分形/book
< 分形
這裡書的意思是
- 未命名（舊）c 程式[1] 由 Claude Heiland-Allen[2]
- 一個 GUI 程式
- 一組控制檯程式
- 關於“如何寫一本關於曼德勃羅集合的書”的 pdf 格式書籍。
- 官方版本
- 這裡有一個 pdf 檔案的非官方版本 ( 68.3 MB )
- 使用流的書籍控制檯程式
- 書籍 GUI 程式現在基本被放棄了。
- 它被移植到 m-perturbator 在 mandelbrot-perturbator 儲存庫
- 另請參閱庫
- sh 檔案（主程式
- bin 目錄中的 c 檔案
- lib 目錄中的 c 和 h 檔案
半徑 定義為“中心和軸對齊檢視矩形的頂部之間的虛數座標差”。半徑沒有初始值。
// / code/bin/mandelbrot_render.c
int retval = 1;
int width = 1280;
int height = 720;
int maximum_iterations = 4096;
long double escape_radius = 512;
mpfr_t radius;
mpfr_init2(radius, 53);
帶有完整 src 程式碼的影像
-
曼德勃羅集合的逃逸時間（= 外部）的水平集 -
連續逃逸時間 -
原子域演算法
所有演算法都在書籍中描述：Claude Heiland-Allen 編著的“如何寫一本關於曼德勃羅集合的書”[6]
請參閱 book/code/gui/main.c 中的 mandelbrot_nucleus 函式。[7] 它使用 牛頓法.
與 mightymandel/src/atom.c 中的 muatom() 函式進行比較[8]
使用牛頓法、mpfr 和 gmp 庫計算外部射線向內，基於庫的程式碼。外部角度從字串中讀取（二進位制分數）
/*
code is from :
https://code.mathr.co.uk/mandelbrot-book
mandelbrot-book/code/examples/ray-in.sh
mandelbrot-book/code/examples/ray-in.c
mandelbrot-book/code/bin/mandelbrot_external_ray_in.c
mandelbrot-book/code/lib/mandelbrot_external_ray_in.c
mandelbrot-book/code/lib/mandelbrot_external_ray_in.h
mandelbrot-book/code/lib/mandelbrot_external_ray_in_native.c
mandelbrot-book/code/lib/mandelbrot_external_ray_in_native.h
mandelbrot-book/code/lib/mandelbrot_binary_angle.c
mandelbrot-book/code/lib/mandelbrot_binary_angle.h
mandelbrot-book/code/lib/pi.c
mandelbrot-book/code/lib/pi.h
--------------------------------------------------------
gcc r.c -std=c99 -Wall -Wextra -pedantic -lm -lgmp -lmpfr
./a.out
*/
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include <complex.h>
#include <gmp.h>
#include <mpfr.h>
#include <stdbool.h>
const float pif = 3.14159265358979323846264338327950288419716939937510f;
const double pi = 3.14159265358979323846264338327950288419716939937510;
const long double pil = 3.14159265358979323846264338327950288419716939937510l;
// ------------------------------------------------------------
struct mandelbrot_binary_angle {
int preperiod;
int period;
char *pre;
char *per;
};
struct mandelbrot_external_ray_in {
mpq_t angle;
mpq_t one;
unsigned int sharpness; // number of steps to take within each dwell band
unsigned int precision; // delta needs this many bits of effective precision
unsigned int accuracy; // epsilon is scaled relative to magnitude of delta
double escape_radius;
mpfr_t epsilon2;
mpfr_t cx;
mpfr_t cy;
unsigned int j;
unsigned int k;
// temporaries
mpfr_t zx, zy, dzx, dzy, ux, uy, vx, vy, ccx, ccy, cccx, cccy;
};
// ----------------------------------------------------------------------------
int depth = 0;
mpq_t angle;
int base = 2; // The base can vary from 2 to 62, or if base is 0 then the leading characters are used: 0x or 0X for hex, 0b or 0B for binary, 0 for octal, or decimal otherwise.
