Difference between revisions of "File:Susinmap8t.jpg"

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[[Complex map]] of the primary approximation of function [[SuSin]] with parameter $M=8$ in the expansion.
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Importing image file
  
  +
  +
$u+\mathrm i v=F_M(x\!+\!x_1 +\mathrm i y)$
  +
  +
where $x_1$ is solution of equation $F(1)=1$.
  +
  +
Function $F_M$ is constructed wish
  +
  +
$\displaystyle F_M(z)=\sqrt{\frac{3}{z}} \left(1+\frac{3 \ln(z)}{10~ z} + \sum_{m=2}^M \frac{P_m(\ln(z))}{z^m} \right)$
  +
  +
$\displaystyle P_m(L)=\sum_{n=0}^m A_{m,n} L^n$
  +
  +
Coefficients $A$ are determined with substitution of
  +
$f(z)=F_M(z)+O\Big(\ln(z)^{M+1} z^{-M-1.5} \Big)$
  +
  +
into equation $~f(z\!+\!1)=\sin(f(z))~$ and the asymptotic analysis at large $z$. Then,
  +
  +
$\displaystyle F(z)=\lim_{k\rightarrow \infty} \arcsin^k(F_M(z\!+\!k))$
  +
  +
The larfer is $M$, the faster is convergence of the limit.
  +
  +
==[[C++]] generator of curves==
  +
// Files [[ado.cin]], [[conto.cin]] and [[susin.cin]] should be loaded to the working directory in order to compile the [[C++]] code below
  +
<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;
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
#include "conto.cin"
  +
// #include"sutran.cin"
  +
  +
//z_type arcsin(z_type z){ return -I* log(I*z + sqrt(1.-z*z) ); }
  +
  +
z_type arcsin(z_type z){
  +
if(Im(z)<0){if(Re(z)>=0){return M_PI/2.-I*log( z + sqrt(z*z-1.) );}
  +
else {return M_PI/2.-I*log( z - sqrt(z*z-1.) );}}
  +
if(Re(z)>=0){return M_PI/2.+I*log( z + sqrt(z*z-1.) );}
  +
else {return M_PI/2.+I*log( z - sqrt(z*z-1.) );} }
  +
  +
// z_type su(z_type z) { int n,m=18.; z_type c,d; c=z+(0.+m); d=sqrt(3./c);
  +
//DO(n,m) d=arcsin(d); return d; }
  +
  +
#include"susin.cin"
  +
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  +
DB e; DB x0=.4;
  +
DO(n,14) { e=Re( susin(x0+z_type(1.,1.e-9)) )-1.; x0+= 4.6*e; printf("%2d %19.16f %19.16f\n",n,x0,e); }
  +
// getchar();
  +
  +
int M=1001,M1=M+1;
  +
int N=1001,N1=N+1;
  +
DB X[M1],Y[N1];
  +
DB *g, *f, *w; // w is working array.
  +
g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
  +
f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
  +
w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
  +
char v[M1*N1]; // v is working array
  +
FILE *o;o=fopen("susinmap8.eps","w"); ado(o,2002,2002);
  +
fprintf(o,"1001 1001 translate\n 100 100 scale\n");
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n");
  +
e=10./sinh(1.);
  +
DO(m,M1) { t=(m-500.5)/500.; X[m]=e*sinh(1.*t);}
  +
e=10./sinh(3.);
  +
DO(n,N1) { t=(n-500.5)/500.; Y[n]=e*sinh(3.*t);}
  +
for(m=-10;m<11;m+=1){M(m,-10) L(m,10) }
  +
for(n=-10;n<11;n+=1){M(-10,n) L(10,n)}
  +
fprintf(o,".006 W 0 0 0 RGB S\n");
  +
DO(m,M1)DO(n,N1){ g[m*N1+n]=999;
  +
f[m*N1+n]=999;}
  +
DO(m,M1){x=X[m]; if(m/10*10==m) printf("x=%6.3f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y); //if(abs(z+2.)>.019)
  +
// c=arcsin(z);
  +
// c=sqrt(3./z);
  +
// c=susin(z);
  +
c=susin0(z+1.4304553465285);
  +
// d=arcsin(su0(z+1.));
  +
// p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.); if(p>0 && p<17) g[m*N1+n]=p;
  +
p=Re(c); q=Im(c); if(p>-19 && p<19 && q>-19 && y<19){ g[m*N1+n]=p;f[m*N1+n]=q;}
  +
}}
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n");
  +
p=.06;q=.06;
  +
/* p=1.;
  +
