Difference between revisions of "File:E1efig09abc1a150.png"

From TORI
Jump to: navigation, search
(Importing image file)
 
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
  +
[[Complex map]]s of [[tetration]] $\mathrm{tet}_b$ to base<br>
−
Importing image file
  
  +
$b\!=\!1.5$ , left, <br>
  +
$b\!=\!\exp(1/\mathrm e)$ , center, and<br>
  +
$b\!=\!\sqrt{2}$ , right.
  +
  +
$f\!=\! \mathrm{tet}(x\!+\!\mathrm i y)$ is shown in the $x,y$ plane with levels
  +
$~p\!=\!\Re(f)\!=\! \mathrm{const}~$ and levels
  +
$~q\!=\!\Im(f)\!=\! \mathrm{const}~$. The integer levels are shown with thick lines.
  +
  +
This image is close to the figure 9 in the article
  +
<ref name="e1e">
  +
http://tori.ils.uec.ac.jp/PAPERS/2011e1e.pdf
  +
H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). (Mathematics of Computation, 2011, in press)
  +
</ref>.
  +
  +
The [[C++]] generators of the pictures used in the figure are available and will be uploaded upon request.
  +
  +
Copyleft 2011 by Dmitrii Kouznetsov. You may copy, modify and/or distribute the image for free but the source should be attributed.
  +
  +
==[[C++]] generator of first map, $b\!=\!1.5$ ==
  +
  +
In order to compile the code below,
  +
the following files should be loaded:
  +
[[ado.cin]],
  +
[[conto.cin]],
  +
[[GLxw2048.inc]],
  +
[[f2048b15.inc]],
  +
[[f15.cin]]
  +
  +
<poem><nomathjax><nowiki>
  +
#include <math.h>
  +
#include <stdio.h>
  +
#include <stdlib.h>
  +
#define DB double
  +
#define DO(x,y) for(x=0;x<y;x++)
  +
using namespace std;
  +
#include<complex>
  +
typedef complex<double> z_type;
  +
  +
// #include <complex.h>
  +
// #define z_type complex<double>
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
#include "conto.cin"
  +
//DB T22=-8.5715740896774235522;
  +
//DB T42= 9.6180745210214273558;
  +
//#include "f21E.cin"
  +
//#include "e1etf.cin"
  +
#include "f15.cin"
  +
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  +
// int M=201,M1=M+1;
  +
// int N=401,N1=N+1;
  +
int M=81,M1=M+1;
  +
int N=161,N1=N+1;
  +
DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
  +
char v[M1*N1]; // v is working array
  +
// FILE *o;o=fopen("f21E.eps","w"); ado(o,0,0,94,92);
  +
// FILE *o;o=fopen("02.eps","w"); ado(o,0,0,214,212);
  +
// FILE *o;o=fopen("be1ezoom.eps","w"); ado(o,0,0,87,87);
  +
FILE *o;o=fopen("b15zoom.eps","w"); ado(o,87,87);
  +
fprintf(o,"46 45 translate\n 10 10 scale\n");
  +
DO(m,M1) X[m]=-4.+.1*(m-.5);
  +
DO(n,N1) Y[n]=-4.+.05*(n-.5);
  +
for(m=-4;m<5;m++) {M(m,-4)L(m,4)}
  +
for(n=-4;n<5;n++) {M( -4,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
  +
fprintf(o,"/adobe-Roman findfont .6 scalefont setfont\n");
  +
for(m=-2;m<0;m+=2) {M(-4.7,m-.2) fprintf(o,"(%1d)s\n",m);}
  +
for(m= 0;m<3;m+=2) {M(-4.4,m-.2) fprintf(o,"(%1d)s\n",m);}
  +