char *s;
struct mandelbrot_binary_angle *a ;
mpfr_t cre, cim;
struct mandelbrot_external_ray_in *ray; // = mandelbrot_external_ray_in_new(angle);
// ----------------------------------------------------------------------------
void mandelbrot_binary_angle_delete(struct mandelbrot_binary_angle *a) {
free(a->pre);
free(a->per);
free(a);
}
// FIXME check canonical form, ie no .01(01) etc
struct mandelbrot_binary_angle *mandelbrot_binary_angle_from_string(const char *s) {
const char *dot = strchr(s, '.');
if (! dot) { return 0; }
const char *bra = strchr(dot, '(');
if (! bra) { return 0; }
const char *ket = strchr(bra, ')');
if (! ket) { return 0; }
if (s[0] != '.') { return 0; }
if (ket[1] != 0) { return 0; }
struct mandelbrot_binary_angle *a = malloc(sizeof(struct mandelbrot_binary_angle));
a->preperiod = bra - dot - 1;
a->period = ket - bra - 1;
fprintf( stderr , "external angle \n");
fprintf( stderr , "\thas preperiod = %d \t period = %d\n", a->preperiod, a->period);
if (a->period < 1) {
free(a);
return 0;
}
a->pre = malloc(a->preperiod + 1);
a->per = malloc(a->period + 1);
memcpy(a->pre, dot + 1, a->preperiod);
memcpy(a->per, bra + 1, a->period);
a->pre[a->preperiod] = 0;
a->per[a->period] = 0;
for (int i = 0; i < a->preperiod; ++i) {
if (a->pre[i] != '0' && a->pre[i] != '1') {
mandelbrot_binary_angle_delete(a);
return 0;
}
}
for (int i = 0; i < a->period; ++i) {
if (a->per[i] != '0' && a->per[i] != '1') {
mandelbrot_binary_angle_delete(a);
return 0;
}
}
return a;
}
void mandelbrot_binary_angle_to_rational(mpq_t r, const struct mandelbrot_binary_angle *t) {
mpq_t pre, per;
mpq_init(pre);
mpq_init(per);
mpz_set_str(mpq_numref(pre), t->pre, 2);
mpz_set_si(mpq_denref(pre), 1);
mpz_mul_2exp(mpq_denref(pre), mpq_denref(pre), t->preperiod);
mpq_canonicalize(pre);
mpz_set_str(mpq_numref(per), t->per, 2);
mpz_set_si(mpq_denref(per), 1);
mpz_mul_2exp(mpq_denref(per), mpq_denref(per), t->period);
mpz_sub_ui(mpq_denref(per), mpq_denref(per), 1);
mpz_mul_2exp(mpq_denref(per), mpq_denref(per), t->preperiod);
mpq_canonicalize(per);
mpq_add(r, pre, per);
mpq_clear(pre);
mpq_clear(per);
}
// -------------------------------------------------------------
struct mandelbrot_external_ray_in *mandelbrot_external_ray_in_new(mpq_t angle) {
struct mandelbrot_external_ray_in *r = malloc(sizeof(struct mandelbrot_external_ray_in));
mpq_init(r->angle);
mpq_set(r->angle, angle);
mpq_init(r->one);
mpq_set_ui(r->one, 1, 1);
r->sharpness = 4;
r->precision = 4;
r->accuracy = 4;
r->escape_radius = 65536.0;
mpfr_init2(r->epsilon2, 53);
mpfr_set_ui(r->epsilon2, 1, GMP_RNDN);
double a = 2.0 * pi * mpq_get_d(r->angle);
mpfr_init2(r->cx, 53);
mpfr_init2(r->cy, 53);
mpfr_set_d(r->cx, r->escape_radius * cos(a), GMP_RNDN);
mpfr_set_d(r->cy, r->escape_radius * sin(a), GMP_RNDN);
r->k = 0;
r->j = 0;
// initialize temporaries
mpfr_inits2(53, r->zx, r->zy, r->dzx, r->dzy, r->ux, r->uy, r->vx, r->vy, r->ccx, r->ccy, r->cccx, r->cccy, (mpfr_ptr) 0);
return r;
}
int mandelbrot_external_ray_in_step(struct mandelbrot_external_ray_in *r) {
if (r->j >= r->sharpness) {
mpq_mul_2exp(r->angle, r->angle, 1);
if (mpq_cmp_ui(r->angle, 1, 1) >= 0) {
mpq_sub(r->angle, r->angle, r->one);
}
r->k++;
r->j = 0;
}
// r0 <- er ** ((1/2) ** ((j + 0.5)/sharpness))
// t0 <- r0 cis(2 * pi * angle)
double r0 = pow(r->escape_radius, pow(0.5, (r->j + 0.5) / r->sharpness));
double a0 = 2.0 * pi * mpq_get_d(r->angle);
double t0x = r0 * cos(a0);
double t0y = r0 * sin(a0);
// c <- r->c
mpfr_set(r->ccx, r->cx, GMP_RNDN);
mpfr_set(r->ccy, r->cy, GMP_RNDN);
for (unsigned int i = 0; i < 64; ++i) { // FIXME arbitrary limit
// z <- 0
// dz <- 0
mpfr_set_ui(r->zx, 0, GMP_RNDN);
mpfr_set_ui(r->zy, 0, GMP_RNDN);
mpfr_set_ui(r->dzx, 0, GMP_RNDN);
mpfr_set_ui(r->dzy, 0, GMP_RNDN);
// iterate
for (unsigned int p = 0; p <= r->k; ++p) {
// dz <- 2 z dz + 1