conto(o,g,w,v,X,Y,M,N,(15.3 ),-p,p); fprintf(o,".1 W .4 1 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,(15. ),-p,p); fprintf(o,".2 W 1 0 1 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,(14.7 ),-p,p); fprintf(o,".1 W 1 .5 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,(14. ),-p,p); fprintf(o,"2 W .2 .2 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,(13. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,(12. ),-p,p); fprintf(o,"6 W 0 0 1 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,(11. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,(10. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (9. ),-p,p); fprintf(o,"6 W 0 1 1 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (8. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (7. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (6. ),-p,p); fprintf(o,"6 W 0 .5 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (5. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (4. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (3. ),-p,p); fprintf(o,"6 W 1 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (2. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, (1. ),-p,p); fprintf(o,"4 W .5 0 0 RGB S\n");
  +
*/
  +
for(m=-8;m<8;m++)for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".016 W 0 .6 0 RGB S\n");
  +
for(m=0;m<8;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".016 W .9 0 0 RGB S\n");
  +
for(m=0;m<8;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".016 W 0 0 .9 RGB S\n");
  +
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".03 W .8 0 0 RGB S\n");
  +
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".03 W 0 0 .8 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".03 W .5 0 .5 RGB S\n");
  +
for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n");
  +
fprintf(o,"0 setlinejoin 0 setlinecap\n");
  +
M(-10,0) L(-1.4304553465285,0) fprintf(o,"1 1 1 RGB .01 W S\n");
  +
DO(n,45) {x=-1.4304553465285-.2*n; M(x-.01,0) L(x-.09,0) } fprintf(o,"0 0 0 RGB .03 W S\n");
  +
//#include "plofu.cin"
  +
  +
fprintf(o,"showpage\n");
  +
fprintf(o,"%c%cTrailer\n",'%','%');
  +
fclose(o); free(f); free(g); free(w);
  +
system("epstopdf susinmap8.eps");
  +
system( "open susinmap8.pdf"); //for macintosh
  +
getchar(); system("killall Preview"); // For macintosh
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\usepackage{geometry}
  +
\usepackage{graphics}
  +
\usepackage{rotating}
  +
\paperwidth 2078pt
  +
\paperheight 2064pt
  +
\topmargin -100pt
  +
\oddsidemargin -72pt
  +
\textwidth 2200pt
  +
\textheight 2200pt
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\pagestyle{empty}
  +
\begin{document}
  +
\begin{picture}(2016,2044)
  +
\put(48,40){\includegraphics{susinmap8}}
  +
%\put(48,40){\includegraphics{susinmap1}}
  +
\put(4,2018){\sx{5.4}{$y$}}
  +
\put(4,1824){\sx{5}{$8$}}
  +
\put(4,1624){\sx{5}{$6$}}
  +
\put(4,1424){\sx{5}{$4$}}
  +
\put(4,1224){\sx{5}{$2$}}
  +
\put(4,1024){\sx{5}{$0$}}
  +
\put(-28,824){\sx{5}{$-2$}}
  +
\put(-28,624){\sx{5}{$-4$}}
  +
\put(-28,424){\sx{5}{$-6$}}
  +
\put(-28,224){\sx{5}{$-8$}}
  +
\put(-8,-9){\sx{5}{$-10$}}
  +
\put(200,-8){\sx{5}{$-8$}}
  +
\put(400,-8){\sx{5}{$-6$}}
  +
\put(600,-8){\sx{5}{$-4$}}
  +
\put(800,-8){\sx{5}{$-2$}}
  +
\put(1040,-8){\sx{5}{$0$}}
  +
\put(1241,-8){\sx{5}{$2$}}
  +
\put(1442,-8){\sx{5}{$4$}}
  +
\put(1643,-8){\sx{5}{$6$}}
  +
\put(1843,-8){\sx{5}{$8$}}
  +
\put(2028,-8){\sx{5.2}{$x$}}
  +
  +
\put(1436,930){\sx{7}{\rot{89}$u\!=\!0.7$\ero}}
  +
\put(1640,930){\sx{7}{\rot{89}$u\!=\!0.6$\ero}}
  +
\put(1980,930){\sx{7}{\rot{89}$u\!=\!0.5$\ero}}
  +
  +
\put(1520,1660){\sx{7}{\rot{68}$v\!=\!-0.2$\ero}}
  +
\put(1790,1406){\sx{7}{\rot{36}$v\!=\!-0.1$\ero}}
  +
  +
\put(250,1030){\sx{5.2}{\bf cut}}
  +
  +
\put(928,1148){\sx{7}{\rot{0.}$u\!=\!1$\ero}}
  +
%\put(668,1080){\sx{7}{\rot{0.}$v\!=\!-1$\ero}}
  +
%\put(688,960){\sx{7}{\rot{0.}$v\!=\!1$\ero}}
  +
  +
\put(1680,1024){\sx{7}{$v\!=\!0$}}
  +
\put(1740,664){\sx{7}{\rot{-35}$v\!=\!0.1$\ero}}
  +
\put(1478,418){\sx{7}{\rot{-68}$v\!=\!0.2$\ero}}
  +
\end{picture}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
  +
[[Category:Approximation]]
  +
[[Category:Complex map]]
  +
[[Category:Sin]]
  +
[[Category:Superfunction]]
  +
[[Category:Super sin]]
  +
[[Category:SuSin]]
  +
[[Category:C++]]
  +
[[Category:Latex]]
  +
[[Category:Generator]]
  +
[[Category:Book]]

