-gnu/libexpat.so.1 (0x00007fb8f6336000) libXau.so.6 => /usr/lib/x86_64-linux-gnu/libXau.so.6 (0x00007fb8f6132000) libXdmcp.so.6 => /usr/lib/x86_64-linux-gnu/libXdmcp.so.6 (0x00007fb8f5f2b000) objdump -p m-render | grep NEEDED NEEDED libmandelbrot-graphics.so NEEDED libcairo.so.2 NEEDED libmandelbrot-numerics.so NEEDED libpthread.so.0 NEEDED libc.so.6 objdump -p m-stretching-cusps | grep NEEDED NEEDED libmandelbrot-graphics.so NEEDED libcairo.so.2 NEEDED libmandelbrot-numerics.so NEEDED libm.so.6 NEEDED libgmp.so.10 NEEDED libpthread.so.0 NEEDED libc.so.6 git git clone https://code.mathr.co.uk/mandelbrot-graphics.git and in the directory containing mandelbrot-graphics: make -C mandelbrot-graphics/c/lib prefix=${HOME}/opt install make -C mandelbrot-graphics/c/bin prefix=${HOME}/opt install hen to run do: export LD_LIBRARY_PATH=${HOME}/opt/lib check : echo $LD_LIBRARY_PATH result : /home/a/opt/lib or export PATH=${HOME}/opt/bin:${PATH} check : echo $PATH To set it permanently change file : .profilestackoverflow question how-to-permanently-set-path-on-linux /etc/ld.so.conf.d/*.confubuntu environment Variables update git From console opened in the mandelbrot-graphics directory : git pull If you made some local changes you can undu them : git checkout -f then git pull Now install again recompile new version bash script : #!/bin/bash cd ~ make -C mandelbrot-graphics/c/lib prefix=${HOME}/opt install make -C mandelbrot-graphics/c/bin prefix=${HOME}/opt install export LD_LIBRARY_PATH=${HOME}/opt/lib export PATH=${HOME}/opt/bin:${PATH} cd /home/a/mandelbrot-graphics/c/bin names m prefix m is from Mandelbrot r/d prefix r or d in name describes precision d = double precision r = arbitrary precision examples: m_d_attractor(double _Complex *z_out, double _Complex z_guess, double _Complex c, int period, int maxsteps) m_r_attractor(mpc_t z_out, const mpc_t z_guess, const mpc_t c, int period, int maxsteps) How to use it ? prelude Haskell program: let c = nucleus 100 . (!! (8 * 2 * 100)) . exRayIn 8 . fromQ . fst . addressAngles . pAddress $ "1 7/12 5/9 100" ; r = 2 * magnitude (size 100 c) in putImage c r 10000 It gives : low resolution image in terminal with block graphics characters center, size and iteration number -0.5664388911664133 + -0.4792791697756855 i @ 2.810e-8 (10000 iterations) procedures in lib directory C source should *only* have #include <mandelbrot-numerics.h> compile and link with pkg-config: see mandelbrot-numerics/c/bin/Makefile for an example quickest way to get started is to just put your file in mandelbrot-numerics/c/bin and run make m_d_transform_rectangular m_d_transform *rect = m_d_transform_rectangular(w, h, c, r); // where : w = width in pixels h = height in pixels c = center of the image ( complex number ) r = radius of the image ( double number m_d_interior find points c of the Mandelbrot set, given a particular hyperbolic component and the desired internal angle. It involves Newton's method in two complex variables to solveInterior coordinates in the Mandelbrot set by Claude Heiland-Allen where p is the period of the target component the desired internal angle r is internal radius . When r = 1.0 point is on the boundary. When r = 0 point is in the center of component ( = nucleus) is a multiplier of point c The hyperbolic component is described by period nucleus Syntax extern m_newton m_d_interior(double _Complex *z_out, double _Complex *c_out, double _Complex z_guess, double _Complex c_guess, double _Complex interior, int period, int maxsteps) Input: z_guess c_guess ( usually nucleus of choosen hyperbolic component) interior ( multiplier) period maxstep Output: c is the coordinates of the point ( c_out) z is periodic point ( z_out) result (m_newton) describes how Newton algorithm has ended : m_failed, m_stepped, m_converged. It is deined in ~/mandelbrot-numerics/c/include/mandelbrot-numerics.h Examples of use: m_d_interior(&z, &half, nucleus, nucleus, -1, period, 64); m_d_interior(&z, &cusp, nucleus, nucleus, 1, period, 64); m_d_interior(&z, &third2, -1, -1, cexp(I * twopi / 3), 2, 64); programs in bin directory List : ~/mandelbrot-graphics/c/bin$ ls -1a *.c result : m-cardioid-warping.c m-render.c m-subwake-diagram-b.c m-dense-misiurewicz.c m-stretching-cusps.c m-subwake-diagram-c.c m-feigenbaum-zoom.c m-subwake-diagram-a.c m-warped-midgets ./m-warped-midgets Result: 4 -1.565201668337550256e-01 + 1.032247108922831780e+00 i @ 