for(m=-2;m<0;m+=2) {M(m-.3,-4.48) fprintf(o,"(%1d)s\n",m);}
  +
for(m= 0;m<3;m+=2) {M(m-.16,-4.48) fprintf(o,"(%1d)s\n",m);}
  +
/*
  +
fprintf(o,"/Times-Italic findfont 1 scalefont setfont\n");
  +
//fprintf(o,"/adobe-italic findfont 1 scalefont setfont\n");
  +
M(-4.7, 4.5) fprintf(o,"(y)s\n");
  +
M( 4.6,-4.8) fprintf(o,"(x)s\n");
  +
M(-4,0)L(4.1,0) M(0,-4)L(0,4.1) fprintf(o,".01 W 1 0 1 RGB S\n");
  +
*/
  +
  +
//z_type tm,tp,F[M1*N1];;
  +
DO(m,M1)DO(n,N1){ g[m*N1+n]=9999;
  +
f[m*N1+n]=9999;}
  +
DB b=sqrt(2);
  +
DO(m,M1){x=X[m]; printf("x=%6.3f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y);
  +
if(abs(z+2.)>.04)
  +
{
  +
// c=F21E(z);
  +
// c=E1ETF(z);
  +
c=F15(z);
  +
p=Re(c); q=Im(c);
  +
if(p>-9999 && p<9999) g[m*N1+n]=p;
  +
if(q>-9999 && q<9999 && fabs(q)>1.e-14) f[m*N1+n]=q;
  +
}
  +
}}
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n");
  +
p=2.; q=1.1;;
  +
#include "plofu.cin"
  +
  +
fprintf(o,"0 setlinejoin 0 setlinecap\n");
  +
// p=1.e-15;
  +
// for(n=-10;n<11;n++) {q=p*n; z=z_type(q,0.); printf("%19.15f %19.15f\n",q, Re(TQ2E(z)) )
  +
; }
  +
//y=2*M_PI/log(2.);
  +
// y=M_PI/log(log(2));
  +
//y=9.064720284;
  +
  +
// M(-2,0)L(-10.1, 0) fprintf(o,"0.05 W 1 1 1 RGB S\n");
  +
//
  +
// DO(n,20){ M(-2.-.4*n,0)L(-2-.4*(n+.5),0) }
  +
// fprintf(o,".11 W 0 0 0 RGB S\n");
  +
  +
//M(-2,0)L(-4.1, 0) fprintf(o,".1 W 0 0 0 RGB [.14 .14] 0 setdash S\n"); //fails at some pri
  +
nters
  +
  +
//M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n");
  +
fprintf(o,"showpage\n");
  +
fprintf(o,"%cTrailer\n",'%');
  +
fclose(o);
  +
// system( "gv b15zoom.eps &"); //for UNIX
  +
system( "open b15zoom.eps"); //for macintosh
  +
system("epstopdf b15zoom.eps");
  +
// system( "xpdf be1ezoom.pdf"); // for LINUX
  +
getchar(); system("killall Preview"); // For macintosh
  +
}
  +
  +
</nowiki></nomathjax></poem>
  +
  +
==[[C++]] generator of second map==
  +
<poem><nomathjax><nowiki>
  +
</nowiki></nomathjax></poem>
  +
  +
==[[C++]] generator of third map, $b\!=\!\sqrt{2}\!\approx\!1.41$==
  +
Files required:
  +
[[ado.cin]],
  +
[[conto.cin]],
  +
[[sqrt2f21e.cin]]
  +
<poem><nomathjax><nowiki>
  +
#include <math.h>
  +
#include <stdio.h>
  +
#include <stdlib.h>
  +
#define DB double
  +
#define DO(x,y) for(x=0;x<y;x++)
  +
#include<complex>
  +
typedef std::complex<double> z_type;
  +
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
#include "conto.cin"
  +
DB T22=-8.5715740896774235522;
  +
DB T42= 9.6180745210214273558;
  +
//#include "f21E.cin"
  +
#include "sqrt2f21e.cin"
  +
//#include "e1etf.cin"
  +
//#include "f15.cin"
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  +
// int M=201,M1=M+1;
  +
// int N=401,N1=N+1;
  +
int M=81,M1=M+1;
  +
int N=161,N1=N+1;
  +
DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