Latest revision as of 08:53, 1 December 2018

Complex map of the primary approximation of function SuSin with parameter $M=8$ in the expansion.

$u+\mathrm i v=F_M(x\!+\!x_1 +\mathrm i y)$

where $x_1$ is solution of equation $F(1)=1$.

Function $F_M$ is constructed wish

$\displaystyle F_M(z)=\sqrt{\frac{3}{z}} \left(1+\frac{3 \ln(z)}{10~ z} + \sum_{m=2}^M \frac{P_m(\ln(z))}{z^m} \right)$

$\displaystyle P_m(L)=\sum_{n=0}^m A_{m,n} L^n$

Coefficients $A$ are determined with substitution of $f(z)=F_M(z)+O\Big(\ln(z)^{M+1} z^{-M-1.5} \Big)$

into equation $~f(z\!+\!1)=\sin(f(z))~$ and the asymptotic analysis at large $z$. Then,

$\displaystyle F(z)=\lim_{k\rightarrow \infty} \arcsin^k(F_M(z\!+\!k))$

The larfer is $M$, the faster is convergence of the limit.

C++ generator of curves

// Files ado.cin, conto.cin and susin.cin should be loaded to the working directory in order to compile the C++ code below


#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;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
// #include"sutran.cin"

//z_type arcsin(z_type z){ return -I* log(I*z + sqrt(1.-z*z) ); }

z_type arcsin(z_type z){
if(Im(z)<0){if(Re(z)>=0){return M_PI/2.-I*log( z + sqrt(z*z-1.) );}
                   else {return M_PI/2.-I*log( z - sqrt(z*z-1.) );}}
           if(Re(z)>=0){return M_PI/2.+I*log( z + sqrt(z*z-1.) );}
                  else {return M_PI/2.+I*log( z - sqrt(z*z-1.) );} }

// z_type su(z_type z) { int n,m=18.; z_type c,d; c=z+(0.+m); d=sqrt(3./c);
//DO(n,m) d=arcsin(d); return d; }

#include"susin.cin"

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
DB e; DB x0=.4;
DO(n,14) { e=Re( susin(x0+z_type(1.,1.e-9)) )-1.; x0+= 4.6*e; printf("%2d %19.16f %19.16f\n",n,x0,e); }
// getchar();

