Difference between revisions of "File:QFacMapT.jpg"
(Importing image file) |
|||
| Line 1: | Line 1: | ||
| + | [[Complex map]] of [[square root of factorial]], id est, half iterate of [[factorial]]: |
||
| − | Importing image file |
||
| + | |||
| + | $u\!+\!\mathrm i v\!=\!\mathrm{Factorial}^{1/2}(x\!+\!\mathrm i y)= |
||
| + | \mathrm{SuFac}\left( \frac{1}{2} + \mathrm{AuFac}(x\!+\!\mathrm i y)\right)$ |
||
| + | |||
| + | ==[[C++]] generator of map== |
||
| + | |||
| + | [[ado.cin]], [[conto.cin]], [[fac.cin]], [[facp.cin]], [[afacc.cin]], [[superfactorial.cin]], |
||
| + | [[abelfac.cin]] should be loaded. |
||
| + | |||
| + | <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 "fac.cin" |
||
| + | //#include "sinc.cin" |
||
| + | #include "facp.cin" |
||
| + | #include "afacc.cin" |
||
| + | #include "superfactorial.cin" |
||
| + | #include "abelfac.cin" |
||
| + | #include "conto.cin" |
||
| + | |||
| + | int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; |
||
| + | int M=401,M1=M+1; |
||
| + | int N=401,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("QFacMap.eps","w");ado(o,402,402); |
||
| + | fprintf(o,"201 201 translate\n 20 20 scale\n"); |
||
| + | DO(m,M1)X[m]=-8.04+.04*(m+.5); |
||
| + | // DO(m,M1){t=-1.+.022*m; X[m]=.2+t-1.11*exp(-1.9*t);} |
||
| + | DO(n,N1)Y[n]=-8.04+.04*(n+.5); |
||
| + | // DO(n,N1){t=-8.04+.04*(n+.5); t*=.97; Y[n]=t-.25*sin(0.6127874523307*t);} |
||
| + | for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}} |
||
| + | for(n=-8;n<9;n++){ M( -8,n)L(8,n)} |
||
| + | fprintf(o,".008 W 0 0 0 RGB S\n"); |
||
| + | DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;} |
||
| + | DO(m,M1){x=X[m]; //printf("%5.2f\n",x); |
||
| + | DO(n,N1){y=Y[n]; z=z_type(x,y); |
||
| + | // c=afacc(z); |
||
| + | // c=fac(z); |
||
| + | c=abelfac(z); |
||
| + | c=superfac(.5+c); |
||
| + | // p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.)-1.; |
||
| + | p=Re(c);q=Im(c); |
||
| + | if(p>-60 && p<60 && |
||
| + | // (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) && |
||
| + | q>-60 && q<60 && fabs(q)> 1.e-16 |
||
| + | ) |
||
| + | {g[m*N1+n]=p;f[m*N1+n]=q;} |
||
| + | }} |
||
| + | //fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1.8;q=.7; |
||
| + | fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.4;q=.8; |
||
| + | for(m=-4;m<4;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".01 W 0 .5 0 RGB S\n"); |
||
| + | for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".01 W .8 0 0 RGB S\n"); |
||
| + | for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".01 W 0 0 .8 RGB S\n"); |
||
| + | for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".04 W .8 0 0 RGB S\n"); |
||
| + | for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 .8 RGB S\n"); |
||
| + | conto(o,f,w,v,X,Y,M,N, (0. ),-9,9); fprintf(o,".04 W .5 0 .5 RGB S\n"); |
||
| + | for(m=-8;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n"); |
||
| + | m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-9,9); fprintf(o,".04 W 0 0 0 RGB S\n"); |
||
| + | for(m=1;m<9;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n"); |
||
| + | //#include"plofu.cin" |
||
| + | // x=0.8856031944; |
||