mpfr_mul(r->ux, r->zx, r->dzx, GMP_RNDN);
mpfr_mul(r->uy, r->zy, r->dzy, GMP_RNDN);
mpfr_mul(r->vx, r->zx, r->dzy, GMP_RNDN);
mpfr_mul(r->vy, r->zy, r->dzx, GMP_RNDN);
mpfr_sub(r->dzx, r->ux, r->uy, GMP_RNDN);
mpfr_add(r->dzy, r->vx, r->vy, GMP_RNDN);
mpfr_mul_2ui(r->dzx, r->dzx, 1, GMP_RNDN);
mpfr_mul_2ui(r->dzy, r->dzy, 1, GMP_RNDN);
mpfr_add_ui(r->dzx, r->dzx, 1, GMP_RNDN);
// z <- z^2 + c
mpfr_sqr(r->ux, r->zx, GMP_RNDN);
mpfr_sqr(r->uy, r->zy, GMP_RNDN);
mpfr_sub(r->vx, r->ux, r->uy, GMP_RNDN);
mpfr_mul(r->vy, r->zx, r->zy, GMP_RNDN);
mpfr_mul_2ui(r->vy, r->vy, 1, GMP_RNDN);
mpfr_add(r->zx, r->vx, r->ccx, GMP_RNDN);
mpfr_add(r->zy, r->vy, r->ccy, GMP_RNDN);
}
// c' <- c - (z - t0) / dz
mpfr_sqr(r->ux, r->dzx, GMP_RNDN);
mpfr_sqr(r->uy, r->dzy, GMP_RNDN);
mpfr_add(r->vy, r->ux, r->uy, GMP_RNDN);
mpfr_sub_d(r->zx, r->zx, t0x, GMP_RNDN);
mpfr_sub_d(r->zy, r->zy, t0y, GMP_RNDN);
mpfr_mul(r->ux, r->zx, r->dzx, GMP_RNDN);
mpfr_mul(r->uy, r->zy, r->dzy, GMP_RNDN);
mpfr_add(r->vx, r->ux, r->uy, GMP_RNDN);
mpfr_div(r->ux, r->vx, r->vy, GMP_RNDN);
mpfr_sub(r->cccx, r->ccx, r->ux, GMP_RNDN);
mpfr_mul(r->ux, r->zy, r->dzx, GMP_RNDN);
mpfr_mul(r->uy, r->zx, r->dzy, GMP_RNDN);
mpfr_sub(r->vx, r->ux, r->uy, GMP_RNDN);
mpfr_div(r->uy, r->vx, r->vy, GMP_RNDN);
mpfr_sub(r->cccy, r->ccy, r->uy, GMP_RNDN);
// delta^2 = |c' - c|^2
mpfr_sub(r->ux, r->cccx, r->ccx, GMP_RNDN);
mpfr_sub(r->uy, r->cccy, r->ccy, GMP_RNDN);
mpfr_sqr(r->vx, r->ux, GMP_RNDN);
mpfr_sqr(r->vy, r->uy, GMP_RNDN);
mpfr_add(r->ux, r->vx, r->vy, GMP_RNDN);
// enough_bits = 0 < 2 * prec(eps^2) + exponent eps^2 - 2 * precision
int enough_bits = 0 < 2 * (mpfr_get_prec(r->epsilon2) - r->precision) + mpfr_get_exp(r->epsilon2);
if (enough_bits) {
// converged = delta^2 < eps^2
int converged = mpfr_less_p(r->ux, r->epsilon2);
if (converged) {
// eps^2 <- |c' - r->c|^2 >> (2 * accuracy)
mpfr_sub(r->ux, r->cccx, r->cx, GMP_RNDN);
mpfr_sub(r->uy, r->cccy, r->cy, GMP_RNDN);
mpfr_sqr(r->vx, r->ux, GMP_RNDN);
mpfr_sqr(r->vy, r->uy, GMP_RNDN);
mpfr_add(r->ux, r->vx, r->vy, GMP_RNDN);
mpfr_div_2ui(r->epsilon2, r->ux, 2 * r->accuracy, GMP_RNDN);
// j <- j + 1
r->j = r->j + 1;
// r->c <- c'
mpfr_set(r->cx, r->cccx, GMP_RNDN);
mpfr_set(r->cy, r->cccy, GMP_RNDN);
return 1;
}
} else {
// bump precision
mpfr_prec_t prec = mpfr_get_prec(r->cx) + 32;
mpfr_prec_round(r->cx, prec, GMP_RNDN);
mpfr_prec_round(r->cy, prec, GMP_RNDN);
mpfr_prec_round(r->epsilon2, prec, GMP_RNDN);
mpfr_set_prec(r->ccx, prec);
mpfr_set_prec(r->ccy, prec);
mpfr_set_prec(r->cccx, prec);
mpfr_set_prec(r->cccy, prec);
mpfr_set_prec(r->zx, prec);
mpfr_set_prec(r->zy, prec);
mpfr_set_prec(r->dzx, prec);
mpfr_set_prec(r->dzy, prec);
mpfr_set_prec(r->ux, prec);
mpfr_set_prec(r->uy, prec);
mpfr_set_prec(r->vx, prec);
mpfr_set_prec(r->vy, prec);
i = 0;
// c <- r->c
mpfr_set(r->cccx, r->cx, GMP_RNDN);
mpfr_set(r->cccy, r->cy, GMP_RNDN);
}
mpfr_set(r->ccx, r->cccx, GMP_RNDN);
mpfr_set(r->ccy, r->cccy, GMP_RNDN);
}
return 0;
}
void mandelbrot_external_ray_in_get(struct mandelbrot_external_ray_in *r, mpfr_t x, mpfr_t y) {
mpfr_set_prec(x, mpfr_get_prec(r->cx));
mpfr_set(x, r->cx, GMP_RNDN);
mpfr_set_prec(y, mpfr_get_prec(r->cy));
mpfr_set(y, r->cy, GMP_RNDN);
}
void mandelbrot_external_ray_in_delete(struct mandelbrot_external_ray_in *r) {
mpfr_clear(r->epsilon2);
mpfr_clear(r->cx);
mpfr_clear(r->cy);
mpq_clear(r->angle);
mpq_clear(r->one);
mpfr_clears(r->zx, r->zy, r->dzx, r->dzy, r->ux, r->uy, r->vx, r->vy, r->ccx, r->ccy, r->cccx, r->cccy, (mpfr_ptr) 0);
free(r);
}
// -------------------------------------------------------------------------------------------------------
void PrintCInfo (void)
{
fprintf(stderr,"gcc version: %d.%d.%d\n", __GNUC__, __GNUC_MINOR__, __GNUC_PATCHLEVEL__); // https://stackoverflow.com/questions/20389193/how-do-i-check-my-gcc-c-compiler-version-for-my-eclipse
