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 +
==Experiments==
  
−
The envelope of rays reflected or refracted by a given curve. The caustic of reflected rays is called a catacaustic, that of refracted rays — a diacaustic.
+
Note a fine distinction from [http://ada00.math.uni-bielefeld.de/MW1236/index.php/User:Boris_Tsirelson/sandbox#Experiments Ada]:
  
 
<center><asy>
 
<center><asy>
 +
fill( box((-1,-1),(1,1)), white );
 +
draw( (-1.2,-0.5)--(1.2,-0.5) );
 +
label("Just a text",(0,0));
 +
filldraw( box((-0.7,-1),(0.7,1)), white, opacity(0) );
 +
shipout(scale(15)*currentpicture);
 +
</asy></center>
 +
 +
I guess, the reason is that there Asy generates pdf file (converted into png afterwards), and here something else (probably ps).
 +
 +
No, it seems, it generates eps, both here and there. Then, what could be the reason?
 +
 +
More.
 +
 +
<center><asy>
 +
label("Just a text",(0,0));
 +
fill( box((-2,-1),(2,1)), white );
 +
//draw( box((-2,-1),(2,1)), green );
 +
shipout(scale(15)*currentpicture);
 +
</asy></center>
 +
 +
 +
<center><asy>
 +
label("Just a text",(0,0));
 +
fill( box((-2,-1),(2,1)), white );
 +
draw( box((-2,-1),(2,1)), green );
 +
shipout(scale(15)*currentpicture);
 +
</asy></center>
 +
 +
Mysterious.
 +
 +
==Three dimensions==
 +
 +
<center><asy>
 +
settings.render = 0;
 +
 +
unitsize(100);
 +
 +
import three;
 +
import tube;
 +
 +
import graph;
 +
path unitCircle = Circle((0,0),1,35);
 +
 +
currentprojection = perspective((900,-350,-650));
 +
currentlight=light(gray(0.4),specularfactor=3,viewport=false,(-0.5,-0.5,-0.75),(0,-0.5,0.5),(0.5,0.5,0.75));
 +
// currentlight=light(gray(0.4),specularfactor=3,viewport=false,(-0.5,-0.5,-0.75),(0.5,-0.5,0.5),(0.5,0.5,0.75));
 +
 +
triple horn_start=(0,-1,0.6);
 +
triple horn_end=(0,0.4,0.2);
 +
real horn_radius=0.2;
 +
 +
real ratio=horn_end.z/(-horn_start.y);    // fractal levels ratio
 +
 +
transform3 implode_right = shift(horn_end) * scale3(ratio) * rotate(-90,X) * shift(-horn_start.y*Y);
 +
transform3 left_right = reflect(O,X,Z)*rotate(90,Y);
 +
 +
path[] cover_with_holes = scale(horn_radius/ratio)*unitCircle^^
 +
  shift((horn_start.z,0))*scale(0.9horn_radius)*reverse(unitCircle)^^
 +
  shift((-horn_start.z,0))*scale(0.9horn_radius)*reverse(unitCircle);
 +
surface cover = surface(cover_with_holes,ZXplane);
 +
surface cover_left = shift((horn_start.x,horn_start.y,0))*cover;
 +
surface two_covers = surface(cover_left,left_right*cover_left);
 +
 +
path3 horn_axis = horn_start..horn_start+(0,0.01,0)..(0,0,0.7)..(0,0.2,0.6)..horn_end+(0,0,0.01)..horn_end;
 +
 +
surface horn = tube( horn_axis, scale(horn_radius)*unitCircle );
 +
surface two_horns = surface(horn,reflect(O,X,Y)*horn);
 +
surface two_horns = surface(horn,reflect(O,X,Y)*horn);
 +
surface four_horns = surface(two_horns,left_right*two_horns,two_covers);
 +
 +
surface four_small_horns = implode_right*four_horns;
 +
surface eight_small_horns = surface(four_small_horns,left_right*four_small_horns);
 +
 +
surface big_surface = surface(four_horns,eight_small_horns);
 +
 +
real R = horn_radius/ratio;
 +
 +
