File:Extreme points illustration.png

function main()

   clf; hold on; axis equal; axis off;                     % prepare the screen
   linewidth=3; i=sqrt(-1); N=50; height=0.3; V=[]; M=[];  % initialize variables
   white=0.99*[1, 1, 1]; red=[1, 0, 0]; lightblue=[135 206 250]/256;

   Z=[0, 1, 0.5+i*sqrt(3)/2.0]; L=length(Z);

   [V, M]=corner (N, V, M, Z(L), Z(1), Z(2), height);
   for k=2:(L-1)
      [V, M]=corner (N, V, M, Z(k-1), Z(k), Z(k+1), height);
   end
   [V, M]=corner (N, V, M, Z(L-1), Z(L), Z(1), height);

   H=fill(real(V), imag(V), lightblue); set(H, 'LineWidth', linewidth/2, 'EdgeColor', lightblue)

   for k=1:L
      W=M(k, :);
      plot(W, 'color', red, 'linewidth', linewidth)
   end

   % force the window to be a bit bigger than the picture
   Winxmin=min(real(V));  Winxmax=max(real(V));
   Winymin=min(imag(V));  Winymax=max(imag(V));
   bd=0.1;
   plot(Winxmin-bd, Winymin-bd, 'color', white);
   plot(Winxmax+bd, Winymax+bd, 'color', white);

saveas(gcf, 'Extreme_points_illustration.eps', 'psc2')

% make a rounded corner at b with edges going towards a and c (all complex numbers)
function [V, M]=corner (N, V, M, a, b, c, height)

   i=sqrt(-1);
   v=-i*height*(b-a)/abs(b-a); w=-i*height*(c-b)/abs(c-b);
   theta1=atan2(imag(v), real(v));  theta2=theta1+atan2(imag(w/v), real(w/v));
   Theta=linspace(theta1, theta2, N);
   C=b+height*exp(i*Theta);

   M=[M', C']'; V=[V, C];