Imagen:Simpsons method illustration.png

Uploaded by Audriusa 16:24, 20 December 2005 (UTC) to use in the non-English Wikipedia versions. Original text follows:

Simpson's method illustration. Done by myself.

I, the author of this work, hereby release it into the public domain. This applies worldwide. In case this is not legally possible: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law. Afrikaans | Alemannisch | Aragonés | العربية | Български | Català | Česky | Cymraeg | Dansk | Deutsch | Ελληνικά | English | Español | Esperanto | فارسی | Français | Galego | 한국어 | हिन्दी | Hrvatski | Ido | Bahasa Indonesia | Íslenska | Italiano | עברית | Latina | Lietuvių | Magyar | Bahasa Melayu | Nederlands | Norsk (bokmål) | Norsk (nynorsk) | 日本語 | Polski | Português | Ripoarish | Română | Русский | Slovenčina | Slovenščina | Српски | Svenska | ไทย | Türkçe | Українська | Tiếng Việt | Walon | 简体中文 | 繁體中文 | 粵語 | +/-

[edit] Source code (carefully documented)

function simpson() % draw an illustration for Simpson's rule

% prepare the scrreen and define some parameters   
clf; hold on; axis equal; axis off; 
fontsize=25; thick_line=3; thin_line=2; black=[0, 0, 0]; red=[1, 0, 0];
arrowsize=0.1; arrow_type=1; arrow_angle=30; % (angle in degrees)
circrad=0.015; % radius of ball showing up in places

% the function formula and its graph
f=inline('0.45*sin(3.3*(x+0.18))+1'); X=-0.6:0.01:0.8; Y=f(X); 

% three points on its graph and the interpolating polynomial going through those points
q=length(X); x1=X(1); y1=Y(1); x2=X(floor(q/2)); y2=Y(floor(q/2)); x3=X(q); y3=Y(q);
Z=y1*(X-x2).*(X-x3)./((x1-x2)*(x1-x3))+y2*(X-x1).*(X-x3)./((x2-x1)*(x2-x3))+y3*(X-x1).*(X-x2)./((x3-x1)*(x3-x2));

% plot the x and y axes
arrow([-0.9 0], [1, 0],          thin_line, arrowsize, arrow_angle, arrow_type, black) 
arrow([-0.8, -0.1], [-0.8, 1.6], thin_line, arrowsize, arrow_angle, arrow_type, black) 

% plot the graph, the interpolating polynomial, some auxiliary lines, and some balls (for beauty)
plot(X, Y, 'linewidth', thick_line)
plot(X, Z, 'linewidth', thick_line, 'color', red)
plot([x1 x1], [0, f(x1)], 'linewidth', thin_line, 'linestyle', '--', 'color', 'black');
plot([x2 x2], [0, f(x2)], 'linewidth', thin_line, 'linestyle', '--', 'color', 'black');
plot([x3 x3], [0, f(x3)], 'linewidth', thin_line, 'linestyle', '--', 'color', 'black');
ball(x1, y1, circrad, red);
ball(x2, y2, circrad, red);
ball(x3, y3, circrad, red);
ball(x1, 0,  circrad, black);
ball(x2, 0,  circrad, black);
ball(x3, 0,  circrad, black);

% place text
tiny=0.1; p0=(x1+x2)/2; q0=(x2+x3)/2; 
H=text(x1, -tiny,  'a');          set(H, 'fontsize', fontsize, 'HorizontalAlignment', 'c')
H=text(x2, -tiny,  'm');          set(H, 'fontsize', fontsize, 'HorizontalAlignment', 'c')
H=text(x3, -tiny,  'b');          set(H, 'fontsize', fontsize, 'HorizontalAlignment', 'c')
H=text(p0, 0.43+f(p0),  'P(x)');  set(H, 'fontsize', fontsize, 'HorizontalAlignment', 'c', 'color', 'red')
H=text(q0, 0.15+f(q0),  'f(x)');  set(H, 'fontsize', fontsize, 'HorizontalAlignment', 'c', 'color', 'blue')

saveas(gcf, 'Simpsons_method_illustration.eps', 'psc2') % export to eps

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