# Ellipse Packing Via Hierarchical Optimization

Posted 6 years ago
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 Packing ellipses into a circumscribing circle is more complex than packing circles, since there is no formula for the minimum distance between two ellipses or the maximum distance from the center of the circumscribing circle to an ellipse. This is overcome by embedding constraints using Lagrange multipliers into the problem to locate the most distant point from the origin of an ellipse and the points on pairs of ellipses closest to each other. A result of a packing, showing the ellipses and the determined points is shown here: The complete calculation is in the attached notebook. The optimization was carried out as a local search using MathOptimizer Professionalhttp://www.wolfram.com/products/applications/mathoptproas FindMinimum does not converge for the calculation. Attachments:
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Posted 6 years ago
 20 ellipses takes about 6 minutes, using a Mathematica interface to the COIN-OR local solver Ipopt.
Posted 6 years ago
 latest results for packing ellipses into circles. the calculation for the attached plot took about 12 seconds.
Posted 6 years ago
 I've developed a slightly simpler method for packing ellipses in a circle. For preventing ellipse overlap, the point on one ellipse which minimizes the equation of the second ellipse is found. Here's an example:
Posted 6 years ago
 I've discovered that my approach for packing ellipses into an ellipse does not work for all situations.
Posted 6 years ago
 The notebook showing the packing of three ellipses into the circumscribing ellipse is attached. Pairs of closest points on all pairs of ellipses were determined using embedded Lagrange multiplier equations and then the values of the ellipse equations for the nearest points on the other ellipses were used to ensure no overlap. Attachments:
Posted 6 years ago
 I can now pack 3 ellipses into an ellipse. Details to follow