draw ( circle((0,1,0), 1.005R, Y ), currentpen+2 );
 +
draw ( circle((horn_start.z,1.01,horn_start.x), horn_radius, Y ), currentpen+2 );
 +
draw ( circle((-horn_start.z,1.01,horn_start.x), horn_radius, Y ), currentpen+2 );
 +
 +
draw (big_surface, yellow);
 +
 +
pen blackpen = currentpen+1.5;
 +
 +
draw ( circle((0,-1,0), 1.005R, Y ), blackpen );
 +
draw ( circle(horn_start, 0.98horn_radius, Y ), blackpen );
 +
draw ( circle((horn_start.x,horn_start.y,-horn_start.z), 0.98horn_radius, Y ), blackpen );
 +
 +
real phi=0.9;  // adjust to the projection
 +
triple u = (cos(phi),0,sin(phi));
 +
draw( R*u-Y -- R*u+Y, blackpen );
 +
draw( -R*u-Y -- -R*u+Y, blackpen );
 +
 +
</asy></center>
 +
 +
 +
<center><asy>
 +
settings.render = 0;
 +
 +
size(200);
 +
import graph3;
 +
 +
currentprojection=perspective((2,2,5));
 +
 +
real R=1;
 +
real a=1;
 +
 +
real co=0.6;
 +
real colo=0.3;
 +
 +
triple f(pair t) {
 +
  return ((R+a*cos(t.y))*cos(t.x),(R+a*cos(t.y))*sin(t.x),a*sin(t.y));
 +
}
 +
 +
surface s=surface(f,(0,0),(2pi,2pi),20,20,Spline);
 +
 +
draw(s,rgb(co,co,co),meshpen=rgb(colo,colo,colo));
 +
 +
</asy></center>
 +
 +
==Sinusoid==
 +
 +
<center><asy>
 +
import graph;
 +
size(450);
 +
real f(real x) {return sin(x);};
 +
 +
real f1(real x) {return cos(x);};
 +
draw(graph(f1,-2*pi,2*pi),blue+1,"$\cos(x)$");
 +
draw(graph(f,-2*pi,2*pi),red+1,"$\sin(x)$");
 +
xaxis("$x$",Arrow);
 +
yaxis();
 +
 +
xtick("$\frac{\pi}{6}$",pi/6,N);
 +
xtick("$\frac{\pi}{4}$",pi/4,N);
 +
xtick("$\frac{\pi}{3}$",pi/3,N);
 +
xtick("$\frac{\pi}{2}$",pi/2,N);
 +
xtick("$\frac{3\pi}{2}$",3*pi/2,N);
 +
xtick("$\pi$",pi,N);
 +
xtick("$2\pi$",2*pi,N);
 +
xtick("$-\frac{\pi}{2}$",-pi/2,N);
 +
xtick("$-\frac{3\pi}{2}$",-3*pi/2,N);
 +
xtick("$-\pi$",-pi,N);
 +
xtick("$-2\pi$",-2*pi,N);
 +
 +
ytick("$1/2$",0.5,1,fontsize(8pt));
 +
ytick("$\sqrt{2}/2$",sqrt(2)/2,1,fontsize(8pt));
 +
ytick("$\sqrt{3}/2$",sqrt(3)/2,1,fontsize(8pt));
 +
ytick("$1$",1,1,fontsize(8pt));
 +
ytick("$-1/2$",-0.5,-1,fontsize(8pt));
 +
ytick("$-\sqrt{2}/2$",-sqrt(2)/2,-1,fontsize(8pt));
 +
ytick("$-\sqrt{3}/2$",-sqrt(3)/2,-1,fontsize(8pt));
 +
ytick("$-1$",-1,-1,fontsize(8pt));
 +
 +
attach(legend(),truepoint(E),10E,UnFill);
 +
</asy></center>
 +
 +
==Sinusoidal spiral==
 +
 +
<center><asy>
 +
import graph;
 +
size (200);
 +
 +
real r = 2.3;
 +
real m = 4;
 +
 +
real eps=10.^(-10);
 +
for  (int k=0; k<m; ++k) {
 +
  draw ( polargraph(  new real(real x) {return cos(m*x)^(1/m);}, -(pi/2m)+eps+k*2pi/m, (pi/2m)-eps+k*2pi/m ),
 +
defaultpen+1.5 );
 +
  draw ( -r*expi(-pi/2m+k*2pi/m)..r*expi(-pi/2m+k*2pi/m), dashed );
 +
  draw ( -r*expi(pi/2m+k*2pi/m)..r*expi(pi/2m+k*2pi/m), dashed );
 +
}
 +
label( "$m=4$", (0.58,0.02), fontsize(7pt) );
 +
 +
real eps=10.^(-2);
 +
for  (int k=0; k<m; ++k) {
 +
  draw ( polargraph(  new real(real x) {return cos(m*x)^(-1/m);}, -(pi/2m)+eps+k*2pi/m, (pi/2m)-eps+k*2pi/m ),
 +
defaultpen+1.5 );
 +
}
 +
label( "$m=-4$", (1.55,0.02), fontsize(7pt) );
 +
 +
label( "sinusoidal spiral: $a=1$", (0,2.3) );
 +
draw ( unitcircle, dashed );
 +
</asy></center>
 +
 +
==Power function==
 +
 +
<center><asy>
 +
import graph;
 
