starconvexpolygon.m

n = 6;
th = linspace(-pi, pi, 2*n+1);
xi = (1+rem((1:2*n+1),2)).*cos(th);
yi = (1+rem((1:2*n+1),2)).*sin(th);
clf;
hold on;
grid on
plot( xi, yi, 'b*-' );
axis equal;

sdpvar x y
lambda = sdpvar(length(xi),1)
Model = [sos2(lambda), lambda>=0, sum(lambda)==1,
         x == xi*lambda, y == yi*lambda];
plot(Model,[x;y],[],[],sdpsettings('plot.shade',.1))    

optimize(Model, (x-1.5)^2 + (y-1)^2)
plot(1.5,1,'*r')
plot(value(x),value(y),'ok')


sdpvar x y
lambda = sdpvar(length(xi),1)
Model = [sos2(lambda), lambda>=0,sum(lambda)<=1,
         x == xi*lambda, y == yi*lambda];

n = 6;
th = linspace(-pi, pi, 2*n+1);
xi = 1+(1+rem((1:2*n+1),2)).*cos(th);
yi = 2+(1+rem((1:2*n+1),2)).*sin(th);
clf;
hold on;
grid on
plot( xi, yi, 'b*-' );
axis equal;

xc = mean(xi(1:end-1));
yc = mean(yi(1:end-1));
lambda = sdpvar(length(xi),1)
Model = [sos2(lambda), lambda>=0,sum(lambda)<=1,
         x == xc + (xi-xc)*lambda, 
         y == yc + (yi-yc)*lambda];
plot(Model,[x;y],[],[],sdpsettings('plot.shade',.1))    

xi = [1 2 2 3 3 4 4 1 1];
yi = [1 1 5 5 1 1 0 0 1];
clf
hold on;
grid on
plot( xi, yi, 'b*-' );
plot( xi-mean(xc), yi-mean(yc), 'r*--' );
axis equal;

n = 6;
th = linspace(-pi, pi, 2*n+1);
xi = 1+(1+rem((1:2*n+1),3)).*cos(th).^3;
yi = 2+(1+rem((1:2*n+1),2)).*sin(th);
clf;
hold on;
grid on
plot( xi, yi, 'b*-' );
axis equal;

sdpvar x y
Model = starpolygon(xi,yi,[x;y],[],[],[1;2]);
plot(Model,[x;y],[],[],sdpsettings('plot.shade',.1))    

Model = starpolygon(xi,yi,[x;y],[],.25,[1;2]);
plot(Model,[x;y],'b')  

Model = starpolygon(xi,yi,[x;y],[],0.25,[1;1]);
plot(Model,[x;y],'g') 

Model = starpolygon(xi,yi,[x;y],[-4;0],[],[1;2]);
plot(Model,[x;y],'y')   

% Define the template star
n = 6;
th = linspace(-pi, pi, 2*n+1);
xs = (1+rem((1:2*n+1),2)).*cos(th);
ys = (1+rem((1:2*n+1),2)).*sin(th);
clf
hold on;
grid on
plot( xs, ys, 'b*-' );
axis equal;

% Create the point-cloud
thn = linspace(-pi, pi, 2*3*n+1);
xn = 3+interp1(th,xs,thn)+.01*randn(1,2*3*n+1)
yn = 2+interp1(th,ys,thn)+.01*randn(1,2*3*n+1)
plot( xn, yn, 'r*' );

% Now say that each point in point-cloud is in scaled and translated polygon
sdpvar t
c = sdpvar(2,1);
Model = [];
for i = 1:length(xn)
  Model = [Model, starpolygon(xs,ys,[xn(i);yn(i)],c,t)];
end

% Minimize scale factor
optimize(Model,t)

% ...and plot the scaled and translated star
plot( value(t)*xs+value(c(1)), value(t)*ys+value(c(2)), 'k*-' );