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| Line 1: |
Line 1: |
| − | <center><asy>
| + | ==Golden ratio== |
| − | int M=30;
| |
| − | real a = 0.07;
| |
| − | real a0 = 0.15;
| |
| − | real b = 0.02;
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| − | real c = 0.6;
| |
| − | real d = -0.2;
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| − | | |
| − | real x0 = -0.22;
| |
| − | real u = 0.2;
| |
| − | real v = 0.32;
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| − | | |
| − | draw ((-0.7,0)--(0.3,0),Arrow);
| |
| − | draw ((x0,-0.02)--(x0,1.2),Arrow);
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| − | | |
| − | label("$x$",(0.3,0),E);
| |
| − | label(rotate(90)*"$y$",(x0,1.2),N);
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| − | label("$x_0$",(x0,-0.02),S);
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| − | | |
| − | guide g1; guide g2; guide g3; guide g4; guide g5;
| |
| − | for (int k=floor(-0.7M); k<floor(0.3M); ++k) {
| |
| − | real x = k/M;
| |
| − | real z = 1+3*x^2;
| |
| − | real y1 = 1/(z-2a-a0)+2b*(1+c*x)+d;
| |
| − | real y2 = 1/(z-a-a0)+b*(1+c*x)+d;
| |
| − | real y3 = 1/(z-a0)+d;
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| − | real y4 = 1/(z+a-a0)-b*(1+c*x)+d;
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| − | real y5 = 1/(z+2a-a0)-2b*(1+c*x)+d;
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| − | g1=g1..(x,y1);
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| − | g2=g2..(x,y2);
| |
| − | g3=g3..(x,y3);
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| − | g4=g4..(x,y4);
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| − | g5=g5..(x,y5);
| |
| − | }
| |
| − | draw(g1,defaultpen+1);
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| − | draw(g2,defaultpen+1);
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| − | draw(g3,defaultpen+1);
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| − | draw(g4,defaultpen+1);
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| − | draw(g5,defaultpen+1);
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| − | | |
| − | real x = x0;
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| − | real z = 1+3*x^2;
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| − | real y1 = 1/(z-2a-a0)+2b*(1+c*x)+d;
| |
| − | real y2 = 1/(z-a-a0)+b*(1+c*x)+d;
| |
| − | real y3 = 1/(z-a0)+d;
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| − | real y4 = 1/(z+a-a0)-b*(1+c*x)+d;
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| − | real y5 = 1/(z+2a-a0)-2b*(1+c*x)+d;
| |
| − | path g = (x,y1)..(x-u,y2)..(x-v,y3)..(x-u,y4)..(x,y5);
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| − | draw( g );
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| − | | |
| − | pair w = (0.1,-0.6);
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| − | | |
| − | pair p = point(g,0.5);
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| − | dot ( p );
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| − | draw( p--p-0.5w, dashed );
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| − | label(rotate(90)*"$\Psi_{x_0}(y)$",p-0.5w,N);
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| − | | |
| − | draw( (x,y2)--(x-u,y2) );
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| − | draw( (x,y3)--(x-v,y3) );
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| − | draw( (x,y4)--(x-u,y4) );
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| − | | |
| − | draw( (x-u,y2+0.05)--(x-u,y4-0.05) );
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| − | draw( (x-v,y2+0.1)--(x-v,y4-0.1) );
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| − | | |
| − | real x = -0.15;
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| − | real z = 1+3*x^2;
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| − | real y4 = 1/(z+a-a0)-b*(1+c*x)+d;
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| − | dot( (x,y4) );
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| − | draw( (x,y4)--(x,y4)+w, dashed );
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| − | label("$\scriptstyle \underline f_\alpha(x)$",(x,y4)+w,SE);
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| − | | |
| − | real x = -0.05;
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| − | real z = 1+3*x^2;
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| − | real y3 = 1/(z-a0)+d;
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| − | dot( (x,y3) );
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| − | draw( (x,y3)--(x,y3)+w, dashed );
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| − | label("$\scriptstyle \underline f_1(x)=\overline f_1(x)$",(x,y3)+w,SE);
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| − | | |
| − | real x = 0.05;
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| − | real z = 1+3*x^2;
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| − | real y2 = 1/(z-a-a0)+b*(1+c*x)+d;
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| − | dot( (x,y2) );
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| − | draw( (x,y2)--(x,y2)+w, dashed );
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| − | label("$\scriptstyle \overline f_\alpha(x)$",(x,y2)+w,SE);
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| − | | |
| − | label("\small Fig. a4: Non-precise function",(x0,-0.2));
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| − | | |
| − | shipout(scale(250,120)*currentpicture);
| |
| − | </asy></center>
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| | | | |
| | + | Strangely, the figure in EoM is erroneous! ED=EB, not BD=EB. |
| | | | |
| | <center><asy> | | <center><asy> |
| − | import gsl;
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| − |
| |
| − | int M=30;
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| − |
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| − | picture whole;
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| − |
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| − | draw ((-0.05,0)--(1.05,0),Arrow);
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| − | draw ((0,0)--(0,1.2),Arrow);
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| − |
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| − | label("$x$",(1.05,0),E);
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| − | label("$\scriptstyle \xi_i(x)$",(0,1.2),N);
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| − |
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| − | draw((-0.05,1)--(1.05,1));
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| − | label("$1$",(-0.05,1),W);
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| − |
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| − | draw ((0.3,0)--(0.7,0),defaultpen+2);
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| − | draw ((0.3,0)--(0.3,1),dashed);
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| − | draw ((0.7,0)--(0.7,1),dashed);
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| − |
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| − | label("$K_j$",(0.5,0),S);
