1

OPTIMAL CLUSTERING PROBLEMS

Optimal clustering problem. Given two (irregular) objects bounded by circular arcs

and/or straight line segments where free continuous rotations of the objects are permitted, find

the minimal sizes of a given containing region (rectangle, circle, or convex polygon) according

to a given objective (a polynomial function) and placement parameters of two objects such that

the objects are placed completely inside the containing region without overlap and taking in to

account allowable distances between objects. We consider a number of frequently occurring

objectives, i.e. minimum area, perimeter, and homothetic coefficient of a given container.

We assume that the object frontier is given by an ordered collection of frontier elements

1 2, , ..., nl l l , (in counter clockwise order). Each element il is given by tuple ( , , , , )i ii i i c cx y r x y

if il is an arc or by tuple ( , , )i i ix y r if il is a line segment, where ( , )i ix y is the starting point of

il , ( , )i ic cx y is the centre point of an arc. We assume that element il is a line segment, if 0ir ;

il is a "convex" arc, if 0ir ; il is a "concave" arc, if 0ir .

We consider the following containing regions:

a) an axis-parallel rectangle: = {( , ) : 0 , 0 }R x y x a y b of variable a and b in fixed

position (Fig. 1 a),

b) a circle of variable radius r: 2 2= {( , ) : }C x y x y r with origin at the center point

(Fig. 1b),

c) a convex polygon K :

K is given by its variable sides ie , 1, ...,i m (Fig. 1c), where each side

1[ , ]i i ie v v of variable length it is defined by two variable vertices

( , )i i iv x y and 1 1 1( , )i i iv x y , 1 cosi i i ix x t , 1 sini i i iy y t .

Therefore, each side ie may be given by variable vector ( , , , )i i i ix y t ,

K is given by its verticies ( , )i i iv x y , 1, ...,i m , where is a variable

homothetic coefficient and ix and iy are constant (Fig. 1d), subject to for

original polygon 1 .

2

Fig. 1 Containing region of variable metrical characteristics

We consider six realizations of the optimal clustering problem, denoted by P1, .., P6, with

respect to the shape of the containing region and the form of the objective function ( )F u :

P1: R , 1( ) =F u a b (area of R),

P2: R , 2 ( ) =F u a b (half-perimeter of R),

P3: C , 3( ) =F u r (radius of C),

P4: K , 41

( )m

ii

F u t

(perimeter of K),

P5: K , 5 1 11

( ) ( )m

i i i ii

F u x y y x

s.t. 1 1m , (doubled area of K),

P6: K , 6( ) =F u (homotetic coefficient of K).

3

Computational experiments In all cases the input data of the example has been provided in Appendix. For local

optimisation in our programs we use IPOPT (https://projects.coin-or.org/Ipopt) developed by

[1]. We use computer AMD Athlon 64 X2 5200+.

Example 1. We consider two triangles A and B and problem P2. The task is to find the

enclosing rectangle of minimal perimeter, i.e. ( ) =F u a b .

Example 1.1. Non-rotatable case. In this example we demonstrate the approach for

computing the global solution. We use Algorithm 1 to relise model (6)-(7). Figure 2 shows the

optimal arrangments of A and B which coorespond to six local minima of problem (4)-(5)

arising from the solution tree. Since all subproblems (7) are linear, we can find the global

Fig. 2 Arrangments of A and B of Example 1.1, corresponding to points *su , 1, , 6s

4

minima *( )F u of problem (6)-(7). In Figure 5, each solution point *su is the global minimum

of subproblem (5), 1, , 6s . Solution * 4*u u is the point of the global minimum of problem

(4)-(5). * 1* 2* 3* 4* 4*( ) min{ ( ), ( ), ( ), ( )} ( ) 7.6667.F u F u F u F u F u F u Running time is

0.06 sec.

Example 1.2. Continuous rotations are allowed, *( )F u a b = 6.3640. Running time is 0.109

sec, see Figure 3. We use Algorithm 1 to relize model (6)-(7).

Fig. 3 Arrangment of polygons A and B corresponding to point *u , Example 1.2

Example 2. We consider two rotated objects A and B, see Figure 4. We use Algorithm 2.

Example 2.1 Clustering of objects A and B into a rectangle R which looks like the optimal,

problem P1 (Figure 4a) , * * *( ) = 23.2253F u a b . Running time is 0.431 sec.