picture whole;
 
picture whole;
  
−
int N=3;
+
real sc=0.8;
−
int M=30;
+
 
−
real c=0.6;
+
draw ( graph( new real(real x) {return x;}, -2, 2), red+1.2, "$y=x$" );
 +
draw ( graph( new real(real x) {return 2x;}, -1, 1), blue+1.2, "$y=2x$" );
 +
draw ( graph( new real(real x) {return x/2;}, -2, 2), green+1.2, "$y=x/2$" );
 +
 
 +
xaxis(-2.1,2.1, LeftTicks(Label(fontsize(8pt)),Step=1,step=0.2,Size=2,size=1,NoZero));
 +
yaxis(-2,2, RightTicks(Label(fontsize(8pt)),Step=0.5,step=0.1,Size=2,size=1,NoZero));
 +
labelx("$x$",(2.3,0.25));
 +
labely("$y$",(0.15,2.3));
 +
 
 +
add(scale(0.72sc,1.2sc)*legend(),(0.5,-0.75));
 +
 
 +
real mrg=1.3;
 +
draw( scale(mrg)*box((-2,-2),(2,2)), white );
 +
 
 +
add (whole,shift(-sc*230,0)*currentpicture.fit(sc*mrg*6.5cm));
 +
erase();
 +
 
 +
 
 +
draw ( graph( new real(real x) {return 1/x;}, -4, -0.25), red+1.2, "$y=1/x$" );
 +
draw ( graph( new real(real x) {return 1/x;}, 0.25, 4), red+1.2 );
 +
draw ( graph( new real(real x) {return 2/x;}, -4, -0.5), blue+1.2, "$y=2/x$" );
 +
draw ( graph( new real(real x) {return 2/x;}, 0.5, 4), blue+1.2 );
 +
draw ( graph( new real(real x) {return 1/(2x);}, -4, -0.125), green+1.2, "$y=1/(2x)$" );
 +
draw ( graph( new real(real x) {return 1/(2x);}, 0.125, 4), green+1.2 );
 +
 
 +
xaxis(-4.2,4.2, LeftTicks(Label(fontsize(8pt)),Step=2,step=0.5,Size=2,size=1,NoZero));
 +
yaxis(-4,4, RightTicks(Label(fontsize(8pt)),Step=1,step=0.2,Size=2,size=1,NoZero));
 +
labelx("$x$",(4.6,0.5));
 +
labely("$y$",(0.3,4.6));
 +
 
 +
add(scale(0.75sc,0.75sc)*legend(),(0.95,-1.2));
 +
 
 +
real mrg=1.3;
 +
draw( scale(mrg)*box((-4,-4),(4,4)), white );
 +
 
 +
add (whole,shift(0,0)*currentpicture.fit(sc*mrg*6.5cm,mrg*6.5cm,false));
 +
erase();
 +
 
 +
 
 +
draw ( graph( new real(real x) {return x^3;}, -4^(1/3), 4^(1/3)), red+1.2, "$y=x^3$" );
 +
draw ( graph( new real(real x) {return x^2;}, -2, 2), blue+1.2, "$y=x^2$" );
 +
draw ( graph( new real(real x) {return sqrt(x);}, 0, 4), green+1.2, "$y=x^{1/2}$" );
 +
draw ( graph( new real(real x) {return -sqrt(x);}, 0, 4), green+1.2 );
 +
 
 +
xaxis(-4.2,4.2, LeftTicks(Label(fontsize(8pt)),Step=2,step=0.5,Size=2,size=1,NoZero));
 +
yaxis(-4,4, RightTicks(Label(fontsize(8pt)),Step=1,step=0.2,Size=2,size=1,NoZero));
 +
labelx("$x$",(4.6,0.5));
 +
labely("$y$",(0.3,4.6));
 +
 