  +
char v[M1*N1]; // v is working array
  +
// FILE *o;o=fopen("f21E.eps","w"); ado(o,0,0,94,92);
  +
// FILE *o;o=fopen("02.eps","w"); ado(o,0,0,214,212);
  +
// FILE *o;o=fopen("bq2zoom.eps","w"); ado(o,87,87);
  +
FILE *o;o=fopen("e1e09q2.eps","w"); ado(o,87,87);
  +
fprintf(o,"46 45 translate\n 10 10 scale\n");
  +
DO(m,M1) X[m]=-4.+.1*(m-.5);
  +
DO(n,N1) Y[n]=-4.+.05*(n-.5);
  +
for(m=-4;m<5;m++) {M(m,-4)L(m,4)}
  +
for(n=-4;n<5;n++) {M( -4,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
  +
/*
  +
fprintf(o,"/adobe-Roman findfont .6 scalefont setfont\n");
  +
for(m=-2;m<0;m+=2) {M(-4.7,m-.2) fprintf(o,"(%1d)s\n",m);}
  +
for(m= 0;m<3;m+=2) {M(-4.4,m-.2) fprintf(o,"(%1d)s\n",m);}
  +
for(m=-2;m<0;m+=2) {M(m-.3,-4.48) fprintf(o,"(%1d)s\n",m);}
  +
for(m= 0;m<3;m+=2) {M(m-.16,-4.48) fprintf(o,"(%1d)s\n",m);}
  +
*/
  +
/*
  +
fprintf(o,"/Times-Italic findfont 1 scalefont setfont\n");
  +
//fprintf(o,"/adobe-italic findfont 1 scalefont setfont\n");
  +
M(-4.7, 4.5) fprintf(o,"(y)s\n");
  +
M( 4.6,-4.8) fprintf(o,"(x)s\n");
  +
M(-4,0)L(4.1,0) M(0,-4)L(0,4.1) fprintf(o,".01 W 1 0 1 RGB S\n");
  +
*/
  +
  +
//z_type tm,tp,F[M1*N1];;
  +
DO(m,M1)DO(n,N1){ g[m*N1+n]=9999;
  +
f[m*N1+n]=9999;}
  +
DB b=sqrt(2);
  +
DO(m,M1){x=X[m]; printf("x=%6.3f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y);
  +
if(abs(z+2.)>.04)
  +
{
  +
c=F21E(z);
  +
//c=E1ETF(z);
  +
// c=F15(z);
  +
p=Re(c); q=Im(c);
  +
if(p>-9999 && p<9999) g[m*N1+n]=p;
  +
if(q>-9999 && q<9999 && fabs(q)>1.e-14) f[m*N1+n]=q;
  +
}
  +
}}
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n");
  +
p=2.; q=1.1;;
  +
#include "plofu.cin"
  +
  +
fprintf(o,"0 setlinejoin 0 setlinecap\n");
  +
// p=1.e-15;
  +
// for(n=-10;n<11;n++) {q=p*n; z=z_type(q,0.); printf("%19.15f %19.15f\n",q, Re(TQ2E(z)) ); }
  +
//y=2*M_PI/log(2.);
  +
// y=M_PI/log(log(2));
  +
//y=9.064720284;
  +
  +
// M(-2,0)L(-10.1, 0) fprintf(o,"0.05 W 1 1 1 RGB S\n");
  +
//
  +
// DO(n,20){ M(-2.-.4*n,0)L(-2-.4*(n+.5),0) }
  +
// fprintf(o,".11 W 0 0 0 RGB S\n");
  +
  +
//M(-2,0)L(-4.1, 0) fprintf(o,".1 W 0 0 0 RGB [.14 .14] 0 setdash S\n"); //fails at some printers
  +
  +
//M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n");
  +
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
  +
system("epstopdf e1e09q2.eps");
  +
system( "open e1e09q2.pdf");
  +
getchar(); system("killall Preview"); // For macintosh
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
<poem><nomathjax><nowiki>
  +
</nowiki></nomathjax></poem>
  +
  +
==References==
  +
<references/>
  +
  +
[[Category:E1e]]
  +
[[Category:Base sqrt(2)]]
  +
[[Category:Complex maps]]
  +
[[Category:C++]]
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:Holomorphic functions]]
  +
[[Category:Latex]]
  +
[[Category:Tetration]]