1.697e-02 8 4.048996651751222142e-01 + 1.458203637665893004e-01 i @ 2.743e-03 16 2.925037532341934199e-01 + 1.492506899834379792e-02 i @ 3.484e-04 32 2.602618199285007261e-01 + 1.667791320926505921e-03 i @ 4.113e-05 64 2.524934589775105209e-01 + 1.971526796077277045e-04 i @ 4.920e-06 128 2.506132008410751344e-01 + 2.396932642510365294e-05 i @ 5.997e-07 256 2.501519680089798192e-01 + 2.954962325906873815e-06 i @ 7.398e-08 512 2.500378219137852631e-01 + 3.668242052764783887e-07 i @ 9.185e-09 1024 2.500094340031833728e-01 + 4.569478652064606379e-08 i @ 1.144e-09 2048 2.500023558032561377e-01 + 5.701985912706822671e-09 i @ 1.428e-10 4096 2.500005886128087162e-01 + 7.121326948562671441e-10 i @ 1.783e-11 8192 2.500001471109009610e-01 + 8.897814201389663379e-11 i @ 2.228e-12 Periodicity scan Periodicity scanperiodicity scan by Claude Heiland-Allen : labelling a picture of parameter plane with the periods of the Mandelbrot set components can provide insights into its deeper structure. Plik : m-period.scan.c Run console program ./m-period-scan usage: ./m-period-scan out.png width height creal cimag radius maxiters mingridsize minfontsize maxfontsize maxatoms periodmod periodneq Example ./m-period-scan out1.png 1500 1000 0.0 0.0 1.5 10000 100 0.1 30.0 100 3 1 Moebius Moebius transformation Code video description ./moebius find point c of component with period = 2 multiplier = -0.4999999999999998+0.8660254037844387 located near c= -1.0000000000000000+0.0000000000000000 find point c of component with period = 4 multiplier = -1.0000000000000000+0.0000000000000000 located near c= -1.3107026413368328+0.0000000000000000 find point c of component with period = 4 multiplier = -0.4999999999999998+0.8660254037844387 located near c= -1.3107026413368328+0.0000000000000000 find point c of component with period = 8 multiplier = -1.0000000000000000+0.0000000000000000 located near c= -1.3815474844320617+0.0000000000000000 find point c of component with period = 8 multiplier = -0.4999999999999998+0.8660254037844387 located near c= -1.3815474844320617+0.0000000000000000 find point c of component with period = 2 multiplier = -0.5000000000000004-0.8660254037844384 located near c= -1.0000000000000000+0.0000000000000000 find point c of component with period = 2 multiplier = -0.8090169943749476-0.5877852522924730 located near c= -1.0000000000000000+0.0000000000000000 find point c of component with period = 2 multiplier = -0.7071067811865477-0.7071067811865475 located near c= -1.0000000000000000+0.0000000000000000 find point c of component with period = 2 multiplier = -0.6548607339452852-0.7557495743542582 located near c= -1.0000000000000000+0.0000000000000000 find point c of component with period = 3 multiplier = -1.0000000000000000+0.0000000000000000 located near c= -1.7548776662466927+0.0000000000000000 find point c of component with period = 3 multiplier = 1.0000000000000000+0.0000000000000000 located near c= -1.7548776662466927+0.0000000000000000 find point c of component with period = 6 multiplier = -1.0000000000000000+0.0000000000000000 located near c= -1.7728929033816239+0.0000000000000000 find point c of component with period = 6 multiplier = -0.4999999999999998+0.8660254037844387 located near c= -1.7728929033816239+0.0000000000000000 find point c of component with period = 12 multiplier = -1.0000000000000000+0.0000000000000000 located near c= -1.7782668211110817+0.0000000000000000 find point c of component with period = 12 multiplier = -0.4999999999999998+0.8660254037844387 located near c= -1.7782668211110817+0.0000000000000000 m-furcation-rainbow For non-real C you can plot all the limit-cycle Z on one image, chances of overlap are small. You can colour according to the position along the path. In attached I have coloured using hue red at roots, going through yellow towards the next bond point in a straight line through the interior coordinate space (interior coordinate is derivative of limit cycle). I have just plotted points, so there are gaps. Perhaps it could be improved by drawing line segments between Z values, but I'm not 100% sure if the first Z value found will always correspond to the same logical line, and keeping track of a changing number of "previous Z" values isn't too fun either. Claudefractalforums.org : tri-furcation-and-more Run: /m-furcation-rainbow 13.png "1/3" "1/3" "1/3" m-dense-misiurewicz thumb|Zoom around principal Misiurewicz point for periods