Example 2.2 Clustering of objects A and B into a circle C, which looks like the optimal,

problem P3 (Figure 4b) , * *( ) = = 3.2599F u r . Running time is 0.387 sec.

Example 2.3 Clustering of objects A and B into a rectangle R taking into account minimal

allowable distance =0.6 between objects, which looks like the optimal, problem P1 (Figure 4c).

* * *( ) = 27.685F u a b . Running time is 0.407sec.

Example 2.4 Polygonal approximation to the minimal convex hull of objects A and B , problem

P5 (Figure 4d) . * *( ) = = 42.6835F u S . Running time is 0.589 sec.

Example 2.5 Clustering of objects A and B into a convex pentagon K of minimal homotetic

coefficient, which looks like the optimal, problem P6 (Figure 7e). * *( ) = = 0.5259F u .

Running time is 0.401 sec.

A

B

5

(a) (b)

(c) (d) (e)

Fig. 4. Arrangement of objects A and B as described in Example 2: a) minimal enclosing

rectangle, b) minimal enclosing circle, c) minimal enclosing rectangle taking into account

distance constraints, d) minimal enclosing m-polygon, e) minimum homottetic coefficient

Example 3. We consider two irregular objects A and B , see Figure 5. We use Algorithm 2.

Example 3.1 . Optimal clustering of objects A and B in a circle of minimal radius, which looks

like the optimal, problem P3, (Figure 5a). F(u*)=8.5826.Running time is 0.531 sec

Example 3.2. Clustering of objects A and B in a circle of minimal radius with allowable

distance 0.3 , which looks like the optimal, problem P3, (Figure 5b). F(u*)=r*=11.3709.

Running time is 1.235 sec

Example 3.3. Clustering of objects A and B in a convex polygon of minimal area, which looks

like the optimal, problem P5, (Figure 5c). F(u*)=S*= 439.8638. Running time is 5.06 sec.

A

B

B

A

B

A

A B

A

B

6

(a) (b) (c)

Fig 5. Minimal enclosing container of objects A and B as described in Example 3: a) circle,

without distance constraints , b) circle, with distance constraints, c) convex m-polygon

Exanple 4. We consider two irregular objects A and B , see Figure 6. We use Algorithm 2.

Example 4.1 Clustering of objects A and B in a circle of minimal radius, which looks like the

optimal, problem P3, (Figure 6a). F(u*)=17.7674. Running time is 5.031 sec.

Example 4.2 Clustering of objects A and B in a rectangle of minimal area, which looks like

optimal, problem P1, (Figure 6b). F(u*)=1121.6867. Running time is 2.938 sec.

Example 4.3 Optimal clustering of objects A and B in a convex m-polygon, which looks like

optimal, problem P5, (Figure 6c). F(u*)=1736.6091. Running time is 10.375 sec.

(a) (b) (c)

Fig 6. Minimal enclosing regions for objects A and B of Example 4: a) rectangle, b)

circle, c) convex m-polygon

Example 5. The convex hull of two rotated convex polygons A and B , the optimal solution,

problem P5, see Figure 10a. F(u*) =387.5215. Running time is 0.14 sec. We use Algorithm 1.

B

A A

B

B

A

B

A

7

(a) (b)

Fig. 7 The convex hull for objects A and B: (a) two convex polygons, Example 5, (b) two non-

convex arc objects, Example 6.

Example 6. The convex hull for two rotated objects A and B, which looks like optimal, problem

P5, see Figure 9b. F(u*) =51.0228. Running time is 0.484 sec. We use Algorithm 2.

Example 7. An approximation of the convex hull for two convex polygons considering distance

constraints, which looks like the optimal, problem P5, see Figure 8. Data of polygons A and B,

are given for Example 1 in Appendix D. F(u*) =11.3211. Running time is 0.58 sec. We use

Algorithm 1.

Fig. 8. An approximation of the m-polygonal convex hull for two convex polygons of Example1

Example 8. An approximation of the convex hull for two non-convex objects, see Figure 12. We

use Algorithm 2.

