Enclosing Ellipse of a set of points

Hello dear community,

I have several points of specific material values and want to enclose them with an ellipse. I have tried it with least squares approaches and also with simpler approaches but without success... The datasets contain three or more points. The ellipse should also be rotated in order to be the best fit for the data. The ellipse as well as the points should be plotted afterwards. Below I have attached one set of my data as an example.

%%Aluminum Data

alu = cell(2,4);

% Aluminum Al 99,5 F7 Merkel
alu{1,1} = 'Al 99,5 F7';
alu{1,2} = 2.7;
alu{1,3} = 0.120;
alu{1,4} = 68.670;
% Aluminum Al 99,5 F17 Merkel
alu{2,1} = 'Al 99,5 F17';
alu{2,2} = 2.7;
alu{2,3} = 0.180;
alu{2,4} = 68.670;
% Aluminum 7075 T6 ASM
alu{3,1} = 'Al 7075 T6';
alu{3,2} = 2.81;
alu{3,3} = 0.572;
alu{3,4} = 71.700;
% Aluminum 6061 T6 ASM
alu{4,1} = 'Al 6061 T6';
alu{4,2} = 2.7;
alu{4,3} = 0.310;
alu{4,4} = 68.900;
% Aluminum 2024 T8 Alcoa
alu{5,1} = 'Al 2024 T8';
alu{5,2} = 2.78;
alu{5,3} = 0.462;
alu{5,4} = 72.400;
% Aluminum 2090 T83 Alcoa
alu{5,1} = 'Al 2090 T83';
alu{5,2} = 2.59;
alu{5,3} = 0.531;
alu{5,4} = 76.000;
% Aluminum 2199 T8 Alcoa
alu{6,1} = 'Al 2199 T8';
alu{6,2} = 2.59;
alu{6,3} = 0.428;
alu{6,4} = 76.000;
for i=1:1:6
    al(i,1) = alu{i,3}/alu{i,2};
    al(i,2) = alu{i,4}/alu{i,2};
end

The ellipse should enclose the data of array al.

I would appreciate, if you could help me. Thanks in advance!

Georg

Answer 1

[John D'Errico]

I recall playing with this general problem some time ago. The ad hoc scheme I came up with was to use an optimization that would vary the angle of the major axis, as well as the eccentricity of the ellipse. That yields a coordinate transformation to turn the problem into that of finding a minimum radius enclosing circle, which I have code to do. Given the circle and other parameters, recover the classic parameters of an ellipse.

The above scheme should work reasonably well, though developed without any search to see if the problem has been solved before. I'm sure it has.

I'd also suggest a search on the file exchange, as I think I recall seeing a minimum enclosing ellipse tool there. A quick search with the string "enclosing ellipse" finds this as a start:

http://www.mathworks.com/matlabcentral/fileexchange/9542-minimum-volume-enclosing-ellipsoid

There are probably others.