from 2 to 1024 Program is based on m-render.c from mandelbrot-graphics. It draws series of png images m-island-zoom m-island-zoom Makes 150 png images showing zoom to the 3 island ( biggest islands of the wake) one on the main antenna ( period 3) with center c = -1.754877666246693 +0.000000000000000 i in the 1/2 wake period 4 with center c = -0.156520166833755 +1.032247108922832 i , in 1/3 wake period 5 with addres 1-> 2-(1/3)-> 6 and center c = -1.256367930068181 +0.380320963472722 i cardioid warping thumb|right|Conformal Warping Around The Cardioid In The Mandelbrot Set The exterior of the cardioid in the Mandelbrot set is warped to give the appearance of rotation. The transformation is built up from smaller components, including: mapping of the cardioid to a circle Moebius transform of the circle to a straight line linear translation (which is animated) the inverses of the linear translation the inverse of Moebius transform of the circle to a straight line These transformations and their derivatives (for distance estimator colouring) are described here: https://mathr.co.uk/blog/2013-12-16_stretching_cusps.html The program to render the animation was implemented in C using the mandelbrot-graphics library found here: https://code.mathr.co.uk/mandelbrot-graphics The program is found in the repository as c/bin/m-cardioid/warping.c https://code.mathr.co.uk/mandelbrot-graphics/blob/60adc5ab8f14aab1be479469dfcf5ad3469feea0:/c/bin/m-cardioid-warping.c What it the relation between x and internal angle ? Hairness thumb|right|Zoom About Feigenbaum Point In The Mandelbrot Set Showing Hairiness m-stretching-cusps thumb|right|Unrolled main cardioid of Mandelbrot set One can add usage description : if (! (argc == 7)) { printf("no input \n"); printf("example usage : \n"); printf("%s re(nucleus) im(nucleus) period t_zero t_one t_infinity \n", argv[0] ); printf("%s 0 0 1 1/2 1/3 0 \n", argv[0] ); return 1; } example usage : m-stretching-cusps 0 0 1 1/2 1/3 0 Input parent component re(nucleus) im(nucleus) period internal angles of 3 child components: t0 t1 tinfinity Test result: P0 = -7.5000000000000000e-01 1.2246467991473532e-16 P1 = -1.2499999999999981e-01 6.4951905283832900e-01 Pinf = 2.5000000000000000e-01 0.0000000000000000e+00 and image out.png duble r = 0.5; // proportional to the number of components on the strip, /* r = 0.5 gives 4 prominent components counted from period 1 to one side only r = 1.0 gives 10 components r = 1.5 gives 15 r = 2.0 gives 20 ( one can see 2 sides of cardioid ?? because it is near cusp) r = 2.5 gives 26 r = 5.0 gives 50 It uses: determinants (m_d_mat2 from mandelbrot-numerics library)) for computing the coefficients a,b,c,d of the Moebius transformationExplicit determinant formula for Moebius transformation from wikipedia m_d_transform_moebius3 function for Moebius transformation defined by 3 points m-stretching-cusps 0 0 1 1/2 1/3 0 parent component with period = 1 and nucleus = 0.0000000000000000e+00 0.0000000000000000e+00 child component with with internal angle tzero = 1/2 and nucleus c = zero = -7.5000000000000000e-01 1.2246467991473532e-16 child component with with internal angle tone = 1/3 and nucleus c = one = -1.2499999999999981e-01 6.4951905283832900e-01 child component with with internal angle tinfinity = 0 and nucleus c = infinity = 2.5000000000000000e-01 0.0000000000000000e+00 Moebius coefficients a = -0.5000000000000002 ; -0.8660254037844387 b = 1.4999999999999998 ; -0.8660254037844390 c = 0.5000000000000002 ; 0.8660254037844387 d = 1.4999999999999998 ; -0.8660254037844388 image 1_0.500000.png saved filename = period_r m-misiurewicz-basins m-misiurewicz-basins usage: m-misiurewicz-basins out.png width height creal cimag radius maxiters preperiod period m-render It is a base program for others. This fragment of code describes how to use it : int main(int argc, char **argv) { if (argc != 8) { fprintf(stderr, "usage: %s out.png width height creal cimag radius maxiters\n", argv[0]); return 1; } Examples m-render a.png 1000 1000 -0.75 0 1.5 10000 The result is Mandelbrot set boundary using DEM thumb|right|Baoundary of Mandelbrot set m-render 1995.png 7680 4320 -0.5664388911664133 -0.4792791697756855 3e-8 10000 1 m-streching-feigenbaum.c m-subwake-diagram-a thumb|right|m-subwake-diagram-a m-subwake-diagram-b thumb|right|m-subwake-diagram-b m-subwake-diagram-c thumb|right|m-subwake-diagram-c See also Dictionary References