Example 8. 1. An approximation of the convex hull of minimal area for two non-convex objects,

which looks like the optimal, problem P5, (Figure 9 a ), F(u*) =373.5249. Running time is 0.56

sec

Example 8. 2. An approximation of the convex hull of minimal perimeter for two non-convex

objects, which looks like the optimal, problem P4, (Figure 9b), F(u*) =55.0508 . Running time is

0.57 sec

A B

B

A

A

B

8

(a) (b)

Fig 9. An approximation of the m-polygonal convex hull for two non-convex objects of Example

8: (a) area of the convex hull, m=6 , (b) perimeter of the convex hull, m=8

[1]. Wachter, A., Biegler, L. T.: On the implementation of an interior-point filter line-search

algorithm for large-scale nonlinear programming.Mathematical Programming.106, 1, 25-57 (2006)

APPENDIX: Objects data and output data for examples Example_1. INPUT DATA

OBJECT A EX_1

Al (2, -1, 0, 0, 2, 0, -2, 0, 0)

OBJECT B EX_1

Bl (0, 0, 0, 3, 2, 0, 0, 2, 0)

Example_1 OUTPUT DATA

Example 1.1. * * * * * * *1 1 2 2= ( , , , , , )u a b x y x y = (4.0, 3.6667, 2.0, 1.0, 0.0 , 1.6667) .

Example 1.2. * * * * * * * * *1 1 1 2 2 2= ( , , , , , , , )u a b x y x y = (3.5355, 2.8284, 2.1213, 1.4142, 2.3562,

0.0791, 0.7591, 6.3087)

Example_2. INPUT DATA

OBJECT A EX_2 Al (-1.605, -2.125, -2.693, 0.829, -3.278, 1.892, -0.804, 0, 2.039, 1.369, 0, -0.2372, 2.0661, 0)

OBJECT B EX_2 Bl (2.022, -1.281, 1.843, 1.1539, 0.3449, 0.708, 2.133, 12.743, 7.836, -8.429,

-2.934, -1.619, -3.632, -0.276, -4.0936).

Example_2. OUTPUT DATA

Example 2.1.

A B

B

A

9

* * * * * * * * *1 1 1 2 2 2= ( , , , , , , , )u a b x y x y = (6.0977, 3.8089, 4.1637, 2.9426, 1.2554, 2.8937,

1.2500, -2.3398)

Example 2.2. * * * * * * * *

1 1 1 2 2 2= ( , , , , , , )u r x y x y = (3.2599 -0.2514 1.4905 -5.4582,-0.1020, -0.7134,

-9.01344)

Example 2.3. * * * * * * * * *

1 1 1 2 2 2= ( , , , , , , , )u a b x y x y (4.4249, 6.2566, 0.8663, 4.3227, 5.9678, 3.1659, 2.8858,

2.3858)

Example 2.4. m=11, * * * * * * * * * * * * * * *1 1 1 1 11 11 11 11= ( , , , , ..., , , , , , , , , , )A A A B B Bu x y t x y t x y x y =(3.7724,

0.0000, 2.3683, 0.8404, 3.1711, -0.5870, 2.8165, 0.8404, 2.3747, -0.8554, -3.0185, 0.8404,

1.5408, -0.7522, -2.5702, 0.8404, 0.8339, -0.2977, -2.1220, 1.1793, 0.2163, 0.70686, -1.6199,

4.4084, 0.0000, 5.1099, -0.0673, 3.5617, 3.5537, 5.3494, 0.5327, 0.0000, 3.5537, 5.3494,

1.3184, 1.3214, 3.8837, 4.0698, 1.5251, 2.6428, 4.0045, 1.4298, 1.7318, 1.4485, 0.9580 , 3.2641,

5.9187, 2.7233, 2.1026, 2.3254)

Example 2.5.

Pentagon K is given by a vector of coordinates of its vertices:

(7.0190, 1.4637, 0, 1.8053, 7.2809, 0, -5.3382, 4.1200, 0, -4.5396 -3.6507, 0, 3.0977 -5.2924, 0)

m=5, * * * * * * * *= ( , , , , , , )A A A B B Bu x y x y ( 0.5259 , 1.6035 1.1955 8.1947 -0.3486 -0.0779

10.8685)

