Here we provide the instances used in the experiments presented in our paper as well as the solutions found by our methods. The descriptions of the solutions for the two- and three-dimensional problems are available in the following formats, respectively.

d1 d2
m
a1 b1 x1 y1 t1
a2 b2 x2 y2 t2
...
am bm xm ym tm

d1 d2 d3
m
a1 b1 c1 x1 y1 z1 t1 u1 v1
a2 b2 c2 x2 y2 z2 t2 u2 v2
...
am bm cm xm ym zm tm um vm

The first line has the dimensions of the container. In the two-dimensional case, it can be the length and width of a rectangle, the lengths of the semi-axes of an ellipse, or the radius of a circle (in which case this line has only one number). For three-dimensional instances, it can be the length, width and height of a cuboid, or the radius of a sphere (in which case this line has only one number). The container is centered at the origin. The second line contains the number m of ellipsoids packed. Then, each one of the next m lines contains the lengths of the semi-axes (a, b) or (a, b, c), the center (x, y) or (x, y, z), and the rotations angles t or (t, u, v) of an ellipsoid. In the three-dimensional case, we have adopted a left-handed coordinate system and the ellipsoids were rotated through an angle t about the x-axis, through an angle u about the y-axis, and through an angle v about the z-axis, in this order.

Three-dimensional instances

In the three-dimensional space, we have considered the problem of packing a given set of ellipsoids within a minimum volume container. The types of containers we have considered are ball and cuboid.

Minimizing the volume of the ball

The problem consists in packing a given set of ellipsoids within a ball of minimum volume. We considered instances defined by 100, 200, 300, 400, and 500 ellipsoids with semi-axis lengths 1, 0.75, and 0.5.

100 ellipsoids within a ball

Number of ellipsoids: 100
Semi-axis lengths: (1, 0.75, 0.5)
Container's radius: 3.83276
Container's volume: 235.84476
Density: 0.66602
Solution description

200 ellipsoids within a ball

Number of ellipsoids: 200
Semi-axis lengths: (1, 0.75, 0.5)
Container's radius: 4.80956
Container's volume: 466.02054
Density: 0.67413
Solution description

300 ellipsoids within a ball

Number of ellipsoids: 300
Semi-axis lengths: (1, 0.75, 0.5)
Container's radius: 5.48411
Container's volume: 690.88831
Density: 0.68207
Solution description

400 ellipsoids within a ball

Number of ellipsoids: 400
Semi-axis lengths: (1, 0.75, 0.5)
Container's radius: 6.04303
Container's volume: 924.38839
Density: 0.67971
Solution description

500 ellipsoids within a ball

Number of ellipsoids: 500
Semi-axis lengths: (1, 0.75, 0.5)
Container's radius: 6.53057
Container's volume: 1166.65841
Density: 0.67320
Solution description

Minimizing the volume of the cuboid

The problem consists in packing a given set of ellipsoids within a cuboid of minimum volume. We considered instances defined by 100, 200, 300, 400, and 500 ellipsoids with semi-axis lengths 1, 0.75, and 0.5.

100 ellipsoids within a cuboid

Number of ellipsoids: 100
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (4.41635, 6.35827, 8.72504)
Container's volume: 245.00312
Density: 0.64113
Solution description

200 ellipsoids within a cuboid

Number of ellipsoids: 200
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (6.31431, 7.73739, 9.76770)
Container's volume: 477.21441
Density: 0.65831
Solution description

300 ellipsoids within a cuboid

Number of ellipsoids: 300
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (6.97424, 8.92550, 11.38372)
Container's volume: 708.62093
Density: 0.66500
Solution description

400 ellipsoids within a cuboid

Number of ellipsoids: 400
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (8.14568, 9.75001, 11.78354)
Container's volume: 935.85630
Density: 0.67138
Solution description

500 ellipsoids within a cuboid

Number of ellipsoids: 500
Semi-axis lengths: (1, 0.75, 0.5)
Cuboid side lengths: (9.24587, 10.29180, 12.22661)
Container's volume: 1163.44518
Density: 0.67506
Solution description

Two-dimensional instances

In the two-dimensional space, we have considered the problem of packing a given set of ellipses within a minimum area container. The types of containers we have considered are circle, ellipse, and rectangle.