 +
add(scale(0.5sc,0.75sc)*legend(),(0.6,-2.5));
 +
 
 +
real mrg=1.3;
 +
draw( scale(mrg)*box((-4,-4),(4,4)), white );
 +
 
 +
add (whole,shift(sc*230,0)*currentpicture.fit(sc*mrg*6.5cm,mrg*6.5cm,false));
 +
erase();
 +
 
 +
shipout(whole);
 +
</asy></center>
 +
 
 +
==Kolmogorov test==
 +
 
 +
<center><asy>
 +
 
 +
srand(2014011);
  
−
draw (arc((0,0),1,-90,90),defaultpen+1.3 );
+
import stats;
  
−
guide g;
+
int size = 13;
−
for (int k=-M*N+1; k<M*N; ++k) {
+
real [] sample = new real[size+1];
−
  real y=k/(M*N);
+
real lambda = 1.3/size;
−
  pair z=(sqrt(1-y^2),y);
+
real width = 2.0;
−
  pair w=(3z-z^3)/4;
+
 
−
  g=g..w;
+
for (int k=0; k<size; ++k) {
−
  if (k%M==0) {
+
  sample[k] = Gaussrand();
−
    draw((-0.5,y)--z);
 
−
    draw((-0.5,y)--(-0.1,y),Arrow);
 
−
    draw(z--z-c*z^2);
 
−
    draw(z--z-0.3c*z^2,Arrow);
 
−
  }
 
 
}
 
}
−
draw(g,defaultpen+1.3);
+
sample[size] = 10;
  
−
add ( whole, shift(-50,0)*scale(30)*currentpicture );
+
sample = sort(sample);
−
erase();
+
 
 +
// for (real x : sample ) {
 +
//  write(x);
 +
// }
  
−
// recursive procedure, draws the fractal within the rectangle [x0,x1]×[y0,y1]
+
real x0 = -10;
−
void f(real x0, real x1, real y0, real y1, int n) {
+
int k = 0;
−
   if (n!=0) {
+
for (real x : sample ) {
−
    real dx = (x1-x0)/3;
+
  filldraw( box( (x0,k/size-lambda), (x,k/size+lambda) ), rgb(0.8,0.8,0.8) );
−
    real dy = (y1-y0)/2;
+
   draw( (x0,k/size-lambda)..(x,k/size-lambda), currentpen+1.5 );
−
    draw( (x0+dx,y0+dy)--(x0+2*dx,y0+dy), defaultpen+1 );
+
  draw( (x0,k/size)..(x,k/size), currentpen+1.5 );
−
    f(x0,x0+dx,y0,y0+dy,n-1);
+
  draw( (x0,k/size+lambda)..(x,k/size+lambda), currentpen+1.5 );
−
    f(x0+2*dx,x1,y0+dy,y1,n-1);
+
  k += 1;
−
  }
+
  x0 = x;
 +
  draw( (x,(k-1)/size-lambda)..(x,k/size+lambda) );
 
}
 
}
  
−
f(0,1,0,1,9);
+
clip( box((-width,-0.005),(width,1.005)) );
−
draw( (0,0)--(1,0), arrow=Arrow );
+
 
−
draw( (0,0)--(0,1), arrow=Arrow );
+
draw ((-width,0)--(width,0),Arrow);
 +
draw ((0,-0.1)--(0,1.3),Arrow);
 +
draw ((-width,1)--(width,1));
 +
 
 +
draw ((sample[2],0)..(sample[2],2/size));
 +
draw ((sample[size-1],0)..(sample[size-1],0.48), dashed);
 +
draw ((sample[size-1],0.7)..(sample[size-1],1-1/size), dashed);
 +
 
 +
label("$x$",(width,0),S);
 +
label("$y$",(0,1.3),W);
 +
label("$0$",(0,0),SW);
 +
label("$1$",(0,1),NE);
 +
 
 +
label("$X_{(1)}$",(sample[0],0),S);
 +
label("$X_{(2)}$",(sample[1],0),S);
 +
label("$X_{(3)}$",(sample[2],0),S);
 +
label("$X_{(n)}$",(sample[size-1],0),S);
 +
 
 +
label("$F_n(x)+\lambda_n(\alpha)$",(-1.55,0.35));
 +
draw ((-1.35,0.25)..(-1.2,1/size+lambda));
 +
dot((-1.2,1/size+lambda));
 +
 