Example_3 INPUT DATA

OBJECT A EX_3

Al ( 4.326, 6.395, -1.433, 2.914, 6.639, 1.56, 6.169, -2.405, 3.738, 7.189, 1.333, 7.212, 1.507, -0.143, 6.908, -0.553, 8.358, 3.78, 3.226, 8.266, 2.139, 4.645, -1.335, 1.574, 3.436, 0.749, 2.385, -41.479, 26.033, 35.267, -4.337, 7.015, 0.69, -4.999, 6.826, -4.974, 6.137, -11.293, -10.967, -3.435, -1.594, 2.865, -2.1027, -3.484, 1.944, -1.523, 1.186, -3.278, -1.925, 4.439, -4.36, 2.245, 18.437, -17.211, -10.975, -8.242, 5.133, -19.365, -19.496, -10.626, -3.431, 0.187, -0.727, -4.157, 0.218, -4.551, -0.393, -6.038, -1.879, 5.022, -6.914, 1.689, 1.485, -8.213, 0.971, -9.274, 2.01, -1.738, -8.923, 0.308, -8.055, -1.197, 7.273, -2.074, 2.942)

OBJECT B EX_3

Bl (2.493, 6.764, 1.771, 2.143, 5.028, 0.38, 5.191, -4.788, 1.364, 0.506, 4.149, 4.399, -1.876, 2.274, 4.368, 1.795, 2.554, 10.681, -0.594, -7.857, -1.702, 2.767, 8.905, 2.509, 10.614, 4.229, 1.877, -0.496, 4.671, 1.651, 4.781, 1.167, -16.555, 1.111, 17.31, -1.293, 0.931, 0.944, -1.623,

10

0.047, -1.214, -0.804, -10.767, 1.31, -11.271, 3.834, -0.804, -1.324, 3.524, -2.091, 3.361, -3.405, -40.094, 8.293, 36.385, -2.239, -2.301, 0.723, -2.193, -3.023, -2.003, -3.72, -4.687, -1.968, -8.407, 1.231, -4.981, -2.155, -0.867, -5.474, 0.837, -6.794, -1.946, -0.701, -5.602, -2.239, -6.794, 1.103, -3.112, -7.469, -3.423, -8.528, 7.762, 0.541, -1.854, 2.0200 -9.4743 8.4740 -0.1576 -1.2849).

Example 3. OUTPUT DATA

Example 3.1. * * * * * * * *1 1 1 2 2 2= ( , , , , , , )u r x y x y = (8.5826, 2.6036, 4.1595, -1.9849, 1.1292, -

0.5965 -4.4319)

Example 3.2. * * * * * * * *1 1 1 2 2 2= ( , , , , , , )u r x y x y = (5.6452, 4.7846, -1.3617, -1.1846, -3.5867, -

3.6332)

Example 3.3.

m=24, * * * * * * * * * * * * * * *1 1 1 1 24 24 24 24= ( , , , , ..., , , , , , , , , , )A A A B B Bu x y t x y t x y x y (-5.3369, -8.5664, -

2.6532, 2.4302, -7.4829, -7.4261, -2.3683, 2.4302, -9.2220, -5.7287, -2.0835, 2.4302, -10.4141, -

3.6120, -1.7987, 2.4302, -10.9631, -1.2436, -1.5138, 2.4302, -10.8247, 1.1826, -1.2290, 2.4302,

-10.0101, 3.4722, -0.9441, 1.2151, -9.2975, 4.4564, -0.8760, 0.9460, -8.6918, 5.1831, -0.6335,

1.8920, -7.1669, 6.3030, -0.3909, 1.8920, -5.4177, 7.0239, -0.2271, 1.3731, -4.0798, 7.3330,

0.0182, 4.0874, 0.00688, 7.2587, 0.3034, 2.2215, 2.1269, 6.5950, 0.6065, 2.2215, 3.9522,

5.3288, 0.9096, 2.2215, 5.3164, 3.5755, 1.2127, 2.2215, 6.0950, 1.4949, 1.5158, 2.22150,

6.2171, -0.7232, 1.8189, 2.2215, 5.6716, -2.8767, 2.1220, 2.2215, 4.5081, -4.7695, 2.3619,

4.8634, 1.0496, -8.1884, 2.6565, 0.5084, 0.5998, -8.4255, 2.7755, 1.2151, -0.5346, -8.8605,

3.0603, 2.4302, -2.9569, -9.0577, -2.9380, 2.4302, 2.1774, 1.4609, 4.9044, -1.7436, -1.6084,

2.4573)

