I’ve been experimenting with what I call “finite element packing”, as shown in the figures.
I start by packing ellipses into a 10-sided polygon which is approximately ellipse shaped, using global
optimization, using MathOptimizer Professional. I then use that result as the input for a local search for packing into a 20-sided polygon
and then repeat the process to go to a 40-sided polygon. I believe this method should work for other circumscribing
curves as well. Packing an ellipse into a polygon is easier than into a curve, as there is a symbolic expression for the maximum
extent of an ellipse in any direction.
4 Likes
Cool.
By “easier” do mean that if you can compute the packing of k ellipses in an ellipse, then you can do n+k ellipses in a n-gon with about the same effort? That can’t be right, but do you have a sense of what might be?
The reason it’s easier to pack ellipses into a polygon is that there is an symbolic expression for the extent of an ellipse in any direction, which can be used for constructing the constraints that keep the ellipses inside the polygons. I haven’t been able to come up the simple constraints to keep ellipses inside an ellipse, so I’ve used embedded Lagrange multipliers. However, that method doesn’t work well, as the Lagrange multiplier equations can have 2 or 4 real solutions, depending on the circumstance.