Latest revision as of 08:34, 1 December 2018

Complex maps of tetration $\mathrm{tet}_b$ to base
$b\!=\!1.5$ , left,
$b\!=\!\exp(1/\mathrm e)$ , center, and
$b\!=\!\sqrt{2}$ , right.

$f\!=\! \mathrm{tet}(x\!+\!\mathrm i y)$ is shown in the $x,y$ plane with levels $~p\!=\!\Re(f)\!=\! \mathrm{const}~$ and levels $~q\!=\!\Im(f)\!=\! \mathrm{const}~$. The integer levels are shown with thick lines.

This image is close to the figure 9 in the article [1].

The C++ generators of the pictures used in the figure are available and will be uploaded upon request.

Copyleft 2011 by Dmitrii Kouznetsov. You may copy, modify and/or distribute the image for free but the source should be attributed.

C++ generator of first map, $b\!=\!1.5$

In order to compile the code below, the following files should be loaded: ado.cin, conto.cin, GLxw2048.inc, f2048b15.inc, f15.cin


 #include <math.h>
 #include <stdio.h>
 #include <stdlib.h>
 #define DB double
 #define DO(x,y) for(x=0;x<y;x++)
using namespace std;
#include<complex>
typedef complex<double> z_type;

// #include <complex.h>
// #define z_type complex<double>
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
 #include "conto.cin"
//DB T22=-8.5715740896774235522;
//DB T42= 9.6180745210214273558;
//#include "f21E.cin"
//#include "e1etf.cin"
#include "f15.cin"
 main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
// int M=201,M1=M+1;
// int N=401,N1=N+1;
 int M=81,M1=M+1;
 int N=161,N1=N+1;
 DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
 char v[M1*N1]; // v is working array
// FILE *o;o=fopen("f21E.eps","w"); ado(o,0,0,94,92);
// FILE *o;o=fopen("02.eps","w"); ado(o,0,0,214,212);
// FILE *o;o=fopen("be1ezoom.eps","w"); ado(o,0,0,87,87);
 FILE *o;o=fopen("b15zoom.eps","w"); ado(o,87,87);
 fprintf(o,"46 45 translate\n 10 10 scale\n");
  DO(m,M1) X[m]=-4.+.1*(m-.5);
  DO(n,N1) Y[n]=-4.+.05*(n-.5);
 for(m=-4;m<5;m++) {M(m,-4)L(m,4)}
 for(n=-4;n<5;n++) {M( -4,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
 fprintf(o,"/adobe-Roman findfont .6 scalefont setfont\n");
 for(m=-2;m<0;m+=2) {M(-4.7,m-.2) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<3;m+=2) {M(-4.4,m-.2) fprintf(o,"(%1d)s\n",m);}
 for(m=-2;m<0;m+=2) {M(m-.3,-4.48) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<3;m+=2) {M(m-.16,-4.48) fprintf(o,"(%1d)s\n",m);}
/*
 fprintf(o,"/Times-Italic findfont 1 scalefont setfont\n");
 //fprintf(o,"/adobe-italic findfont 1 scalefont setfont\n");
 M(-4.7, 4.5) fprintf(o,"(y)s\n");
 M( 4.6,-4.8) fprintf(o,"(x)s\n");
 M(-4,0)L(4.1,0) M(0,-4)L(0,4.1) fprintf(o,".01 W 1 0 1 RGB S\n");
*/
 
//z_type tm,tp,F[M1*N1];;
 DO(m,M1)DO(n,N1){ g[m*N1+n]=9999;
  f[m*N1+n]=9999;}
DB b=sqrt(2);
 DO(m,M1){x=X[m]; printf("x=%6.3f\n",x);
 DO(n,N1){y=Y[n]; z=z_type(x,y);
if(abs(z+2.)>.04)
{
// c=F21E(z);
// c=E1ETF(z);
         c=F15(z);
  p=Re(c); q=Im(c);
if(p>-9999 && p<9999) g[m*N1+n]=p;
  if(q>-9999 && q<9999 && fabs(q)>1.e-14) f[m*N1+n]=q;
}
  }}
fprintf(o,"1 setlinejoin 2 setlinecap\n");
p=2.; q=1.1;;
#include "plofu.cin"

fprintf(o,"0 setlinejoin 0 setlinecap\n");
// p=1.e-15;
// for(n=-10;n<11;n++) {q=p*n; z=z_type(q,0.); printf("%19.15f %19.15f\n",q, Re(TQ2E(z)) )
; }
//y=2*M_PI/log(2.);
// y=M_PI/log(log(2));
//y=9.064720284;

// M(-2,0)L(-10.1, 0) fprintf(o,"0.05 W 1 1 1 RGB S\n");
//
// DO(n,20){ M(-2.-.4*n,0)L(-2-.4*(n+.5),0) }
// fprintf(o,".11 W 0 0 0 RGB S\n");

//M(-2,0)L(-4.1, 0) fprintf(o,".1 W 0 0 0 RGB [.14 .14] 0 setdash S\n"); //fails at some pri
nters

//M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n");
 fprintf(o,"showpage\n");
 fprintf(o,"%cTrailer\n",'%');
 fclose(o);
// system( "gv b15zoom.eps &"); //for UNIX
system( "open b15zoom.eps"); //for macintosh
  system("epstopdf b15zoom.eps");
// system( "xpdf be1ezoom.pdf"); // for LINUX
getchar(); system("killall Preview"); // For macintosh
}

C++ generator of second map

C++ generator of third map, $b\!=\!\sqrt{2}\!\approx\!1.41$

Files required: ado.cin, conto.cin, sqrt2f21e.cin


 #include <math.h>
 #include <stdio.h>
 #include <stdlib.h>
 #define DB double
 #define DO(x,y) for(x=0;x<y;x++)
#include<complex>
typedef std::complex<double> z_type;

