legal path and corners

check_corner.m

+function corner = check_corner(i0,i1,A,y)
+m = size(y,1);
+C = zeros(m);
+C(1:m-2,1:m-2) = A;
+D = C'+C;
+
+% going from x0 to x1
+[~,nodes] = find(D(i1,:)); 
+nodes = nodes(nodes~=i0); %remove existing edge if it exists
+j = 1;
+theta = zeros(length(nodes),1);
+for k=nodes
+    theta0 = acos(dot(y(i0,:)-y(i1,:), y(k,:)-y(i1,:))/(sqrt(sum(abs(y(i0,:)-y(i1,:)).^2))*D(k,i1)));
+    thecross = cross([y(i0,:)-y(i1,:),0], [y(k,:)-y(i1,:),0]);
+    thesign = sign(thecross(3));
+    if thesign<0
+        theta(j) = 2*pi-theta0;
+    else
+        theta(j) = theta0;
+    end
+    j=j+1;
+end
+if isempty(nodes)
+    corner = 0;
+else
+    corner = (max(theta) - min(theta))>pi;
+end
+
+end
+

legal_path.m

+function [B,x] = legal_path(A,x )
+% We generate n uniformly random points in [0,1]x[0,1]
+% x = rand(n,2);
+n = size(x,1);
+% Initialise Adjacency mantrix
+B = zeros(n+2);
+% B(1:n,1:n) = A; % Only do upper triangular since symmetric
+x(n+1,:) = [0,0];
+x(n+2,:) = [0.5,0.5];
+% A = A+A';
+% First Line
+% B(1,2) = norm(x(1,:)-x(2,:));
+% Construct the graph
+for i=1:n+2
+    distances = sqrt(sum(abs(x-x(i,:)).^2,2));
+    [distances, nodes] = sort(distances);
+    crosses = 1;
+    j = 1;
+    for l=setdiff(1:n+2,i)
+        node = nodes(j);
+        d_min = distances(j);
+        [I,J] = find(A);
+        % Remove node included in current segment
+        J = J(I ~= node & I~=i);
+        I = I(I ~= node & I~=i);
+        I = I(J ~= node & J~=i);
+        J = J(J ~= node & J~=i);
+        for k=1:length(I)
+              crosses = checkCrossedLines(x(i,:),x(node,:),x(I(k),:),x(J(k),:));
+              if crosses==1
+                  break
+              end
+        end
+        if crosses == 1
+            corners = 1;
+        else
+           % check if outgoing nodes are corners
+           node_corner = check_corner(i,node,A+A',x);
+           outgoing_corner = check_corner(node,i,A+A',x);
+           corners = node_corner | outgoing_corner;
+        end
+            
+        if (crosses==0 && corners==0) || isempty(I)
+            B(node,i) = d_min;
+        end  
+        j = j+1;
+    end
+   
+    
+    
+    
+end
+
+
+end
+

make_plot.m

-G = graph(A+A');
-plot(G, 'XData',x(:,1),'YData',x(:,2))
+figure
+G = graph(B+B');
+plot(G, 'XData',y(:,1),'YData',y(:,2))
+hold on
+G2 = graph(A+A');
+plot(G2, 'XData',x(:,1),'YData',x(:,2))
 xlim([0,1])
 ylim([0,1])
\ No newline at end of file

shortest_path.m

@@ -10,11 +10,11 @@ A(1,2) = norm(x(1,:)-x(2,:));
 for i=3:n
     distances = sqrt(sum(abs(x(1:i-1,:)-x(i,:)).^2,2));
     [distances, nodes] = sort(distances);
-    cross = 1;
+    crosses = 1;
     node = nodes(1);
     d_min = distances(1);
     j = 1;
-    while cross == 1
+    while crosses == 1
 
         [I,J] = find(A(1:i-1,1:i-1));
         % Remove node included in current segment
@@ -23,12 +23,12 @@ for i=3:n
         I = I(J ~= node);
         J = J(J ~= node);
         for k=1:length(I)
-              cross = checkCrossedLines(x(i,:),x(node,:),x(I(k),:),x(J(k),:));
-              if cross==1
+              crosses = checkCrossedLines(x(i,:),x(node,:),x(I(k),:),x(J(k),:));
+              if crosses==1
                   break
               end
         end
-        if cross==1 && ~isempty(I)
+        if crosses==1 && ~isempty(I)
             j = j+1;
             node = nodes(j);
             d_min = distances(j);