File:Analytic continuation along a curve.png
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![File:Analytic continuation along a curve.png](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Analytic_continuation_along_a_curve.png/243px-Analytic_continuation_along_a_curve.png?20070408032652)
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Made by myself with MATLAB.
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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. |
% illustrate the analytic continuation along a curve
function main()
lw=2; % line width
fs=20; % font size
h=1/100;
tinyrad = 0.04;
red = [1, 0, 0];
figure(1); clf; hold on; axis equal; axis off;
% generate the curve on which the analytic continuation will take place
XX=[-0.1, 0.3, 0.4 0.28 0.0]; YY=1.4*[0, 1, 1.5 2 2.8];
Y=YY(1):h:YY(length(YY)); X=spline(YY, XX, Y);
N = length(X);
% plot the circles of analytic continuation
rad=0.8;
spacing = 1; gap = spacing/h;
k=1;
while k <= N
plot_circle(X(k), Y(k), rad, lw);
k = k+gap;
end
plot_circle(X(N), Y(N), rad, lw);
% plot the curve
plot(X, Y, 'color', red, 'linewidth', lw);
% plot the text
tiny=0.003*fs;
plot_text(X(1), Y(1), -2*tiny, -3.5*tiny, '\gamma(0)', fs, tinyrad, red);
t=0.5; k=floor(N*t);
plot_text(X(k), Y(k), tiny, tiny, '\gamma(\it{t})', fs, tinyrad, red);
plot_text(X(N), Y(N), -4*tiny, 4.5*tiny, '\gamma(1)', fs, tinyrad, red);
% plot arrows showing the direction along the curve
thickness = lw;
arrow_size = 0.27;
sharpness=25;
arrow_type=1;
t= 0.2; s=0.1; k = floor(N*t); l = floor(N*(t+s));
arrow([X(k), Y(k)], [X(l), Y(l)], thickness, arrow_size, sharpness, arrow_type, red)
t= 0.8; s=0.1; k = floor(N*t); l = floor(N*(t+s));
arrow([X(k), Y(k)], [X(l), Y(l)], thickness, arrow_size, sharpness, arrow_type, red)
% plot a phony box around the graph to avoid a bug in matlab
% with truncating around the edges when exporting to eps
white=0.99*[1 1 1]; factor=1.1;
plot(X(1)-factor*rad, Y(1)-factor*rad, '*', 'color', white);
plot(X(N)+factor*rad, Y(N)+factor*rad, '*', 'color', white);
saveas(gcf, 'analytic_continuation_along_a_curve.eps', 'psc2');
function plot_circle(x, y, r, lw)
N=100;
Theta=0:(1/N):2.1*pi;
X=r*cos(Theta);
Y=r*sin(Theta);
plot(x+X, y+Y, 'linewidth', lw);
function plot_text(x, y, shiftx, shifty, str, fs, tinyrad, color)
text(x+shiftx, y+shifty, str, 'fontsize', fs);
ball(x, y, tinyrad, color);
function ball(x, y, r, color)
Theta=0:0.1:2*pi;
X=r*cos(Theta)+x;
Y=r*sin(Theta)+y;
H=fill(X, Y, color);
set(H, 'EdgeColor', 'none');
function arrow(start, stop, thickness, arrow_size, sharpness, arrow_type, color)
% Function arguments:
% start, stop: start and end coordinates of arrow, vectors of size 2
% thickness: thickness of arrow stick
% arrow_size: the size of the two sides of the angle in this picture ->
% sharpness: angle between the arrow stick and arrow side, in degrees
% arrow_type: 1 for filled arrow, otherwise the arrow will be just two segments
% color: arrow color, a vector of length three with values in [0, 1]
% convert to complex numbers
i=sqrt(-1);
start=start(1)+i*start(2); stop=stop(1)+i*stop(2);
rotate_angle=exp(i*pi*sharpness/180);
% points making up the arrow tip (besides the "stop" point)
point1 = stop - (arrow_size*rotate_angle)*(stop-start)/abs(stop-start);
point2 = stop - (arrow_size/rotate_angle)*(stop-start)/abs(stop-start);
if arrow_type==1 % filled arrow
% plot the stick, but not till the end, looks bad
t=0.5*arrow_size*cos(pi*sharpness/180)/abs(stop-start); stop1=t*start+(1-t)*stop;
plot(real([start, stop1]), imag([start, stop1]), 'LineWidth', thickness, 'Color', color);
% fill the arrow
H=fill(real([stop, point1, point2]), imag([stop, point1, point2]), color);
set(H, 'EdgeColor', 'none')
else % two-segment arrow
plot(real([start, stop]), imag([start, stop]), 'LineWidth', thickness, 'Color', color);
plot(real([stop, point1]), imag([stop, point1]), 'LineWidth', thickness, 'Color', color);
plot(real([stop, point2]), imag([stop, point2]), 'LineWidth', thickness, 'Color', color);
end
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 03:26, 8 April 2007 | ![]() | 408 × 1,006 (32 KB) | Oleg Alexandrov (talk | contribs) | Made by myself with MATLAB. {{PD-self}} |
03:24, 8 April 2007 | ![]() | 408 × 1,006 (32 KB) | Oleg Alexandrov (talk | contribs) | Made by myself with MATLAB. {{PD-self}} | |
03:20, 8 April 2007 | ![]() | 408 × 1,006 (32 KB) | Oleg Alexandrov (talk | contribs) | Made by myself with MATLAB. {{PD-self}} | |
03:17, 8 April 2007 | ![]() | 408 × 1,006 (33 KB) | Oleg Alexandrov (talk | contribs) | Made by myself with MATLAB. {{PD-self}} | |
03:15, 8 April 2007 | ![]() | 408 × 1,006 (33 KB) | Oleg Alexandrov (talk | contribs) | Made by myself with MATLAB. {{PD-self}} |
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