 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
 #include "conto.cin"
DB T22=-8.5715740896774235522;
DB T42= 9.6180745210214273558;
//#include "f21E.cin"
#include "sqrt2f21e.cin"
//#include "e1etf.cin"
//#include "f15.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
// int M=201,M1=M+1;
// int N=401,N1=N+1;
 int M=81,M1=M+1;
 int N=161,N1=N+1;
 DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
 char v[M1*N1]; // v is working array
// FILE *o;o=fopen("f21E.eps","w"); ado(o,0,0,94,92);
// FILE *o;o=fopen("02.eps","w"); ado(o,0,0,214,212);
// FILE *o;o=fopen("bq2zoom.eps","w"); ado(o,87,87);
 FILE *o;o=fopen("e1e09q2.eps","w"); ado(o,87,87);
 fprintf(o,"46 45 translate\n 10 10 scale\n");
  DO(m,M1) X[m]=-4.+.1*(m-.5);
  DO(n,N1) Y[n]=-4.+.05*(n-.5);
 for(m=-4;m<5;m++) {M(m,-4)L(m,4)}
 for(n=-4;n<5;n++) {M( -4,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
/*
 fprintf(o,"/adobe-Roman findfont .6 scalefont setfont\n");
 for(m=-2;m<0;m+=2) {M(-4.7,m-.2) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<3;m+=2) {M(-4.4,m-.2) fprintf(o,"(%1d)s\n",m);}
 for(m=-2;m<0;m+=2) {M(m-.3,-4.48) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<3;m+=2) {M(m-.16,-4.48) fprintf(o,"(%1d)s\n",m);}
*/
/*
 fprintf(o,"/Times-Italic findfont 1 scalefont setfont\n");
 //fprintf(o,"/adobe-italic findfont 1 scalefont setfont\n");
 M(-4.7, 4.5) fprintf(o,"(y)s\n");
 M( 4.6,-4.8) fprintf(o,"(x)s\n");
 M(-4,0)L(4.1,0) M(0,-4)L(0,4.1) fprintf(o,".01 W 1 0 1 RGB S\n");
*/
 
//z_type tm,tp,F[M1*N1];;
 DO(m,M1)DO(n,N1){ g[m*N1+n]=9999;
  f[m*N1+n]=9999;}
DB b=sqrt(2);
 DO(m,M1){x=X[m]; printf("x=%6.3f\n",x);
 DO(n,N1){y=Y[n]; z=z_type(x,y);
if(abs(z+2.)>.04)
{
c=F21E(z);
//c=E1ETF(z);
                // c=F15(z);
  p=Re(c); q=Im(c);
if(p>-9999 && p<9999) g[m*N1+n]=p;
  if(q>-9999 && q<9999 && fabs(q)>1.e-14) f[m*N1+n]=q;
}
  }}
fprintf(o,"1 setlinejoin 2 setlinecap\n");
p=2.; q=1.1;;
#include "plofu.cin"

fprintf(o,"0 setlinejoin 0 setlinecap\n");
// p=1.e-15;
// for(n=-10;n<11;n++) {q=p*n; z=z_type(q,0.); printf("%19.15f %19.15f\n",q, Re(TQ2E(z)) ); }
//y=2*M_PI/log(2.);
// y=M_PI/log(log(2));
//y=9.064720284;

// M(-2,0)L(-10.1, 0) fprintf(o,"0.05 W 1 1 1 RGB S\n");
//
// DO(n,20){ M(-2.-.4*n,0)L(-2-.4*(n+.5),0) }
// fprintf(o,".11 W 0 0 0 RGB S\n");

//M(-2,0)L(-4.1, 0) fprintf(o,".1 W 0 0 0 RGB [.14 .14] 0 setdash S\n"); //fails at some printers

//M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n");
 fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
  system("epstopdf e1e09q2.eps");
  system( "open e1e09q2.pdf");
getchar(); system("killall Preview"); // For macintosh
}

Latex generator of labels

References

  1. ↑ http://tori.ils.uec.ac.jp/PAPERS/2011e1e.pdf H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). (Mathematics of Computation, 2011, in press)

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current17:50, 20 June 2013Thumbnail for version as of 17:50, 20 June 20132,234 × 711 (883 KB)Maintenance script (talk | contribs)Importing image file
  • You cannot overwrite this file.

The following page links to this file:

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it. If the file has been modified from its original state, some details may not fully reflect the modified file.
Retrieved from "https://mizugadro.mydns.jp/t/index.php?title=File:E1efig09abc1a150.png&oldid=18888"