| + | // conto(o,g,w,v,X,Y,M,N,0.8856031944,-p,p); fprintf(o,".004 W .2 .2 0 RGB S\n"); |
||
| + | M(1,0)L(-8.1,0) fprintf(o,"0 setlinejoin 0 setlinecap .05 W 1 1 1 RGB S\n"); |
||
| + | DO(m,23){ M(1-.4*(m+.2),0)L(1-.4*(m+.4),0);} fprintf(o,".09 W 1 .4 0 RGB S\n"); |
||
| + | DO(m,23){ M(1-.4*(m+.7),0)L(1-.4*(m+.9),0);} fprintf(o,".09 W 0 .4 1 RGB S\n"); |
||
| + | //M(x,0)L(-8.1,0) fprintf(o,"[.19 .21]0 setdash .05 W 0 0 0 RGB S\n"); |
||
| + | // May it be, that, some printers do not interpret well the dashing ? |
||
| + | fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); |
||
| + | system("epstopdf QFacMap.eps"); |
||
| + | system( "open QFacMap.pdf"); //for LINUX |
||
| + | // getchar(); system("killall Preview");//for mac |
||
| + | } |
||
| + | </nowiki></nomathjax></poem> |
||
| + | |||
| + | ==[[Latex]] generator of labels== |
||
| + | <poem><nomathjax><nowiki> |
||
| + | \documentclass[12pt,a4paper,oneside]{book} |
||
| + | %\newcommand \EN[1] {{#1}} % make the English version |
||
| + | \newcommand \EN[1] {{}} % suppress the English version |
||
| + | \newcommand \RU[1] {{#1}} % make the Russian version (in this document not supported) |
||
| + | %\newcommand \RU[1] {{}} % suppress the Russian version |
||
| + | %The Japanese version is not yet supported. While \JP is used to suppress several lines at once. |
||
| + | \newcommand \JP[1] {{}} % ореsuppress some text |
||
| + | %\usepackage[space]{cite }% If exist. |
||
| + | \usepackage[utf8]{inputenc} |
||
| + | \usepackage[T2A]{fontenc} |
||
| + | \usepackage[russian]{babel} |
||
| + | \usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx} |
||
| + | |||
| + | \usepackage{rotating} |
||
| + | \usepackage{hyperref} |
||
| + | \usepackage{wrapfig} |
||
| + | \usepackage{geometry} |
||
| + | %\renewcommand\baselinestretch{.94} |
||
| + | %\renewcommand\baselinestretch{.94} |
||
| + | %\renewcommand\baselinestretch{1.15} |
||
| + | %\renewcommand\baselinestretch{1.2} |
||
| + | %\renewcommand\baselinestretch{1.2} |
||
| + | |||
| + | \paperwidth150mm |
||
| + | %\paperheight 257mm |
||
| + | %\paperheight 216mm |
||
| + | \paperheight 148mm |
||
| + | %\textwidth 175mm |
||
| + | \textwidth 150mm |
||
| + | %\oddsidemargin 5mm |
||
| + | \oddsidemargin -72pt |
||
| + | %\topmargin -17mm |
||
| + | \topmargin -103pt |
||
| + | \textheight 257mm |
||
| + | \pagestyle{empty} |
||
| + | \large |
||
| + | |||
| + | \usepackage{color} |
||
| + | \definecolor{rose}{rgb}{1,.9,1} |
||
| + | \renewcommand\thesection{\arabic{section}} |
||
| + | \renewcommand\figurename{Рис.} |
||
| + | %\renewcommand\refname{References} |
||
| + | %\renewcommand\figurename{Fig.} |
||
| + | \newcommand \be {\begin{eqnarray}} |
||
| + | \newcommand \ee {\end{eqnarray} } |
||
| + | \newcommand \sx {\scalebox} |
||
| + | \newcommand \rme {{\rm e}} %%makes the base of natural logarithms Roman font |
||
| + | %\newcommand \rme {{e}} %%makes the base of natural logarithms Italics font; choose one of these |
||
| + | \newcommand \rmi {{\rm i}} %%imaginary unity is always roman font |
||
| + | \newcommand \ds {\displaystyle} |
||
| + | \newcommand \bN {\mathbb{N}} |
||
| + | \newcommand \bC {\mathbb{C}} |
||
| + | \newcommand \bR {\mathbb{R}} |
||
| + | \newcommand \cO {\mathcal{O}} |