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| − |
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| − | guide g;
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| − | for (int k=floor(-3M); k<floor(3M); ++k) {
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| − | real x = k/M;
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| − | real y = sqrt(2pi) * pdf_gaussian(x);
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| − | g=g..(x,y);
| |
| − | }
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| − | draw(shift(0.33,0)*scale(0.03,1)*g,defaultpen+1);
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| − | draw(shift(0.5,0)*scale(0.045,1)*g,defaultpen+1);
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| − | draw(shift(0.72,0)*scale(0.04,1)*g,defaultpen+1);
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| − |
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| − | label("\small Fig. a3: Non-precise observations and a class of a histogram. is a class of a histogram and is a characterizing function ",(30,-20));
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| − |
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| − | shipout(scale(170,50)*currentpicture);
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| − | </asy></center>
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| − |
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| − | <center><asy>
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| − | import gsl;
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| − |
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| − | int N=30;
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| − |
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| − | picture whole;
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| − | picture common;
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| − |
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| − | draw ((-0.3,0)--(1.45,0),Arrow);
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| − | draw ((0,0)--(0,1.2));
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| − | draw ((1,0)--(1,1.2));
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| − |
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| − | label("$x$",(1.6,0),E);
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| − |
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| − | add ( common, currentpicture );
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| − | erase();
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| − |
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| − |
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| − | add ( currentpicture, common );
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| − |
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| − | guide g;
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| − | for (int k=floor(-0.2N); k<floor(1.2N); ++k) {
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| − | real x = k/N;
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| − | real y = cdf_gaussian_P(7*(x-0.5));
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| − | y =0.85 y+0.15;
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| − | g=g..(x,y);
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| − | }
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| − | draw(g,defaultpen+1.3);
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| − |
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| − | label("$\scriptstyle g(x)$",(0.33,0.8));
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| − |
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| − | add ( whole, shift(-60,0)*scale(40,36)*currentpicture );
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| − | erase();
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| − |
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| − |
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| − | add ( currentpicture, common );
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| − |
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| − | draw((-0.2,1)--(1.2,1));
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| − | label("$1$",(-0.2,1),W);
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| − |
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| − | guide g1;
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| − | guide g2;
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| − | for (int k=floor(-0.2N); k<floor(1.2N); ++k) {
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| − | real x = k/N;
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| − | real y = sqrt(2pi) * pdf_gaussian(7*(x-0.5));
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| − | g1=g1..(x,y);
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| − | g2=g2..(x,1.3y);
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| − | }
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| − | draw(g1,defaultpen+1.3);
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| − | draw(g2,defaultpen+1.3);
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| − |
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| − | label("$\scriptstyle \xi(x)$",(0.5,0.4));
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| − | label("$\scriptstyle g'(x)$",(0.8,1.3));
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| − |
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| − | add ( whole, shift(60,0)*scale(40)*currentpicture );
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| − | erase();
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| − |
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| − | label(whole,"\small Fig. a2: Characterizing function obtained from a gray intensity ",(30,-20));
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| − |
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| − | shipout(scale(1.2)*whole);
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| − |
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| − | </asy></center>
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| − |
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| − |
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| − | <center><asy>
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| − |
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| − | picture whole;
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| − | picture common;
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| − |
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| − | draw ((-0.3,0)--(1.6,0),Arrow);
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| − | draw ((0,-0.2)--(0,1.3),Arrow);
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| − | draw((-0.2,1)--(1.5,1));
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| − | label("$x$",(1.6,0),E);
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| − | label("$\xi(x)$",(0,1.3),N);