// OpenMP version is displayed in the console : export OMP_DISPLAY_ENV="TRUE"
fprintf(stderr,"__STDC__ = %d\n", __STDC__);
fprintf(stderr,"__STDC_VERSION__ = %ld\n", __STDC_VERSION__);
fprintf(stderr,"c dialect = ");
switch (__STDC_VERSION__)
{ // the format YYYYMM
case 199409L:
fprintf(stderr,"C94\n");
break;
case 199901L:
fprintf(stderr,"C99\n");
break;
case 201112L:
fprintf(stderr,"C11\n");
break;
case 201710L:
fprintf(stderr,"C18\n");
break;
//default : /* Optional */
}
//gmp_printf (" GMP-%s \n ", gmp_version );
mpfr_printf("Versions: MPFR-%s \n GMP-%s \n", mpfr_version, gmp_version );
}
void PrintInfo(void){
//
fprintf(stderr, "console C program ( CLI ) for computing external ray of Mandelbrot set (parametric ray)\n");
fprintf(stderr, "using Newton method described by Tomoki Kawahira \n");
fprintf(stderr, "tracing inward ( from infinity toward Mandelbrot set) = ray-in\n");
fprintf(stderr, "arbitrary precision (gmp, mpfr) with dynamic precision adjustment.\n Based on the code by Claude Heiland-Allen: https://mathr.co.uk/web/\n");
fprintf(stderr, "usage: ./a.out angle depth\n outputs space-separated complex numbers on stdout\n example command \n./a.out 1/3 25\n or\n ./a.out .(01) 25\n");
fprintf( stderr , "\n");
fprintf( stderr , "\tsharpness = %d\n", ray->sharpness);
fprintf( stderr , "\tprecision = %d\n", ray->precision);
fprintf( stderr , "\taccuracy = %d\n", ray->accuracy);
fprintf( stderr , "\tescape_radius = %.0f\n", ray->escape_radius);
//
PrintCInfo();
}
int setup(void){
// 2 parameters of external ray : depth, angle
depth = 35;
// external angle
s = ".(01)"; // landing point c = -0.750000000000000 +0.000000000000000 i period = 10000
// mpq
mpq_init(angle);
// mpq init from string or 2 integers
a = mandelbrot_binary_angle_from_string(s); // set a from string
// gmp_fprintf (stderr, "\tas a binary fraction = %0b\n", a); // 0b or 0B for binary // not works
if (a) {
mandelbrot_binary_angle_to_rational(angle, a); // convert binary string to decimal rational number
mandelbrot_binary_angle_delete(a); } // use only rational form so delete string form
mpq_canonicalize (angle); // It is the responsibility of the user to canonicalize the assigned variable before any arithmetic operations are performed on that variable.
gmp_fprintf (stderr, "\tas a decimal rational number = %Qd\n", angle); //
fprintf( stderr , "\tas a formated binary string = 0%s\n", s);
// ---------------------------------------------------
mpfr_init2(cre, 53);
mpfr_init2(cim, 53);
return 0;
}
int compute_ray(mpq_t angle){
ray = mandelbrot_external_ray_in_new(angle);
if (ray ==NULL ){ fprintf(stderr, " ray error \n"); return 1;}
for (int i = 0; i < depth * 4; ++i) { // FIXME 4 = implementation's sharpness
//fprintf( stderr , "i = %d\n", i);
mandelbrot_external_ray_in_step(ray); //fprintf(stderr, " ray step ok \n");
mandelbrot_external_ray_in_get(ray, cre, cim); //fprintf(stderr, " ray get ok \n");
mpfr_out_str(stdout, 10, 0, cre, GMP_RNDN);
putchar(' ');
mpfr_out_str(stdout, 10, 0, cim, GMP_RNDN);
putchar('\n');
}
return 0;
}
int end(void){
PrintInfo();
fprintf(stderr, " allways free memory (deallocate ) to avoid memory leaks \n"); // wikipedia C_dynamic_memory_allocation
mpq_clear (angle);
mpfr_clear(cre);
mpfr_clear(cim);
mandelbrot_external_ray_in_delete(ray);
return 0;
}
int main(void){
setup();
compute_ray(angle);
end();
return 0;
}
- OpenGl
- GL/glew.h ( libglew-dev )
- Gtk ( GUI
sudo apt install mesa-utils sudo apt install libglew-dev sudo apt install freeglut3-dev sudo apt-get install libgtk-3-dev
首先克隆官方 git 倉庫
git clone https://code.mathr.co.uk/mandelbrot-book
或舊的（已棄用）
git clone http://code.mathr.co.uk/book.git
或者你可以使用非官方倉庫
git clone https://gitlab.com/adammajewski/my-book
go to the program directory and install :
cd book # or my_book cd code make -C lib make -C bin make cd gui make cd ..