||
| + | \newcommand \cF {\mathcal{F}} |
||
| + | \newcommand \rot {\begin{rotate}} |
||
| + | \newcommand \ero {\end{rotate}} |
||
| + | \newcommand \nS {\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!} |
||
| + | \newcommand \pS {{~}~{~}} |
||
| + | \newcommand \fac {\mathrm{Factorial}} |
||
| + | \newcommand {\rf}[1] {(\ref{#1})} |
||
| + | \newcommand{\iL}[1] {~\label{#1}\pS \rm[#1]\nS} %make the labels visible |
||
| + | %\newcommand{\iL}[1] {\label{#1}} %make the labels invisible; choose one of these options |
||
| + | \newcommand \eL[1] {\iL{#1}\ee} |
||
| + | \newcommand \ing \includegraphics |
||
| + | \newcommand \tet {\mathrm{tet}} |
||
| + | \parindent 0pt |
||
| + | \begin{document} |
||
| + | \hskip -28pt |
||
| + | %{\sx{.72}{\begin{picture}(362,330) |
||
| + | {\sx{1.25}{\begin{picture}(362,363) |
||
| + | \put(0,0){\ing{QFacMap}} |
||
| + | \put(30,357){\sx{1.3}{$y$}} |
||
| + | \put(30,317){\sx{1.3}{$6$}} |
||
| + | \put(30,277){\sx{1.3}{$4$}} |
||
| + | \put(30,237){\sx{1.3}{$2$}} |
||
| + | \put(29,196){\sx{1.3}{$0$}} |
||
| + | \put(20,156){\sx{1.3}{$-2$}} |
||
| + | \put(20,116){\sx{1.3}{$-4$}} |
||
| + | \put(20,76){\sx{1.3}{$-6$}} |
||
| + | \put(20,36){\sx{1.2}{$-8$}} |
||
| + | \put(70,30){\sx{1.1}{$-6$}} |
||
| + | \put(111,30){\sx{1.1}{$-4$}} |
||
| + | \put(151,30){\sx{1.1}{$-2$}} |
||
| + | \put(199,30){\sx{1.1}{$0$}} |
||
| + | \put(239,30){\sx{1.1}{$2$}} |
||
| + | \put(279,30){\sx{1.1}{$4$}} |
||
| + | \put(319,30){\sx{1.1}{$6$}} |
||
| + | \put(355,30){\sx{1.1}{$x$}} |
||
| + | \put(70,198){\sx{1.3}{\bf cut}} %central |
||
| + | \put(47,162){\sx{1.3}{\rot{72}$u\!=\!0$\ero}} |
||
| + | \put(99,161){\sx{1.3}{\rot{80}$v\!=\!0$\ero}} |
||
| + | \put(139,162){\sx{1.3}{\rot{89}$u\!=\!0$\ero}} |
||
| + | \put(164,204){\sx{1.3}{\rot{77}$v\!=\!0$\ero}} |
||
| + | \put(177,200){\sx{1.2}{\rot{56}$u\!=\!0$\ero}} |
||
| + | \put(82,116){\sx{1.1}{\rot{0.}$u\!=\!0.4$\ero}} |
||
| + | \put(86,102){\sx{1.1}{\rot{0.}$u\!=\!0.6$\ero}} |
||
| + | \put(137,128){\sx{1.1}{\rot{0.}$v\!=\!0.4$\ero}} |
||
| + | \put(332,252){\sx{1.3}{\rot{23}$u\!=\!8$\ero}} |
||
| + | \put(331,220){\sx{1.3}{\rot{-12}$v\!=\!8$\ero}} |
||
| + | \put(321,174){\sx{1.3}{\rot{9}$v\!=\!-8$\ero}} |
||
| + | \put(319,102){\sx{1.3}{\rot{-7}$u\!=\!-8$\ero}} |
||
| + | \end{picture}}} |
||
| + | \end{document} |
||
| + | |||
| + | </nowiki></nomathjax></poem> |
||
| + | |||
| + | ==References== |
||
| + | <references/> |
||
| + | |||
| + | [[Category:Abel function]] |
||
| + | [[Category:AuFac]] |
||
| + | [[Category:Book]] |
||
| + | [[Category:BookMap]] |
||
| + | [[Category:C++]] |
||
| + | [[Category:Complex map]] |
||
| + | [[Category:Factorial]] |
||
| + | [[Category:Iterate]] |
||
| + | [[Category:Latex]] |
||
| + | [[Category:SuFac]] |
||
| + | [[Category:Superfunction]] |
||
Latest revision as of 08:48, 1 December 2018
Complex map of square root of factorial, id est, half iterate of factorial:
$u\!+\!\mathrm i v\!=\!\mathrm{Factorial}^{1/2}(x\!+\!\mathrm i y)= \mathrm{SuFac}\left( \frac{1}{2} + \mathrm{AuFac}(x\!+\!\mathrm i y)\right)$
C++ generator of map
ado.cin, conto.cin, fac.cin, facp.cin, afacc.cin, superfactorial.cin, abelfac.cin should be loaded.