Example_4 INPUT DATA

OBJECT A EX_4

Al (0.916, -3.2835, -2.1141, -1.1925, -3.4331, -3.1599, -4.2069, 1.5176, -4.5722, -4.7624, -6.0118, -5.243, -1.9194, -7.8323, -5.8509, -9.7247, -5.5302, 6.5362, -16.1691, -4.4383, -10.9427, -0.5133, 0.531, -11.3672, -0.8321, -11.8691, -0.6587, 6.4935, -5.7318, -2.7797, -11.614, -5.5302, -10.3158, -20.9587, -9.8998, -10.6432, -9.9873, 1.0287, -9.8552, -10.6486, -9.5622, -11.6347, 12.1386, -19.1497, -4.1899, -7.9948, -8.9767, -0.8426, -7.1675, -9.1368, -6.3926, -8.8058, 1.509, -5.0049, -8.213, -3.5952, -8.7513, -2.7401, -1.0353, -9.7286, -0.3511, -7.0753, -2.5557, 0.6609, -9.4221, 2.6318, -7.7951, 3.383, 5.924, -7.0167, 5.1053 ,-3.7343, -2.0191, 5.7366, -1.8164, 4.5707, -0.1679, 5.8141, -0.784, 2.0974, 4.2101, 5.0745, 2.1588, 2.3558, 3.9691, 2.1899, 6.1215, -0.8636, 2.1236, 6.9825, 1.721, 7.7466, 1.7119, 0.9229, 9.2611, 0.7867, 10.9676,-1.2333, 0.6886,12.1970,1.1388,13.3452,2.0740, 1.8957, 15.2761, 3.2381, 16.8571, 2.0329, 1.9223,

11

15.3074, -0.0714, 15.7046 -0.8439,-0.8991,15.8695,-1.5973,15.3956,1.5839,-2.9079, 14.5062, -3.3626, 12.9889, -1.2291, -3.7155, 11.8116, -2.5921, 11.3130,-1.0093,-3.5146, 11.7225, -4.2379, 11.0185, 0.9691, -4.9323, 10.3426, -4.8298, 9.3789, -1.7325, -4.6465, 7.6562, -4.5749, 5.9252, 2.3942, -4.4760, 3.5330, -4.0691, 1.1736, -3.8135, -3.4210, -2.5844, -0.6502, 0.0357, -5.8961, -4.9343, -4.0153)

OBJECT B EX_4

Bl (-3.9051, 15.6754, 3.6105, -6.3238, 12.9949, -5.4522, 9.4912, -1.7667, -4.4782, 8.0172, -5.7334, 6.7739, -7.4716, -6.5223, 14.2038, -8.5183, 7.0037, -5.0616, -13.3461, 5.4831, -8.3588, 4.6188, 6.9000, -2.9414, 0.3454, -9.7608, -0.7063, 4.6404, -11.0770, 3.7435, -6.4367, 3.7287, 1.0321, -5.8343, 4.5668, -4.8427, 4.8531, -1.5625, -3.2958, 4.6331, -1.7954, 4.1972, 0.9511, -1.0711, 3.5808, -0.7171, 2.6980, -8.0493, 2.2783, -4.7732, 9.5968, -8.1244, -4.6532, 5.3660, -6.1871, 1.0175, -7.8433, 1.0906, -0.0017, -8.2314, -1.0922, -8.2181, -3.8299, -4.9218, -8.1712, -3.1081, -11.5444, -3.3853, -4.7112, -8.5628, -8.0775, -8.9208, 1.1200, -9.1913, -9.0393, -10.2810, -8.7803, -1.2462, -11.4934, -8.4921, -11.2655, -9.7173, 1.8333, -10.9302, -11.5197, -10.6091, -13.3247, -4.1852, -9.8762, -17.4452, -5.9678, -15.9483, 2.5006, -3.6327, -15.0539, -1.7485, -16.6979, -71.089, 51.8177, -63.4345 1.6738 -13.0436 -2.7890, 3.6411, -15.0206, 5.9870, -13.5121, 1.6693, 7.3910, -12.6092, 9.0343, -12.9031, -12.2743, 21.1169, -15.0634, 13.7224, -5.2665, 3.2491, 11.7650, -2.6732, 13.2536, 0.2149, -4.2809, 15.2149, 4.0201, 11.5659, 6.2586, 1.4523, 10.3279, 7.0180, 10.1594, 8.4605, -7.0745, 9.3387, 15.4872, 4.3461, 10.4751, 0.7968, 3.7838, 9.9105, 3.1741, 9.3975, -1.5498, 1.9882, 8.3997, 1.4863, 9.866, 2.0727, 0.8151, 11.8271, 2.8459, 11.4121, -1.7315, 4.5423, 11.0654, 5.2837, 12.6302, 0.8337, 5.6407, 13.3836, 6.4089, 13.7077, 4.1508, 2.5845, 12.0942, 1.6738, 16.1439, -20.4365, -2.8097, 36.0825)