首先
make clean
然後 再次製作
首先
sudo apt-get install texlive-latex-recommended texlive-latex-extra texlive-fonts-recommended texlive-fonts-extra sudo apt-get install texlive-science
然後製作 pdf 格式的書籍
- 轉到 book/book 目錄
- 構建
cd book make
或使用包含完整 c 程式碼的非官方版本[9]
https://code.mathr.co.uk/mandelbrot-book-images/
從 mandelbrot-numerics 目錄中開啟的控制檯中
git pull
如果你做了一些本地更改，你可以撤消它們
git checkout -f
然後
git pull
現在再次安裝
書籍程式的最後版本有兩種型別
- 控制檯
- gui
要執行 gui 程式
# cd gui ./mandelbrot_gui
要執行控制檯程式，請使用 bash 指令碼，參見示例
./examples/standardview.sh
- 控制檯程式是 bash sh 檔案，使用管道（流）
- 對於 GUI 版本的程式（/gui/mandelbrot_gui，參見 Makefile）從文字檔案讀取引數
./mandelbrot_render float_type cre cim plane_radius escape_radius "string_file_name" image_width image_heght maximum_iterations interior ./mandelbrot_render 3 -5.7941214880704257e-02 8.1326829943642642e-01 8.1000000737099996e-09 10000 "g" 8000 8000
#!/bin/bash
# Radius is defined as "the difference in imaginary coordinate between the center and the top of the axis-aligned view rectangle".
# https://wikibook.tw/wiki/Fractals/Computer_graphic_techniques/2D/plane
#
# in examples directory : chmod +x standardview.sh
# then go to the parent directory : cd ..
# run run these example from the parent directory :
# ./examples/standardview.sh
#
# result :
# ./render using double
# rgba 1.000000 1.000000 1.000000 1.000000
# Image standard.png is saved
# info text file .txt is saved
view="-0.75 0 1.5" # "standard" view of parameter plane : center_re, center_im, radius
#
filename="standard"
pngfilename=$filename".png"
# Heredoc
./render $view && ./colour > out.ppm && ./annotate out.ppm $pngfilename <<EOF
rgba 1 1 1 1
EOF
echo "Image " $pngfilename " is saved"
# info text file
echo "view = re(center) im(center) radius = " $view > $filename".txt"
echo "info text file " $filname".txt is saved"
擴充套件使用
# Radius is defined as "the difference in imaginary coordinate between the center and the top of the axis-aligned view rectangle".
# https://wikibook.tw/wiki/Fractals/Computer_graphic_techniques/2D/plane
#
# in examples directory : chmod +x standardview2.sh
# then go to the parent directory : cd ..
# run run these example from the parent directory :
# ./examples/ian.sh
#
# result :
# ./render using double
# rgba 1.000000 1.000000 1.000000 1.000000
# Image standard.png is saved
# info text file .txt is saved
# from file : / code/bin/mandelbrot_render.c
# int width = 1280;
# int height = 720;
# standard view of parameter plane : center_re, center_im, radius
# plane description : view = center and radius
center_re="-1.86057396001587505"
center_im="-0.00000093437424500"
center="$center_re $center_im"
radius="3.5884751097983668e-09" #$(echo 1.0/$zoom | bc -l) # real radius
view="$center $radius"
# inbits are proportional to zoom
# inbits=$((54+$((4*$zoom))))
# image file names
filename="ian" # filename stem : stem="standard"
ppmfilename=$filename".ppm"
pngfilename=$filename".png"
# escape radius
er="512"
# image size in pixels
w="800"
h="600"
# maximum iterations
n="1500"
# interior rendering
i="1"
# Heredoc
# ./render $view && ./colour "$stem" > "$stem.ppm"
#./render $view "$er" "$filename" "$w" "$h" "$n" "$i" && ./colour "$filename" > "$ppmfilename" && ./annotate "$ppmfilename" $pngfilename <<EOF
#rgba 1 1 1 1
#EOF
echo "Image " $pngfilename " is saved"
# info text file
textfilename=$filename".txt"
ratio=$(echo $w/$h | bc -l)
# http://stackoverflow.com/questions/12882611/how-to-get-bc-to-handle-numbers-in-scientific-aka-exponential-notation
# bash do not use floating point
rim=`echo ${radius} | sed -e 's/[eE]+*/\\*10\\^/'` # conver e to 10
rre=$(echo $ratio*$rim | bc -l) # real radius
cu=$(echo $center_im+$rim | bc -l)
cd=$(echo $center_im-$rim | bc -l)
cl=$(echo $center_re+$rre | bc -l)
cr=$(echo $center_re-$rre | bc -l)
rim="("$rim")" # add () because precedence of operators
zoom=$(echo 1/$rim | bc -l) # zoom = 1/radius
z=$(echo "$zoom" | sed 's/e/*10^/g;s/ /*/' | bc)
echo "part of parameter complex plane " > $textfilename
echo "center of image c = " $center >> $textfilename
echo "radius = (image height/2) = " $rim >> $textfilename
echo "radius = (image height/2) = " $radius >> $textfilename
echo "up corner = center_im+radius = cu =" $cu >> $textfilename
echo "down corner = center_im-radius = cd =" $cd >> $textfilename
echo "left corner = center_re+ratio*radius = cl =" $cl >> $textfilename
echo "right corner = center_im-ratio*radius = cr =" $cr >> $textfilename
echo "image ratio = width/height =" $ratio >> $textfilename