#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 "fac.cin"
//#include "sinc.cin"
#include "facp.cin"
#include "afacc.cin"
#include "superfactorial.cin"
#include "abelfac.cin"
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=401,M1=M+1;
int N=401,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("QFacMap.eps","w");ado(o,402,402);
fprintf(o,"201 201 translate\n 20 20 scale\n");
DO(m,M1)X[m]=-8.04+.04*(m+.5);
// DO(m,M1){t=-1.+.022*m; X[m]=.2+t-1.11*exp(-1.9*t);}
DO(n,N1)Y[n]=-8.04+.04*(n+.5);
// DO(n,N1){t=-8.04+.04*(n+.5); t*=.97; Y[n]=t-.25*sin(0.6127874523307*t);}
for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}}
for(n=-8;n<9;n++){ M( -8,n)L(8,n)}
fprintf(o,".008 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
// c=afacc(z);
// c=fac(z);
c=abelfac(z);
c=superfac(.5+c);
// p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.)-1.;
p=Re(c);q=Im(c);
if(p>-60 && p<60 &&
// (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) &&
q>-60 && q<60 && fabs(q)> 1.e-16
)
{g[m*N1+n]=p;f[m*N1+n]=q;}
}}
//fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1.8;q=.7;
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.4;q=.8;
for(m=-4;m<4;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".01 W 0 .5 0 RGB S\n");
for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".01 W .8 0 0 RGB S\n");
for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".01 W 0 0 .8 RGB S\n");
for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".04 W .8 0 0 RGB S\n");
for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 .8 RGB S\n");
conto(o,f,w,v,X,Y,M,N, (0. ),-9,9); fprintf(o,".04 W .5 0 .5 RGB S\n");
for(m=-8;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n");
m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-9,9); fprintf(o,".04 W 0 0 0 RGB S\n");
for(m=1;m<9;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n");
//#include"plofu.cin"
// x=0.8856031944;
// conto(o,g,w,v,X,Y,M,N,0.8856031944,-p,p); fprintf(o,".004 W .2 .2 0 RGB S\n");
M(1,0)L(-8.1,0) fprintf(o,"0 setlinejoin 0 setlinecap .05 W 1 1 1 RGB S\n");
DO(m,23){ M(1-.4*(m+.2),0)L(1-.4*(m+.4),0);} fprintf(o,".09 W 1 .4 0 RGB S\n");
DO(m,23){ M(1-.4*(m+.7),0)L(1-.4*(m+.9),0);} fprintf(o,".09 W 0 .4 1 RGB S\n");
//M(x,0)L(-8.1,0) fprintf(o,"[.19 .21]0 setdash .05 W 0 0 0 RGB S\n");
// May it be, that, some printers do not interpret well the dashing ?
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
system("epstopdf QFacMap.eps");
system( "open QFacMap.pdf"); //for LINUX
// getchar(); system("killall Preview");//for mac
}
Latex generator of labels
\documentclass[12pt,a4paper,oneside]{book}
%\newcommand \EN[1] {{#1}} % make the English version
\newcommand \EN[1] {{}} % suppress the English version
\newcommand \RU[1] {{#1}} % make the Russian version (in this document not supported)
%\newcommand \RU[1] {{}} % suppress the Russian version
%The Japanese version is not yet supported. While \JP is used to suppress several lines at once.
\newcommand \JP[1] {{}} % ореsuppress some text
%\usepackage[space]{cite }% If exist.