Example 4. OUTPUT DATA

Example 4.1. * * * * * * * *1 1 1 2 2 2= ( , , , , , , )u r x y x y = (17.7674, -1.2785, 5.0441, -2.7258, -0.2362, -

2.6272, 1.8515)

Example 4.2. * * * * * * * * *1 1 1 2 2 2= ( , , , , , , , )u a b x y x y =(32.8975, 34.0964, 22.0452, 19.8732, 4.7927,

15.1417, 16.3673, 3.0874)

Example 4.3. u*= * * * * * * * * * *1 1 1 1( , , , , ..., , , , , , )m m m m A Bx y t x y t u u , m=22, where

* * * * * * * *1 1 1 1( , , , , ..., , , , )m m m mx y t x y t = (32.1154, -1.3741, 2.4936, 9.2842, 24.7132, -6.9780, 2.6752,

0.2644, 24.4770, -7.0969, 2.8567, 10.1271, 14.7581, -9.9431, 3.0798, 0.7279, 14.0316, -9.9880,

-2.9803, 0.7279, 13.3131, -9.8711, -2.7572, 0.7279, 12.6384, -9.5981, -2.5341, 11.4076, 3.2719,

-3.0864, -2.17501, 0.6059, 2.9277, -2.5878, -1.8159, 12.0635, 0.0000, 9.1151, -1.4215, 0.9994,

0.14867, 10.1034, -1.0270, 14.4742, 7.6372, 22.4899, -0.6472, 1.3006, 8.6747, 23.2741, -0.2674,

1.3006, 9.9291, 23.6177, 0.1125, 16.8834, 26.7059, 21.7236, 0.5216, 0.4268, 27.0760, 21.5110,

0.9306, 11.0851, 33.6978, 12.6210, 1.0337, 0.8197, 34.1172, 11.9168, 1.3810, 1.1366, 34.3316,

10.8006, 1.6580, 9.9226, 33.4677, 0.9157, 1.8892, 0.9642, 33.1660, 0.0000, 2.1204, 0.9641,

12

32.6624,-0.82219,2.3517,0.7771), * *( , )A Bu u =(14.6660,12.1616, 3.334, 17.4147, 4.9045, 1.6176).

Example 5. INPUT DATA

OBJECT A EX_5 Al =(-7.2662, 1.5935, 0, -5.9413, -6.8803, 0, -3.2915, -8.7340, 0, 2.3109, -10.6633, 0 6.7020, -

10.6633, 0) OBJECT B EX_5

Bl =(-1.443, -4.819, 0. 2.67, -1.107, 0. 2.089,7 4.41, -4.991, -2.885, 3.884, 0.83, 0.55, 0, -1.443, 2.605, -5.001, -4.7927, -1.107, 8 0.11, -0.12, -5.0, -2.885, 3.884, -2.879, -1.116, 0. 0.21, -1.116, -5.0, -4.7927, -1.109)

Example 5. OUTPUT DATA

m=10, u*= * * * * * * * * * *1 1 1 1( , , , , ..., , , , , , )m m m m A Bx y t x y t u u =(10.0238, -2.2864, 3.0242, 3.2338, 6.8123, -

2.6651, -2.4537, 8.5767, 0.186, 2.7804, -1.5931, 8.3482, 0, 11.1266, -0.9437, 9.1846, 5.3899, 18.5634, -

0.2207, 15.5024, 20.5161, 21.9577, 1.463, 7.9482, 21.3712, 14.0556, 1.8004, 11.4346, 18.7688, 2.9211,

2.4138, 0.0001, 18.7688, 2.921, 2.4138, 4.391, 15.4903, 0, 2.7454, 5.925, 6.6713, 6.4244, 5.5554,

6.6713, 6.4244, 5.5554)