 +
label("$F_n(x)$",(0.4,0.3));
 +
draw ((0.4,0.4)..(0.3,8/size));
 +
dot((0.3,8/size));
 +
 
 +
label("$F_n(x)-\lambda_n(\alpha)$",(1.5,0.6));
 +
draw ((1.6,0.7)..(1.7,1-lambda));
 +
dot((1.7,1-lambda));
  
−
add ( whole, shift(50,-20)*scale(60)*currentpicture );
+
shipout(scale(100,100)*currentpicture);
−
shipout(whole);
 
 
</asy></center>
 
</asy></center>
  
 +
==Golden ratio==
 +
 +
Strangely, the figure in EoM is erroneous! ED=EB, not BD=EB.
 +
 +
<center><asy>
  
−
Figure: c021000a
+
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)));
  
−
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c021000b.gif" />
+
draw( A--B--E--cycle,currentpen+1.5 );
 +
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);
  
−
Figure: c021000b
+
draw( shift(E)*scale(0.5)*unitcircle,currentpen+1 );
 +
draw( shift(A)*scale(0.5*(sqrt(5)-1))*unitcircle,currentpen+1 );
  
−
For example, the catacaustic of a parallel beam of rays reflected from a semi-circle is part of an [[Epicycloid|epicycloid]] (see Fig. a); the diacaustic of a pencil of rays emanating from a point $A$ lying in a denser medium and refracted by a straight line is part of an [[Astroid|astroid]] with cusp $A'$ the distance of which from the line is $1/n$ times that of the point $A$ (where $n$ is the refraction index) (see Fig. b).
+
draw( shift(B)*scale(0.5)*unitcircle, dashed+red );
  
−
====References====
+
clip(A+(-0.15,-0.15)--B+(0.15,-0.15)--E+(0.15,0.15)--A+(-0.15,0.15)--cycle);
−
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)</TD></TR></table>
 
  
 +
label("$A$",A,S); label("$B$",B,S); label("$C$",C,S);
 +
label("$E$",E,N); label("$D$",D,N);
 +
 +
label( "\small Golden Ratio construction", (-0.5,0.8) );
 +
 +
shipout(scale(100)*currentpicture);
 +
</asy></center>
  
  
−
====Comments====
 
−
In terms of purely geometric optics these are curves of light of infinite brightness consisting of points through which infinitely many reflected or refracted light rays pass. In reality they can often be observed as a pattern of pieces of very bright curves. E.g. on a sunny day at the seashore on the bottom beneath a bit of wavy water. Or at the bottom of a cup of tea into which light is shining. Suitably interpreted, caustics are [[Bifurcation|bifurcation]] sets of elementary catastrophes with 3 control variables. To discuss not only the shape of caustics but also their brightness one needs wave optics and this leads to the study of asymptotic solutions of wave equations and oscillatory integrals [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]].
 
  
−
One way of describing the caustic resulting from rays from a point $Q$ reflected by a curve $\gamma$ is as follows. Take a point $T$ on $\gamma$ and draw the tangent $l$; from $Q$ draw the perpendicular to this tangent and let $R$ be the point on the perpendicular on the other side of $l$ at the same distance from $l$ as $Q$. Then the caustic by reflection is the [[Evolute|evolute]] of the curve described by $R$ as $T$ runs over $\gamma$. This result is due to A. Quetelet. It follows readily that the caustic by reflection of the circle is the [[Pascal limaçon|Pascal limaçon]]. There is a similar result for a caustic by refraction.
 
  
  

Latest revision as of 20:14, 12 December 2014

Experiments

Note a fine distinction from Ada:

I guess, the reason is that there Asy generates pdf file (converted into png afterwards), and here something else (probably ps).

No, it seems, it generates eps, both here and there. Then, what could be the reason?

More.


Mysterious.

Three dimensions


Sinusoid

Sinusoidal spiral

Power function

Kolmogorov test

Golden ratio

Strangely, the figure in EoM is erroneous! ED=EB, not BD=EB.





[Calculus: ] the art of numbering and measuring exactly a thing whose existence cannot be conceived. (Voltaire, Letter XVII: On Infinites In Geometry, And Sir Isaac Newton's Chronology)

And what are these fluxions? The velocities of evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities? (Berkeley, The Analyst)


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How to Cite This Entry:
Boris Tsirelson/sandbox2. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox2&oldid=34298
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