echo "zoom level = fractint mag = "$zoom >>$textfilename
echo "ratio = image width/height = " $ratio >> $textfilename
echo "info text file " $textfilename "is saved"
用法
./mandelbrot_box_period cx cy radius maxperiod
示例
./mandelbrot_box_period 3.06095924635699068e-01 2.37427672737983361e-02 2.0000000000000001e-10 1000
下一個示例
./mandelbrot_box_period -0.220857943592514 -0.650752006989254 0.0000020 2000 596
如果核不在上述引數定義的矩形內，則返回 0
用法
./mandelbrot_nucleu bits cx cy period maxiters
示例
./mandelbrot_nucleus 100 3.06095924635699068e-01 2.37427672737983361e-02 68 1000
計算落在根點上的射線的角度，[10] 用法
./mandelbrot_external_angles bits cx cy period
示例
./mandelbrot_external_angles 100 3.06095924635699068e-01 2.37427672737983361e-02 68
新的
./mandelbrot_external_angles 53 -3.9089629378291085e-01 5.8676031775674931e-01 89
結果
.(01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001) .(01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010)
另一個
./mandelbrot_external_angles 100 -1.115234269232491 0.252761823831669 24 .(010110010110010110011001) .(010110010110011001010110)
G Pastor 給出了一個 IEEE 754 精度不足的外射線示例：[11]
（週期 3，落在週期 3 分量 c3 = -0.125000000000000 + 0.649519052838329i 的根點）
可以使用 Claude Heiland-Allen 的程式分析這些角度
./bin/mandelbrot_describe_external_angle ".(001)"
binary: .(001)
decimal: 1/7
preperiod: 0
period: 3
./bin/mandelbrot_describe_external_angle
".(001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001010)"
binary:
.(001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001010)
decimal:
33877456965431938318210482471113262183356704085033125021829876006886584214655562/237142198758023568227473377297792835283496928595231875152809132048206089502588927
preperiod: 0
period: 267
./bin/mandelbrot_describe_external_angle
".(001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001010001)"
binary:
.(001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001010001)
decimal:
33877456965431938318210482471113262183356704085033125021829876006886584214655569/237142198758023568227473377297792835283496928595231875152809132048206089502588927
preperiod: 0
period: 267
./bin/mandelbrot_describe_external_angle
".(0010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010001)"
binary:
.(0010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010001)
decimal:
67754913930863876636420964942226524366713408170066250043659752013773168429311121/474284397516047136454946754595585670566993857190463750305618264096412179005177855
preperiod: 0
period: 268
./bin/mandelbrot_describe_external_angle
".(0010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010)"
binary:
.(0010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010)
decimal:
67754913930863876636420964942226524366713408170066250043659752013773168429311122/474284397516047136454946754595585670566993857190463750305618264096412179005177855
preperiod: 0
period: 268
上述射線的著陸點是具有 **帶角度內地址** 的根（由 Claude Heiland-Allen 描述）
- 上面的那個將是 1 -> 1/3 -> 3 -> 1/(週期/3) -> 週期，因為它是最靠近 1/3 球的底部根尖的球，而 1/3 球的子球具有周期 3 * 分母（內角），即 1 -> 1/3 -> 3 -> 1/89 -> 267
- 下面的那個將是 1 -> floor(週期/3)/週期 -> 週期，因為它是最靠近 1/3 尖的下方球，即 1 -> 89/268 -> 268
- 中間射線 .(001) 落在 1 -> 1/3 -> 3 的根點上，從下側尖開始（在標準未旋轉檢視中位於右側）
size 2000 2000 view 53 3.0609592463186358e-01 2.3742767274521188e-02 7.6870929325598169e-11 text 57 3.060959246472445584e-01 2.374276727158354376e-02 272 text 56 3.06095924633099439e-01 2.3742767284688944e-02 204 text 56 3.06095924629285095e-01 2.37427672645622342e-02 204 text 54 3.06095924643046857e-01 2.374276727237906e-02 136 text 54 3.06095924629536442e-01 2.37427672749394494e-02 68 ray_in 2200 .(00000000000100000000000000110000000000000000000000000100000000000001) ray_in 2200 .(00000000000100000000000000101111111111111100000000000010000000000010)
要了解載入引數檔案，請參見反序列化函式
static int deserialize(const char *filename) {
FILE *in = fopen(filename, "rb");
if (in) {
while (G.anno) {
GtkTreeIter iter;
gpointer *thing;
gtk_tree_model_get_iter_first(GTK_TREE_MODEL(G.annostore), &iter);
if (gtk_tree_model_iter_next(GTK_TREE_MODEL(G.annostore), &iter)) {
gtk_tree_model_get(GTK_TREE_MODEL(G.annostore), &iter, 1, &thing, -1);
if (thing) {
delete_annotation((struct annotation *) thing);
gtk_tree_store_remove(G.annostore, &iter);
}
}
}
char *line = 0;
size_t linelen = 0;
ssize_t readlen = 0;
while (-1 != (readlen = getline(&line, &linelen, in))) {
if (readlen && line[readlen-1] == '\n') {
line[readlen - 1] = 0;
}
if (0 == strncmp(line, "size ", 5)) {
int w = 0, h = 0;
sscanf(line + 5, "%d %d\n", &w, &h);
// resize_image(w, h); // FIXME TODO make this work...
