Thank you Joe, pretty good actually, for a quick attempt. eyeballing the results, looks like it is within less than 1% of theoretical maximum. Pretty good. Thank you.

Of course, typical design, 1,200+ polygons to run through, then I will decide that I don’t like it, so I will tweak it then run it again, and so on.

I can use your approach, but would prefer to find an algorithmic approach for easier inclusion into design flow. here is the matlib code, if anyone is fluent in its’ syntax:

% Section 8.4.1, Boyd & Vandenberghe “Convex Optimization”

% Original version by Lieven Vandenberghe

% Updated for CVX by Almir Mutapcic - Jan 2006

% (a figure is generated)

%

% We find the ellipsoid E of maximum volume that lies inside of

% a polyhedra C described by a set of linear inequalities.

%

% C = { x | a_i^T x <= b_i, i = 1,…,m } (polyhedra)

% E = { Bu + d | || u || <= 1 } (ellipsoid)

%

% This problem can be formulated as a log det maximization

% which can then be computed using the det_rootn function, ie,

% maximize log det B

% subject to || B a_i || + a_i^T d <= b, for i = 1,…,m

% problem data

n = 2;

px = [0 .5 2 3 1];

py = [0 1 1.5 .5 -.5];

m = size(px,2);

pxint = sum(px)/m; pyint = sum(py)/m;

px = [px px(1)];

py = [py py(1)];

% generate A,b

A = zeros(m,n); b = zeros(m,1);

for i=1:m

A(i, = null([px(i+1)-px(i) py(i+1)-py(i)])’;

b(i) = A(i,*.5*[px(i+1)+px(i); py(i+1)+py(i)];

if A(i,:)*[pxint; pyint]-b(i)>0

A(i, = -A(i,:);

b(i) = -b(i);

end

end

% formulate and solve the problem

cvx_begin

variable B(n,n) symmetric

variable d(n)

maximize( det_rootn( B ) )

subject to

for i = 1:m

norm( B*A(i,:)’, 2 ) + A(i,:)*d <= b(i);

end

cvx_end

% make the plots

noangles = 200;

angles = linspace( 0, 2 * pi, noangles );

ellipse_inner = B * [ cos(angles) ; sin(angles) ] + d * ones( 1, noangles );

ellipse_outer = 2*B * [ cos(angles) ; sin(angles) ] + d * ones( 1, noangles );

clf

plot(px,py)

hold on

plot( ellipse_inner(1,:), ellipse_inner(2,:), ‘r–’ );

plot( ellipse_outer(1,:), ellipse_outer(2,:), ‘r–’ );

axis square

axis off

hold off