分形/book
< 分形
这里书的意思是
- 未命名（旧）c 程序[1] 由 Claude Heiland-Allen[2]
- 一个 GUI 程序
- 一组控制台程序
- 关于“如何写一本关于曼德勃罗集合的书”的 pdf 格式书籍。
- 官方版本
- 这里有一个 pdf 文件的非官方版本 ( 68.3 MB )
- 使用流的书籍控制台程序
- 书籍 GUI 程序现在基本被放弃了。
- 它被移植到 m-perturbator 在 mandelbrot-perturbator 存储库
- 另请参阅库
- sh 文件（主程序
- bin 目录中的 c 文件
- lib 目录中的 c 和 h 文件
半径 定义为“中心和轴对齐视图矩形的顶部之间的虚数坐标差”。半径没有初始值。
// / 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);
带有完整 src 代码的图像
-
曼德勃罗集合的逃逸时间（= 外部）的水平集 -
连续逃逸时间 -
原子域算法
所有算法都在书籍中描述：Claude Heiland-Allen 编著的“如何写一本关于曼德勃罗集合的书”[6]
请参阅 book/code/gui/main.c 中的 mandelbrot_nucleus 函数。[7] 它使用 牛顿法.
与 mightymandel/src/atom.c 中的 muatom() 函数进行比较[8]
使用牛顿法、mpfr 和 gmp 库计算外部射线向内，基于库的代码。外部角度从字符串中读取（二进制分数）
/*
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 仓库
git clone https://code.mathr.co.uk/mandelbrot-book
或旧的（已弃用）
git clone http://code.mathr.co.uk/book.git
或者你可以使用非官方仓库
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
然后 再次制作
首先
sudo apt-get install texlive-latex-recommended texlive-latex-extra texlive-fonts-recommended texlive-fonts-extra sudo apt-get install texlive-science
然后制作 pdf 格式的书籍
- 转到 book/book 目录
- 构建
cd book make
或使用包含完整 c 代码的非官方版本[9]
https://code.mathr.co.uk/mandelbrot-book-images/
从 mandelbrot-numerics 目录中打开的控制台中
git pull
如果你做了一些本地更改，你可以撤消它们
git checkout -f
然后
git pull
现在再次安装
书籍程序的最后版本有两种类型
- 控制台
- gui
要运行 gui 程序
# cd gui ./mandelbrot_gui
要运行控制台程序，请使用 bash 脚本，参见示例
./examples/standardview.sh
- 控制台程序是 bash sh 文件，使用管道（流）
- 对于 GUI 版本的程序（/gui/mandelbrot_gui，参见 Makefile）从文本文件读取参数
./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://wikibooks.cn/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 " $filname".txt is saved"
扩展使用
# Radius is defined as "the difference in imaginary coordinate between the center and the top of the axis-aligned view rectangle".
# https://wikibooks.cn/wiki/Fractals/Computer_graphic_techniques/2D/plane
#
# in examples directory : chmod +x standardview2.sh
# then go to the parent directory : cd ..
# run run these example from the parent directory :
# ./examples/ian.sh
#
# result :
# ./render using double
# rgba 1.000000 1.000000 1.000000 1.000000
# Image standard.png is saved
# info text file .txt is saved
# from file : / code/bin/mandelbrot_render.c
# int width = 1280;
# int height = 720;
# standard view of parameter plane : center_re, center_im, radius
# plane description : view = center and radius
center_re="-1.86057396001587505"
center_im="-0.00000093437424500"
center="$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
下一个示例
./mandelbrot_box_period -0.220857943592514 -0.650752006989254 0.0000020 2000 596
如果核不在上述参数定义的矩形内，则返回 0
用法
./mandelbrot_nucleu bits cx cy period maxiters
示例
./mandelbrot_nucleus 100 3.06095924635699068e-01 2.37427672737983361e-02 68 1000
计算落在根点上的射线的角度，[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
结果
.(01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001) .(01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010)
另一个
./mandelbrot_external_angles 100 -1.115234269232491 0.252761823831669 24 .(010110010110010110011001) .(010110010110011001010110)
G Pastor 给出了一个 IEEE 754 精度不足的外射线示例：[11]
（周期 3，落在周期 3 分量 c3 = -0.125000000000000 + 0.649519052838329i 的根点）
可以使用 Claude Heiland-Allen 的程序分析这些角度
./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
上述射线的着陆点是具有 **带角度内地址** 的根（由 Claude Heiland-Allen 描述）
- 上面的那个将是 1 -> 1/3 -> 3 -> 1/(周期/3) -> 周期，因为它是最靠近 1/3 球的底部根尖的球，而 1/3 球的子球具有周期 3 * 分母（内角），即 1 -> 1/3 -> 3 -> 1/89 -> 267
- 下面的那个将是 1 -> floor(周期/3)/周期 -> 周期，因为它是最靠近 1/3 尖的下方球，即 1 -> 89/268 -> 268
- 中间射线 .(001) 落在 1 -> 1/3 -> 3 的根点上，从下侧尖开始（在标准未旋转视图中位于右侧）
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)
要了解加载参数文件，请参见反序列化函数
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;
}
}
输入格式应为
.pre(period)
例如
.010(110)
在GUI程序中，方框表示用户需要标记矩形（如原子核），而不是点击（如键）
- 组件
- Mu-Atom
- 接触点
- 当前点
- 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.
