File:Jordan illustration.png
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Original file (1,064 × 1,006 pixels, file size: 55 KB, MIME type: image/png)
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DescriptionJordan illustration.png |
English: A set (represented in the picture by the region inside the blue curve) is Jordan measurable if and only if it can be well-approximated both from the inside and outside by simple sets (their boundaries are shown in dark green and dark pink respectively). |
Date | |
Source | Own work |
Author | User:Oleg Alexandrov |
Summary
[edit]Made by myself with Matlab.
Public domainPublic domainfalsefalse |
I, the copyright holder of this work, release this work into the public domain. This applies worldwide. In some countries this may not be legally possible; if so: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law. |
Licensing
[edit]Public domainPublic domainfalsefalse |
I, the copyright holder of this work, release this work into the public domain. This applies worldwide. In some countries this may not be legally possible; if so: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law. |
function main()
% the function whose zero level set and inner and outer approximations will be drawn
f = inline('60-real(z).^2-1.2*imag(z).^2-0.006*(real(z)-6).^4-0.01*(imag(z)-5).^4', 'z');
M=10; i=sqrt(-1); lw=2.5;
figure(1); clf; hold on; axis equal; axis off;
if 1==0
for p=-M:M
for q=-M:M
z=p+i*q;
if f(z)>0
plot(real(z), imag(z), 'r.')
else
plot(real(z), imag(z), 'b.')
end
end
end
end
% draw the zero level set of f
h=0.1;
XX = -M:h:M; YY = -M:h:M;
[X, Y] = meshgrid (XX, YY); Z = f(X+i*Y);
[C, H] = contour(X, Y, Z, [0, 0]);
set(H, 'linewidth', lw, 'EdgeColor', [0;0;156]/256);
% plot the outer polygonal curve
Start=5+6*i; Dir=-i; Sign=-1;
plot_poly (Start, Dir, Sign, f, lw, [139;10;80]/256);
% plot the inner polygonal curve
Sign=1; Start=4+5*i;
plot_poly (Start, Dir, Sign, f, lw, [0;100;0]/256);
% a dummy plot to avoid a matlab bug causing some lines to appear too thin
plot(8.5, 7.5, '*', 'color', 0.99*[1, 1, 1]);
plot(-4.5, -5, '*', 'color', 0.99*[1, 1, 1]);
saveas(gcf, 'jordan_illustration.eps', 'psc2');
function plot_poly (Start, Dir, Sign, f, lw, color)
Current_point = Start;
Current_dir = Dir;
Ball_rad = 0.03;
for k=1:100
Next_dir=-Current_dir;
% from the current point, search to the left, down, and right and see where to go next
for l=1:3
Next_dir = Next_dir*(Sign*i);
if Sign*f(Current_point+Next_dir)>=0 & Sign*f(Current_point+(Sign*i)*Next_dir) < 0
break;
end
end
Next_point = Current_point+Next_dir;
plot([real(Current_point), real(Next_point)], [imag(Current_point), imag(Next_point)], 'linewidth', lw, 'color', color);
round_ball(Current_point, Ball_rad, color'); % just for beauty, to round off some rough corners
Current_dir=Next_dir;
Current_point = Next_point;
end
function round_ball(z, r, color)
x=real(z); y=imag(z);
Theta = 0:0.1:2*pi;
X = r*cos(Theta)+x;
Y = r*sin(Theta)+y;
Handle = fill(X, Y, color);
set(Handle, 'EdgeColor', color);
This math image could be re-created using vector graphics as an SVG file. This has several advantages; see Commons:Media for cleanup for more information. If an SVG form of this image is available, please upload it and afterwards replace this template with
{{vector version available|new image name}} .
It is recommended to name the SVG file “Jordan illustration.svg”—then the template Vector version available (or Vva) does not need the new image name parameter. |
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 17:27, 4 February 2007 | 1,064 × 1,006 (55 KB) | Oleg Alexandrov (talk | contribs) | Made by myself with Matlab. {{PD}} | |
17:24, 4 February 2007 | 1,064 × 1,006 (55 KB) | Oleg Alexandrov (talk | contribs) | Made by myself with Matlab. {{PD}} |
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