Minimizing the area of the ellipse

The problem consists in packing a given set of ellipses within an ellipse of minimum area. We considered instances defined by 100, 200, 300, 400, 500, and 1000 ellipses with semi-axis lengths 2 and 1.

100 ellipses

Number of ellipses: 100
Semi-axis lengths: (2, 1)
Container's semi-axis lengths: (14.26421, 16.08800)
Container's area: 720.94153
Density: 0.87152
Solution description

200 ellipses

Number of ellipses: 200
Semi-axis lengths: (2, 1)
Container's semi-axis lengths: (20.56799, 22.11586)
Container's area: 1429.04475
Density: 0.87935
Solution description

300 ellipses

Number of ellipses: 300
Semi-axis lengths: (2, 1)
Container's semi-axis lengths: (25.27320, 26.89813)
Container's area: 2135.66113
Density: 0.88260
Solution description

400 ellipses

Number of ellipses: 400
Semi-axis lengths: (2, 1)
Container's semi-axis lengths: (29.34767, 30.90109)
Container's area: 2849.03263
Density: 0.88214
Solution description

500 ellipses

Number of ellipses: 500
Semi-axis lengths: (2, 1)
Container's semi-axis lengths: (33.21420, 34.05469)
Container's area: 3553.45496
Density: 0.88409
Solution description

1000 ellipses

Number of ellipses: 1000
Semi-axis lengths: (2, 1)
Container's semi-axis lengths: (47.67451, 47.47700)
Container's area: 7110.81559
Density: 0.88360
Solution description

Minimizing the area of the rectangle

The problem consists in packing a given set of ellipses within a rectangle of minimum area. We considered instances defined by 100, 200, 300, 400, 500, and 1000 ellipses with semi-axis lengths 2 and 1.

100 rectangles

Number of ellipses: 100
Semi-axis lengths: (2, 1)
Container's length: 37.10730
Container's width: 19.35748
Container's area: 718.30399
Density: 0.87472
Solution description

200 rectangles

Number of ellipses: 200
Semi-axis lengths: (2, 1)
Container's length: 54.79460
Container's width: 25.98829
Container's area: 1424.01823
Density: 0.88245
Solution description

300 rectangles

Number of ellipses: 300
Semi-axis lengths: (2, 1)
Container's length: 69.44148
Container's width: 30.60858
Container's area: 2125.50589
Density: 0.88682
Solution description

400 rectangles

Number of ellipses: 400
Semi-axis lengths: (2, 1)
Container's length: 78.23603
Container's width: 36.06210
Container's area: 2821.35591
Density: 0.89080
Solution description

500 rectangles

Number of ellipses: 500
Semi-axis lengths: (2, 1)
Container's length: 89.97611
Container's width: 39.17699
Container's area: 3524.99378
Density: 0.89123
Solution description

1000 rectangles

Number of ellipses: 1000
Semi-axis lengths: (2, 1)
Container's length: 126.22572
Container's width: 55.55462
Container's area: 7012.42249
Density: 0.89600
Solution description

Minimizing the area of the circle

We considered an instance for the problem of packing non-identical ellipses within a minimizing area circle. This instance is formed by 231 ellipses whose semi-axis lengths belong to the set { ( 1 + 0.2 α , 1 + 0.2 β ) ∣ α , β ∈ { 0, 1 , … , 20 } , α ≤ β } . A description of this instance is also available at instance-231.txt. The description of the solution depicted in the figure below is available at solution-nonidentical-231.txt.

231 ellipses