} else if (0 == strncmp(line, "view ", 5)) {
int p = 53;
int set_result;
char *xs = malloc(readlen);
char *ys = malloc(readlen);
char *rs = malloc(readlen);
sscanf(line + 5, "%d %s %s %s", &p, xs, ys, rs);
struct view *v = view_new();
mpfr_set_prec(v->cx, p);
mpfr_set_prec(v->cy, p);
set_result = mpfr_set_str(v->cx, xs, 0, GMP_RNDN);
set_result = set_result + mpfr_set_str(v->cy, ys, 0, GMP_RNDN);
set_result = set_result + mpfr_set_str(v->radius, rs, 0, GMP_RNDN);
free(xs);
free(ys);
free(rs);
if (set_result < 0) { printf("view line not valid, check chars\n"); return 1; }
start_render(v);
} else if (0 == strncmp(line, "ray_out ", 8)) {
int p = 53;
int set_result;
char *xs = malloc(readlen);
char *ys = malloc(readlen);
sscanf(line + 8, "%d %s %s", &p, xs, ys);
mpfr_t cx, cy;
mpfr_init2(cx, p);
mpfr_init2(cy, p);
set_result = mpfr_set_str(cx, xs, 0, GMP_RNDN);
set_result = set_result + mpfr_set_str(cy, ys, 0, GMP_RNDN);
free(xs);
free(ys);
double x = 0, y = 0;
param_to_screen(&x, &y, cx, cy);
mpfr_clear(cx);
mpfr_clear(cy);
if (set_result < 0) { printf("ray_out line not valid, check chars\n"); return 1; }
start_ray_out(x, y);
} else if (0 == strncmp(line, "ray_in ", 7)) {
int depth = 1000;
char *as = malloc(readlen);
sscanf(line + 7, "%d %s", &depth, as);
struct mandelbrot_binary_angle *angle = mandelbrot_binary_angle_from_string(as);
struct mandelbrot_binary_angle *angle2 = mandelbrot_binary_angle_canonicalize(angle);
mandelbrot_binary_angle_delete(angle);
start_ray_in(angle2, depth);
free(as);
} else if (0 == strncmp(line, "text ", 5)) {
int p = 53;
int set_result;
char *xs = malloc(readlen);
char *ys = malloc(readlen);
char *ss = malloc(readlen);
sscanf(line + 5, "%d %s %s %s", &p, xs, ys, ss);
struct annotation *anno = calloc(1, sizeof(struct annotation));
anno->tag = annotation_text;
anno->label = ss;
mpfr_init2(anno->u.text.x, p);
mpfr_init2(anno->u.text.y, p);
set_result = mpfr_set_str(anno->u.text.x, xs, 0, GMP_RNDN);
set_result = set_result + mpfr_set_str(anno->u.text.y, ys, 0, GMP_RNDN);
free(xs);
free(ys);
if (set_result < 0) {
free(ss);
mpfr_clear(anno->u.text.x);
mpfr_clear(anno->u.text.y);
printf("text line not valid, check chars \n");
return 1;
}
add_annotation(anno);
}
}
free(line);
fclose(in);
return 0;
} else {
return 1;
}
}
輸入格式應為
.pre(period)
例如
.010(110)
在GUI程式中，方框表示使用者需要標記矩形（如原子核），而不是點選（如鍵）
- 元件
- Mu-Atom
- 接觸點
- 當前點
- 根
"Bond is a " point where two mu-atoms meet. The parent "binds to" the child through the bond, and the two are said to be adjacent. This use of the word 'bond' was introduced by Benoit Mandelbrot in his description of the Mandelbrot set in The Fractal Geometry of Nature." From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015.
在該書程式中，計算鍵的結果是父元件的內角（=組合旋轉數）
"停留時間是逃逸迭代（稱為表示函式）的口語名稱。" 來自羅伯特·穆納福的《曼德布羅特集合詞彙表和百科全書》，(c) 1987-2015。
"停留時間帶是標準檢視中曼德布羅特集合中出現的純色區域；它們也稱為水平集。停留時間帶中的所有點都具有相同的停留時間。它們由“等高線”或勒姆尼斯卡特分隔開."
- 分支型別米休列維奇點
Mu-Atom 的原子核 = 曼德布羅特集合的雙曲分量的中心
"The unique point within any mu-atom which has the property of belonging to its own limit cycle. This point is called the superstable point. This use of the word 'nucleus' was introduced by Benoit Mandelbrot in his description of the Mandelbrot set in The Fractal Geometry of Nature. If you set the polynomial formula for a lemniscate ZN equal to zero and solve for C (to get the roots of the polynomial), the roots are the nuclei of the mu-atoms of period N, plus any mu-atoms of periods that divide evenly into N. This procedure has been used numerically by Jay Hill to find all mu-atoms for periods up to about 16." From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015.