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| − | label("$1$",(-0.2,1),W);
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| − |
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| − | add ( common, currentpicture );
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| − | erase();
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| − |
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| − | add ( currentpicture, common );
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| − | dot ((1,1),currentpen+5);
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| − | draw((1,0)--(1,1),dashed);
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| − | draw((-0.2,0)--(1.2,0),currentpen+1.5);
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| − | filldraw( shift(1,0)*scale(0.06)*unitcircle, white );
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| − | label("$x_0$",(1,0),S);
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| − |
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| − | add ( whole, shift(-120,0)*scale(40)*currentpicture );
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| − | erase();
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| − |
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| − | add ( currentpicture, common );
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| − | dot ((0.4,1),currentpen+5);
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| − | dot ((1,1),currentpen+5);
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| − | draw((0.4,0)--(0.4,1),dashed);
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| − | draw((1,0)--(1,1),dashed);
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| − | draw((-0.2,0)--(0.4,0),currentpen+1.5);
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| − | draw((1,0)--(1.3,0),currentpen+1.5);
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| − | draw((0.4,1)--(1,1),currentpen+1.5);
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| − | filldraw( shift(0.4,0)*scale(0.06)*unitcircle, white );
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| − | filldraw( shift(1,0)*scale(0.06)*unitcircle, white );
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| − | label("$a$",(0.4,0),S);
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| − | label("$b$",(1,0),S);
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| − |
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| − | add ( whole, shift(0,0)*scale(40)*currentpicture );
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| − | erase();
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| − |
| |
| − | add ( currentpicture, common );
| |
| − | draw((-0.2,0)--(0.3,0)--(0.5,1)--(0.7,1)--(1,0)--(1.3,0),currentpen+1.5);
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| − |
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| − | label(whole,"Fig. a1: Some characterizing functions",(30,-25));
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| − | add ( whole, shift(120,0)*scale(40)*currentpicture );
| |
| − | erase();
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| − |
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| − | shipout(scale(1.2)*whole);
| |
| − | </asy></center>
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| − |
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| − | <center><asy>
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| − | picture whole;
| |
| − |
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| − | int N=3;
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| − | int M=30;
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| − | real c=0.6;
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| − |
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| − | draw (arc((0,0),1,-90,90),defaultpen+2 );
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| − |
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| − | guide g;
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| − | for (int k=-M*N+1; k<M*N; ++k) {
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| − | real y=k/(M*N);
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| − | pair z=(sqrt(1-y^2),y);
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| − | pair w=(3z-z^3)/4;
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| − | g=g..w;
| |
| − | if (k%M==0) {
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| − | draw((-0.5,y)--z);
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| − | draw((-0.5,y)--(-0.1,y),Arrow);
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| − | draw(z--z-c*z^2);
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| − | draw(z--z-0.3c*z^2,Arrow);
| |
| − | }
| |
| − | }
| |
| − | draw(g,defaultpen+1.3);
| |
| − |
| |
| − | add ( whole, shift(-120,0)*scale(60)*currentpicture );
| |
| − | erase();
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| − |
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| − |
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| − | real n=1.3;
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| − |
| |
| − | int N=7;
| |
| − | int M=10;
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| − |
| |
| − | real c1=1.2;
| |
| − | real c2=0.6;
| |
| | | | |
| − | draw ((-1.4,0)--(1.4,0),defaultpen+2);
| + | pair A=(-1,0); |
| | + | pair B=(0,0); |
| | + | pair E=(0,0.5); |
| | + | pair C=A+(0.5*(sqrt(5)-1),0); |
| | + | pair D=(-1/sqrt(5), 0.5*(1-1/sqrt(5))); |
| | | | |
| − | guide g;
| + | draw( A--B--E--cycle,currentpen+1.5 ); |
| − | for (int k=-M*N+1; k<M*N; ++k) {
| + | dot(A,currentpen+3.5); dot(B,currentpen+3.5); dot(E,currentpen+3.5); dot(C,currentpen+3.5); dot(D,currentpen+3.5); |
| − | real a=0.5*pi*k/(M*N);
| |
| − | real s=sin(a);
| |
| − | real t=tan(a);
| |
| − | real x=(n^2-1)*t^3;
| |
| − | if (n*abs(s)>=1) { continue; }
| |
| − | real aux=(1-(n*s)^2)/(1-s^2);
| |
| − | real y=-aux^1.5/n;
| |
| − | g=g..(x,y);
| |
| − | if (k%M==0) {
| |
| − | draw((0,-1)--(t,0),Arrow(Relative(0.9)));
| |
| − | draw((t,0)--(t,0)+c1*((x,y)-(t,0)),dashed);
| |
| − | draw((t,0)--(t,0)+c2*((t,0)-(x,y)),Arrow(6,Relative(0.8)));
| |
| − | }
| |
| − | }
| |
| − | draw(g,defaultpen+1.3);
| |
| | | | |
| − | dot((0,-1)); dot((0,-1/n));
| + | draw( shift(E)*scale(0.5)*unitcircle,currentpen+1 ); |
| − | label("$A$",(0,-1),W);
| + | draw( shift(A)*scale(0.5*(sqrt(5)-1))*unitcircle,currentpen+1 ); |
| − | label("$A'$",(0,-1/n),W);
| |
| | | | |
| − | add ( whole, shift(120,20)*scale(80)*currentpicture );
| + | draw( shift(B)*scale(0.5)*unitcircle, dashed ); |
| | | | |
| | + | clip(A+(-0.15,-0.15)--B+(0.15,-0.15)--E+(0.15,0.15)--A+(-0.15,0.15)--cycle); |
| | | | |
| − | label(whole,"Fig. a",(-100,-80)); | + | label("$A$",A,S); label("$B$",B,S); label("$C$",C,S); |
| − | label(whole,"Fig. b",(120,-80)); | + | label("$E$",E,N); label("$D$",D,N); |
| | | | |
| | + | label( "\small Golden Ratio construction", (-0.5,0.8) ); |
| | | | |
| − | shipout(whole); | + | shipout(scale(100)*currentpicture); |
| | </asy></center> | | </asy></center> |
| | | | |