\usepackage[utf8]{inputenc}
\usepackage[T2A]{fontenc}
\usepackage[russian]{babel}
\usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx}
\usepackage{rotating}
\usepackage{hyperref}
\usepackage{wrapfig}
\usepackage{geometry}
%\renewcommand\baselinestretch{.94}
%\renewcommand\baselinestretch{.94}
%\renewcommand\baselinestretch{1.15}
%\renewcommand\baselinestretch{1.2}
%\renewcommand\baselinestretch{1.2}
\paperwidth150mm
%\paperheight 257mm
%\paperheight 216mm
\paperheight 148mm
%\textwidth 175mm
\textwidth 150mm
%\oddsidemargin 5mm
\oddsidemargin -72pt
%\topmargin -17mm
\topmargin -103pt
\textheight 257mm
\pagestyle{empty}
\large
\usepackage{color}
\definecolor{rose}{rgb}{1,.9,1}
\renewcommand\thesection{\arabic{section}}
\renewcommand\figurename{Рис.}
%\renewcommand\refname{References}
%\renewcommand\figurename{Fig.}
\newcommand \be {\begin{eqnarray}}
\newcommand \ee {\end{eqnarray} }
\newcommand \sx {\scalebox}
\newcommand \rme {{\rm e}} %%makes the base of natural logarithms Roman font
%\newcommand \rme {{e}} %%makes the base of natural logarithms Italics font; choose one of these
\newcommand \rmi {{\rm i}} %%imaginary unity is always roman font
\newcommand \ds {\displaystyle}
\newcommand \bN {\mathbb{N}}
\newcommand \bC {\mathbb{C}}
\newcommand \bR {\mathbb{R}}
\newcommand \cO {\mathcal{O}}
\newcommand \cF {\mathcal{F}}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \nS {\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}
\newcommand \pS {{~}~{~}}
\newcommand \fac {\mathrm{Factorial}}
\newcommand {\rf}[1] {(\ref{#1})}
\newcommand{\iL}[1] {~\label{#1}\pS \rm[#1]\nS} %make the labels visible
%\newcommand{\iL}[1] {\label{#1}} %make the labels invisible; choose one of these options
\newcommand \eL[1] {\iL{#1}\ee}
\newcommand \ing \includegraphics
\newcommand \tet {\mathrm{tet}}
\parindent 0pt
\begin{document}
\hskip -28pt
%{\sx{.72}{\begin{picture}(362,330)
{\sx{1.25}{\begin{picture}(362,363)
\put(0,0){\ing{QFacMap}}
\put(30,357){\sx{1.3}{$y$}}
\put(30,317){\sx{1.3}{$6$}}
\put(30,277){\sx{1.3}{$4$}}
\put(30,237){\sx{1.3}{$2$}}
\put(29,196){\sx{1.3}{$0$}}
\put(20,156){\sx{1.3}{$-2$}}
\put(20,116){\sx{1.3}{$-4$}}
\put(20,76){\sx{1.3}{$-6$}}
\put(20,36){\sx{1.2}{$-8$}}
\put(70,30){\sx{1.1}{$-6$}}
\put(111,30){\sx{1.1}{$-4$}}
\put(151,30){\sx{1.1}{$-2$}}
\put(199,30){\sx{1.1}{$0$}}
\put(239,30){\sx{1.1}{$2$}}
\put(279,30){\sx{1.1}{$4$}}
\put(319,30){\sx{1.1}{$6$}}
\put(355,30){\sx{1.1}{$x$}}
\put(70,198){\sx{1.3}{\bf cut}} %central
\put(47,162){\sx{1.3}{\rot{72}$u\!=\!0$\ero}}
\put(99,161){\sx{1.3}{\rot{80}$v\!=\!0$\ero}}
\put(139,162){\sx{1.3}{\rot{89}$u\!=\!0$\ero}}
\put(164,204){\sx{1.3}{\rot{77}$v\!=\!0$\ero}}
\put(177,200){\sx{1.2}{\rot{56}$u\!=\!0$\ero}}
\put(82,116){\sx{1.1}{\rot{0.}$u\!=\!0.4$\ero}}
\put(86,102){\sx{1.1}{\rot{0.}$u\!=\!0.6$\ero}}
\put(137,128){\sx{1.1}{\rot{0.}$v\!=\!0.4$\ero}}
\put(332,252){\sx{1.3}{\rot{23}$u\!=\!8$\ero}}
\put(331,220){\sx{1.3}{\rot{-12}$v\!=\!8$\ero}}
\put(321,174){\sx{1.3}{\rot{9}$v\!=\!-8$\ero}}
\put(319,102){\sx{1.3}{\rot{-7}$u\!=\!-8$\ero}}
\end{picture}}}
\end{document}
References
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 06:13, 1 December 2018 |  | 1,771 × 1,748 (936 KB) | Maintenance script (talk | contribs) | Importing image file |
- You cannot overwrite this file.
File usage
The following 2 pages link to this file:
- File:QQFacMapT.jpg (file redirect)
- File:QQFacMapT.jpg (file redirect)