尋找原子核
週期為 p 的原子核 n 是引數平面的一個點
滿足
其中
克勞德正在使用一個變數中的牛頓法來計算 c=n
- 對 c 的雙曲分量（由原子域包圍）的週期的良好猜測是部分，即 |fpc(0)| 達到新的最小值的 p 值。
- 部分，即 |fpc(0)| 達到新的最小值的 p 值。它與內部距離相關 [12]
- "部分" 是迭代 p，其中 |z_p| 小於所有先前的 |z_n|。它們是可能的週期的良好候選者。
- "部分" 是我對具有 0 < n 的 n 的集合的術語，使得對於所有 0 < m < n，|z_n| < |z_m|，其中 z_{n+1} = z_n^2 + c，z_0 = 0。這與 "原子週期域" 著色相關：http://mrob.com/pub/muency/atomdomain.html
- 另一種方法是檢查 z_p 是否接近零，而不是 z_{p+1} 是否接近 c。它們最終具有相同的效果，但計算起來更容易。
一些例子
-9.82053364278e-2 + 0.87751161636 i 是一個週期為 30、大小約為 8.7e-4 的燈泡的中心，其部分為 1 3 6 12 30
-1.40107054826248 + 1.68078322683058e-2 i 是一個週期為 25、大小約為 2.9e-7 的心形的中心，其部分為 1 2 4 8 25
-6.30751837180080329933379814864882594423413629277790243935409e-02 + 9.97813152226579778761450011018468925066022924931316287706002e-01 i 是一個週期為 660、大小約為 1e-51 的心形的中心，其部分為 1 3 4 5 12 34 133 660
-1.949583466095265215576424927006606703613013182307914337344552997126238598475224082315026579e+00 + -7.76868505224924928703440073040924718407938044210292978384443422263629366944037056031909997e-06 i 是一個週期為 1820、大小約為 3.5e-83 的心形的中心，其部分為 1 2 3 4 9 25 83 359 1820
// gcc -std=c99 -Wall -Wextra -pedantic -O3 -o partials partials.c -lm
// ./partials re im
// http://www.fractalforums.com/help-and-support/period-detection-for-dummies/?topicseen
#include <complex.h>
#include <stdio.h>
#include <stdlib.h>
int main(int argc, char **argv)
{
if (argc != 3)
{
fprintf(stderr, "usage: %s re im\n", argv[0]);
return 1;
}
double re = atof(argv[1]);
double im = atof(argv[2]);
double _Complex c = re + im * I;
double _Complex z = 0;
double minimum = 1.0 / 0.0; // infninity
printf(" n z |z| min\n");
for (int n = 1; n < 100; ++n)
{
z = z * z + c;
double absz = cabs(z);
printf("%4d %+.18f + %+.18f i %+.18f", n, creal(z), cimag(z), absz);
if (absz < minimum)
{
minimum = absz;
printf(" ****");
}
printf("\n");
}
return 0;
}
示例輸出
./partials -0.509 0.576
n z |z| min
1 -0.509000000000000008 + +0.575999999999999956 i +0.768672231838772757 ****
2 -0.581694999999999962 + -0.010368000000000044 i +0.581787391105204277 ****
3 -0.170738422399000056 + +0.588062027520000030 i +0.612346762132562117
4 -0.825665339327633863 + +0.375190434296955644 i +0.906912958643195766
5 +0.031955390579078591 + -0.043563474492556487 i +0.054027060783695097 ****
6 -0.509876629322802200 + +0.573215824315217226 i +0.767170488467169953
7 -0.577602204115791773 + -0.008538704752669046 i +0.577665314588191370
8 -0.175448603279432458 + +0.585863949370871162 i +0.611570747800398218
9 -0.821454354779731055 + +0.370421976742216996 i +0.901110258425790955
10 +0.028574816132972747 + -0.032569491802020845 i +0.043327726841770761 ****
11 -0.509244251679208726 + +0.574138665520425695 i +0.767440496138881434
12 -0.579305499377257949 + -0.008753630166097426 i +0.579371631726838143
13 -0.173481764432350638 + +0.586142052189469798 i +0.611276065240121014
14 -0.822466582754321607 + +0.372630085156343605 i +0.902942113377815159
15 +0.028598099383947528 + -0.036951585539979570 i +0.046725485147749588
16 -0.509547568385544269 + +0.573886509768666397 i +0.767453223683425945
17 -0.578707001646840746 + -0.008844951163781811 i +0.578774590765840591
18 -0.174176439406013128 + +0.586237270335409733 i +0.611564852795244307
19 -0.822336705086155639 + +0.371782559211755903 i +0.902474336402979138
20 +0.029015385197912136 + -0.035460889501387816 i +0.045818852696383125
- final_n = 最後迭代 = z 逃逸時的 n
- final_z = 最後 z
- final angle = 最後 z 的角度
int final_n; // last iteration
complex float final_z; //
if (cabs2(z) > escape_radius2) {
final_n = n;
final_z = z; }
輻條 = 樹的分支，樹狀結構的臂
週期點 z = wucleus
點 c 內週期為 p 的雙曲分量的 wucleus w 滿足
其中
所以 w 是動力學平面（z 平面）上的一個點
git add book.pdf book.tex code/*.c git commit -m "description" git format-patch origin/master
並透過電子郵件將補丁傳送給克勞德。
gcc render.c -E -o 26render.c remove some code manually add include
- ↑ 克勞德·海蘭德-艾倫的 c 程式
- ↑ 克勞德·海蘭德-艾倫 - 部落格
- ↑ 一種繪製曼德布羅特集合外部射線的演算法，作者：河平智樹
- ↑ 曼德布羅特集合中的內部座標
- ↑ 修改後的原子域
- ↑ 曼德布羅特筆記本
- ↑ book/code/gui/main.c
- ↑ mightymandel/src/atom.c 程式碼
- ↑ Claude Heiland-Allen 編著的“如何編寫關於曼德布洛特集的書籍”的非官方版本（包含完整 C 程式碼）
- ↑ Claude Heiland-Allen 編著的“自動查詢外部角度”
- ↑ 解決曼德布洛特集外部射線繪製限制的方法 M. Romera，1 G. Pastor，A. B. Orue，1 A. Martin，M.-F. Danca 和 F. Montoya
- ↑ Claude Heiland-Allen 編著的“實用內部距離渲染”