在该书程序中，计算键的结果是父组件的内角（=组合旋转数）
"停留时间是逃逸迭代（称为表示函数）的口语名称。" 来自罗伯特·穆纳福的《曼德布罗特集合词汇表和百科全书》，(c) 1987-2015。
"停留时间带是标准视图中曼德布罗特集合中出现的纯色区域；它们也称为水平集。停留时间带中的所有点都具有相同的停留时间。它们由“等高线”或勒姆尼斯卡特分隔开."
- 分支类型米休列维奇点
Mu-Atom 的原子核 = 曼德布罗特集合的双曲分量的中心
"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.
寻找原子核
周期为 p 的原子核 n 是参数平面的一个点
满足
其中
克劳德正在使用一个变量中的牛顿法来计算 c=n
- 对 c 的双曲分量（由原子域包围）的周期的良好猜测是部分，即 |fpc(0)| 达到新的最小值的 p 值。
- 部分，即 |fpc(0)| 达到新的最小值的 p 值。它与内部距离相关 [12]
- "部分" 是迭代 p，其中 |z_p| 小于所有先前的 |z_n|。它们是可能的周期的良好候选者。
- "部分" 是我对具有 0 < n 的 n 的集合的术语，使得对于所有 0 < m < n，|z_n| < |z_m|，其中 z_{n+1} = z_n^2 + c，z_0 = 0。这与 "原子周期域" 着色相关：http://mrob.com/pub/muency/atomdomain.html
- 另一种方法是检查 z_p 是否接近零，而不是 z_{p+1} 是否接近 c。它们最终具有相同的效果，但计算起来更容易。
一些例子
-9.82053364278e-2 + 0.87751161636 i 是一个周期为 30、大小约为 8.7e-4 的灯泡的中心，其部分为 1 3 6 12 30
-1.40107054826248 + 1.68078322683058e-2 i 是一个周期为 25、大小约为 2.9e-7 的心形的中心，其部分为 1 2 4 8 25
-6.30751837180080329933379814864882594423413629277790243935409e-02 + 9.97813152226579778761450011018468925066022924931316287706002e-01 i 是一个周期为 660、大小约为 1e-51 的心形的中心，其部分为 1 3 4 5 12 34 133 660
-1.949583466095265215576424927006606703613013182307914337344552997126238598475224082315026579e+00 + -7.76868505224924928703440073040924718407938044210292978384443422263629366944037056031909997e-06 i 是一个周期为 1820、大小约为 3.5e-83 的心形的中心，其部分为 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;
}
示例输出
./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 = 最后迭代 = z 逃逸时的 n
- final_z = 最后 z
- final angle = 最后 z 的角度
int final_n; // last iteration
complex float final_z; //
if (cabs2(z) > escape_radius2) {
final_n = n;
final_z = z; }
辐条 = 树的分支，树状结构的臂
周期点 z = wucleus
点 c 内周期为 p 的双曲分量的 wucleus w 满足
其中
所以 w 是动力学平面（z 平面）上的一个点
git add book.pdf book.tex code/*.c git commit -m "description" git format-patch origin/master
并通过电子邮件将补丁发送给克劳德。
gcc render.c -E -o 26render.c remove some code manually add include
- ↑ 克劳德·海兰德-艾伦的 c 程序
- ↑ 克劳德·海兰德-艾伦 - 博客
- ↑ 一种绘制曼德布罗特集合外部射线的算法，作者：河平智树
- ↑ 曼德布罗特集合中的内部坐标
- ↑ 修改后的原子域
- ↑ 曼德布罗特笔记本
- ↑ book/code/gui/main.c
- ↑ mightymandel/src/atom.c 代码
- ↑ Claude Heiland-Allen 编著的“如何编写关于曼德布洛特集的书籍”的非官方版本（包含完整 C 代码）
- ↑ Claude Heiland-Allen 编著的“自动查找外部角度”
- ↑ 解决曼德布洛特集外部射线绘制限制的方法 M. Romera，1 G. Pastor，A. B. Orue，1 A. Martin，M.-F. Danca 和 F. Montoya
- ↑ Claude Heiland-Allen 编著的“实用内部距离渲染”