int M=1001,M1=M+1;
int N=1001,N1=N+1;
DB X[M1],Y[N1];
DB *g, *f, *w; // w is working array.
g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
char v[M1*N1]; // v is working array
FILE *o;o=fopen("susinmap8.eps","w"); ado(o,2002,2002);
fprintf(o,"1001 1001 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
e=10./sinh(1.);
DO(m,M1) { t=(m-500.5)/500.; X[m]=e*sinh(1.*t);}
e=10./sinh(3.);
DO(n,N1) { t=(n-500.5)/500.; Y[n]=e*sinh(3.*t);}
for(m=-10;m<11;m+=1){M(m,-10) L(m,10) }
for(n=-10;n<11;n+=1){M(-10,n) L(10,n)}
 fprintf(o,".006 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){ g[m*N1+n]=999;
                       f[m*N1+n]=999;}
DO(m,M1){x=X[m]; if(m/10*10==m) printf("x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y); //if(abs(z+2.)>.019)
// c=arcsin(z);
// c=sqrt(3./z);
// c=susin(z);
   c=susin0(z+1.4304553465285);
// d=arcsin(su0(z+1.));
// p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.); if(p>0 && p<17) g[m*N1+n]=p;
 p=Re(c); q=Im(c); if(p>-19 && p<19 && q>-19 && y<19){ g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 2 setlinecap\n");
 p=.06;q=.06;
/* p=1.;
 conto(o,g,w,v,X,Y,M,N,(15.3 ),-p,p); fprintf(o,".1 W .4 1 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(15. ),-p,p); fprintf(o,".2 W 1 0 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14.7 ),-p,p); fprintf(o,".1 W 1 .5 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14. ),-p,p); fprintf(o,"2 W .2 .2 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(13. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(12. ),-p,p); fprintf(o,"6 W 0 0 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(11. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(10. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (9. ),-p,p); fprintf(o,"6 W 0 1 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (8. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (7. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (6. ),-p,p); fprintf(o,"6 W 0 .5 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (5. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (4. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (3. ),-p,p); fprintf(o,"6 W 1 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (2. ),-p,p); fprintf(o,"2 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (1. ),-p,p); fprintf(o,"4 W .5 0 0 RGB S\n");
*/
for(m=-8;m<8;m++)for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".016 W 0 .6 0 RGB S\n");
for(m=0;m<8;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".016 W .9 0 0 RGB S\n");
for(m=0;m<8;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".016 W 0 0 .9 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".03 W .8 0 0 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".03 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".03 W .5 0 .5 RGB S\n");
for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n");
fprintf(o,"0 setlinejoin 0 setlinecap\n");
M(-10,0) L(-1.4304553465285,0) fprintf(o,"1 1 1 RGB .01 W S\n");
DO(n,45) {x=-1.4304553465285-.2*n; M(x-.01,0) L(x-.09,0) } fprintf(o,"0 0 0 RGB .03 W S\n");
//#include "plofu.cin"

fprintf(o,"showpage\n");
fprintf(o,"%c%cTrailer\n",'%','%');
fclose(o); free(f); free(g); free(w);
      system("epstopdf susinmap8.eps");
      system( "open susinmap8.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

Latex generator of labels


\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphics}
\usepackage{rotating}
\paperwidth 2078pt
\paperheight 2064pt
\topmargin -100pt
\oddsidemargin -72pt
\textwidth 2200pt
\textheight 2200pt
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\pagestyle{empty}
\begin{document}
\begin{picture}(2016,2044)
\put(48,40){\includegraphics{susinmap8}}
%\put(48,40){\includegraphics{susinmap1}}
\put(4,2018){\sx{5.4}{$y$}}
\put(4,1824){\sx{5}{$8$}}
\put(4,1624){\sx{5}{$6$}}
\put(4,1424){\sx{5}{$4$}}
\put(4,1224){\sx{5}{$2$}}
\put(4,1024){\sx{5}{$0$}}
\put(-28,824){\sx{5}{$-2$}}
\put(-28,624){\sx{5}{$-4$}}
\put(-28,424){\sx{5}{$-6$}}
\put(-28,224){\sx{5}{$-8$}}
\put(-8,-9){\sx{5}{$-10$}}
\put(200,-8){\sx{5}{$-8$}}
\put(400,-8){\sx{5}{$-6$}}
\put(600,-8){\sx{5}{$-4$}}
\put(800,-8){\sx{5}{$-2$}}
\put(1040,-8){\sx{5}{$0$}}
\put(1241,-8){\sx{5}{$2$}}
\put(1442,-8){\sx{5}{$4$}}
\put(1643,-8){\sx{5}{$6$}}
\put(1843,-8){\sx{5}{$8$}}
\put(2028,-8){\sx{5.2}{$x$}}

\put(1436,930){\sx{7}{\rot{89}$u\!=\!0.7$\ero}}
\put(1640,930){\sx{7}{\rot{89}$u\!=\!0.6$\ero}}
\put(1980,930){\sx{7}{\rot{89}$u\!=\!0.5$\ero}}

\put(1520,1660){\sx{7}{\rot{68}$v\!=\!-0.2$\ero}}
\put(1790,1406){\sx{7}{\rot{36}$v\!=\!-0.1$\ero}}

\put(250,1030){\sx{5.2}{\bf cut}}

\put(928,1148){\sx{7}{\rot{0.}$u\!=\!1$\ero}}
%\put(668,1080){\sx{7}{\rot{0.}$v\!=\!-1$\ero}}
%\put(688,960){\sx{7}{\rot{0.}$v\!=\!1$\ero}}

\put(1680,1024){\sx{7}{$v\!=\!0$}}
\put(1740,664){\sx{7}{\rot{-35}$v\!=\!0.1$\ero}}
\put(1478,418){\sx{7}{\rot{-68}$v\!=\!0.2$\ero}}
\end{picture}
\end{document}

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