File:Boyt.jpg
Approximation of the shape of the sledge runner by [1] with the hundredth iteration of sin [2];
$y=\sin^n(\pi/2)-\sin^n(x)$
with single adjusting parameter $n\!=\!100$. Super sin and the Abel sin functions, id est, SuSin and AuSin, are used to evaluate the high iteration of sin.
C++ generator of curve
#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 "ado.cin"
#include "arcsin.cin"
#include "susin.cin"
#include "ausin.cin"
DB sinni(int n, DB x){ DB s=x; int m; DO(m,n)s=sin(s); return s;}
z_type sinn(z_type n, z_type z){return susin(n+ausin(z));}
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
FILE *o;o=fopen("boy.eps","w"); ado(o,318,30);
#define M(x,y) {fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);}
#define L(x,y) {fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);}
fprintf(o,"1 1 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
for(m=0;m<4;m++){M(m,0) L(m,.2) }
for(n=0;n<3;n++){M( 0,.1*n) L(M_PI,.1*n)}
fprintf(o,".004 W 0 0 0 RGB S\n");
M(M_PI/2.,0); L(M_PI/2.,.2)
M(M_PI,0); L(M_PI,.2)
fprintf(o,".003 W 0 0 0 RGB S\n");
n=100;
p=sinni(100,M_PI/2);
M(0,p) DO(m,315){x=.01*(m+.1);y=sin(x); y=p-sinni(100,x); L(x,y); } fprintf(o,".01 W 0 1 0 RGB S\n");
fprintf(o,"showpage\n");
fprintf(o,"%c%cTrailer\n",'%','%');
fclose(o);
system("epstopdf boy.eps");
system( "open boy.pdf"); //for macintosh
getchar(); system("killall Preview"); // For macintosh
}
Latex generator of labels
File boy05.pdf should be loaded from http://mizugadro.mydns.jp/t/index.php/File:Boy04.jpg to the working directory in order to compile the Latex document below]]
\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphics}
\usepackage{rotating}
%\paperwidth 3230pt
%\paperheight 1700pt
\paperwidth 2460pt
\paperheight 1180pt
\topmargin -100pt
\oddsidemargin -72pt
\textwidth 3200pt
\textheight 1700pt
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\pagestyle{empty}
\begin{document}
\hskip 140pt
\sx{10}{\begin{picture}(248,116)
%\put(4,9){\includegraphics{sinplo1}}
%\put(0,0){\sx{1.6}{\includegraphics{Boy_on_snow_sled,_1945}}}
\put(-16,0){\rot{1}\sx{.14}{\includegraphics{boy05}}\ero}
\put(14,9){\includegraphics{boy}}
\put(3,28){\sx{1.}{$y$}}
\put(0,16){\sx{1.}{$0.1$}}
\put(0, 06){\sx{1.}{$0$}}
%\put(-7, 06){\sx{1.2}{$-2$}}
\put(13,0){\sx{1.}{$0$}}
\put(113,0){\sx{1.}{$1$}}
\put(163,0){\sx{1.}{$\pi/2$}}
\put(213,0){\sx{1.}{$2$}}
%\put(313,0){\sx{1.}{$3$}}
\put(224,0){\sx{1.}{$x$}}
\end{picture}}
\end{document}
References
- ↑ http://en.wikipedia.org/wiki/File:Boy_on_snow_sled,_1945.jpg Father of JGKlein. Boy on snow sled, 1945.
- ↑ http://mizugadro.mydns.jp/PAPERS/2014susin.pdf Dmitrii Kouznetsov. Super sin. December 17, 2013. Iterates of function sin are considered. The superfunction SuSin is constructed as holomorphic solution of the transfer equation sin(SuSin(z))=SuSin(z+1). The Abel function AuSin is constructed as solution of the Abel equation AuSin(sin(z))=AuSin(z)+1; in wide range of values z, the rela- tion SuSin(AuSin(z))=z holds. Iteration of sin is expressed with sinˆn(z)=SuSin(n+AuSin(z)), where the number n of iteration has no need to be integer. ..
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|---|---|---|---|---|---|
| current | 06:10, 1 December 2018 | | 5,105 × 2,449 (1.17 MB) | Maintenance script (talk | contribs) | Importing image file |
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