Example 6. INPUT DATA

OBJECT A EX_6

Al =(-1.4427, -4.8186, 0, 2.6700, -1.1068, 0, 2.089, 4.4005, -1, 0.830000, 0.5500, 0, -1.4427, 2.6050, 0,

-0.1100, -0.1100, -1, -2.8787, -1.1163, -1, 0.2100, -1.1163, 0, -1.4427, -4.8186, -1)

OBJECT B EX_6

Bl =(-1.4427, -4.8186, 0, 2.6700, -1.1068, 0, 2.0887, 4.4005, -1, 0.8300, 0.5500, 0, -1.4427, 2.6050, 0, -

0.1100, -0.1100, -1, -2.8787, -1.1163, -1, 0.2100, -1.1163, 0, -1.4427, -4.8186, -1)

Example 6. OUTPUT DATA

m=6, u*= * * * * * * * * * *1 1 1 1( , , , , ..., , , , , , )m m m m A Bx y t x y t u u = (0.0889, 5.3038, -1.5932, 3.9616, 0, 9.2643,

0.3428, 5.5377, 5.2157, 7.4031, 1.2794, 5.5372, 6.8065, 2.099, 1.5484, 3.9615, 6.896, -1.8612, -2.7988,

5.5379, 1.6797, 0, -1.8622, 5.5372, 3.0618, 5.4759, 5.1604, 3.8337, 1.9273, 2.019)

Example 7.

INPUT DATA

0.2 is allowable distance between polygons A and B (see input data on polygons of example 1).

OUTPUT DATA

13

m=16, u*= * * * * * * * * * *1 1 1 1( , , , , ..., , , , , , )m m m m A Bx y t x y t u u = (5.3365, -0.1823, 2.5860, 2.9023, 2.8708, -

1.7132, 2.7833, 0.2494, 2.6372, -1.8006, 3.0773, 0.0592, 2.5782, -1.8045, 3.3713, 2.0522, 0.5799, -

1.3371, -2.6874, 0.045, 0.5394, -1.3173, -2.463, 0.0451, 0.5043, -1.289, -2.2386, 0.0451, 0.4764, -

1.2536, -2.0142, 0.045, 0.4571, -1.2129, -1.7898, 0.045, 0.4473, -1.169, -1.5655, 0.0451, 0.4475, -

1.1239, -1.341, 3.1296, 1.1603, 1.9235, -0.3583, 0.3955, 1.5307, 2.0621, 0.4747, 4.2656, 5.3246,

0.1123, 1.0025, 0.1081, 5.3827, 0.0212, 1.5303, 0.1081, 5.3871, -0.0868, 2.0581, 0.1081, 3.2376,

0.4146, 3.3713, 2.5949, -1.6030, 17.5085)

Example 8. INPUT DATA

OBJECT A EX_8 and OBJECT B EX_8

Al = Bl =( 2.0, 7.0, 0, -4.0, -3.0, 0, 0, -5.0, -2.5, 1.5, -3.0, 3.0, -5.0, 0, 11.0, 0, -8.0623, 10.0, 8.0)

Example 8. OUTPUT DATA

Example 8.1. m=6, u*= * * * * * * * * * *1 1 1 1( , , , , ..., , , , , , )m m m m A Bx y t x y t u u = (20.2289, -2.57832, 1.4978

4.4721, 20.5550, -7.0386, 3.1454, 11.6619, 8.8932, -6.9943, -2.4381, 11.6619, 0.0000, 0.5497, -

0.7467, 11.4018, 8.3684, 8.2938, 0.6766, 13.2450, 18.6952, 0.0000, 1.0342, 3.0000, 15.9317, -

5.1345, 4.1758, 6.5824, -2.5603, 4.8755)

Example 8.2. m=8, u*= * * * * * * * * * *1 1 1 1( , , , , ..., , , , , , )m m m m A Bx y t x y t u u = ( 0, 4.4189, -0.3471,

10.6193, 9.9858, 8.0318, 0.3941, 9.4340, 18.696, 4.4093, 0.9527, 3, 20.435, 1.9643, 1.4164,

4.4721, 21.123, -2.4546, 2.794, 10.6193, 11.1372, -6.0674, -2.7475, 9.434, 2.4264, -2.445, -

2.1889, 3, 0.688, 0, -1.7252, 4.4721, 16.3601, -0.9331, 4.0943, 4.7629, 2.8974, 0.9527)