A New Objective Function for the Recovery of Gielis Curves
Abstract
:1. Introduction
2. Gielis Curves
3. Background
- Each gene represents a parameter from the superformula, i.e., each gene may be represented by a real number, except p and q, which are restricted to integer numbers.
- Crossover is performed as follows: given two parent solutions, two children solutions are obtained, where each child inherits each gene from one of the parents with a probability of 0.5, using a value between 0 and 1. Since p and q must be relative primes, in case they are taken from different parents, the fulfillment of this condition is verified. If it is not fulfilled, both genes are taken from the same parent, in order for this condition to be fulfilled. Figure 3 shows crossover method: in Figure 3a an example in which the crossover produces a child in which p and q are not relative primes (4 and 2, respectively), and in Figure 3b the correction applied in order for the condition to be fulfilled.
- In order to perform mutation on continuous variables (a, b, , , ), a normal distribution is used as follows:
- Discrete variables (p and q in this case) indicate the number of sectors of the Gielis curve and the maximum number of self-intersections, respectively. When performing mutation, small variations of these variables have no reason for being, since they produce big changes in the adaptation to the sample points. This can be observed in Figure 4, where both Gielis curves have the same parameters, except p, but they have completely different shapes. For this reason, it is suggested to simply generate random values, with the condition that p and q must be relative prime.
- The selection process is performed as follows: the size of the population is kept constant and individuals with best adaptive value remain. In order to face local convergence and premature convergence issues, Fougerolle et al. [11] suggested to measure the standard deviation of the population after each iteration, and if it is less than a user defined threshold, , the population is re-initialized randomly. This is computed with normalized values. The norm used is defined according to Equation (4), denoted as
- If the standard deviation of all variables is less than the defined threshold, population is re-initialized randomly.
4. Proposal
- Obtain a finite amount of points from the generated Gielis curve, to equals intervals of , with a sampling. An example of this is observed in Figure 9, with the points in red.
- For each point = find the sample point that is closest and compute its Euclidean distance. This distance is observed in green in Figure 9.
5. Experimental Tests
5.1. Evaluation Metrics
- Considering that the tests were performed using Gielis curves with known parameters, an amount h of sample points in the original curve is obtained, to equal intervals of . These sample points are obtained by dividing the interval in h equal parts. Figure 13a shows an example where , and since , the values of are . There is in blue the original curve and in black the points obtained.
- For each solution (one for each method), an amount h of sample points is obtained, where values of are the same used for the original curve. Figure 13b shows an example, corresponding to the Gielis curve obtained using objective function in Equation (14). The Gielis curve is in red and the sample points are in black.
- Considering a h-dimensional space, finite representation of the original curve and the solutions obtained with each method are built, with polar distances (, ,..., ) for each of them. We denote as the original curve, as the Gielis curve from the method proposed by Fougerolle et al., as the curve from the method that uses objective function from Equation (13) and for the method that uses our contribution, the objective function from Equation (14). In the example from Figure 13, there are and .
- Finally, the Euclidean distance between and each h-dimensional point corresponding to each solution is calculated. The distance between and will be called ; between and , ; and between and , . The lower the value of the distance, the better the quality of the solution. Continuing with the example given in Figure 13, Figure 14 shows , that is equal to the root square of the sum of squares of the length of the segments in green.
5.2. Parameters Used
- Mutation parameter k = 0.1.
- Mutation probability is 0.15.
- Minimum limit for population standard deviation = 0.05.
- The size of the population is 100 individuals.
- The number of generations is 1000.
- The intervals of the parameters of the superformula are a, b∈ [0.01, 100]; , , ∈ [0.01, 1000]; p, q∈ [1, 11].
5.3. Results
6. Conclusions and Future Work
Author Contributions
Funding
Conflicts of Interest
Appendix A
N | ■ | F1 | F2 | F3 |
---|---|---|---|---|
1 | 1, 1, 2, 1, 1, 1, 1 | 0.484, 0.791, 4, 3, 98.105, 975.495, 129.099 | 0.319, 0.832, 4, 3, 424.050, 114.101, 770.828 | 0.914, 0.159, 4, 1, 325.238, 742.188, 59.849 |
2 | 1, 1, 2, 1, 1, 4, 8 | 4.588, 0.414, 4, 7, 191.792, 144.745, 383.061 | 1.042, 93.605, 4, 1, 279.456, 290.473, 67.824 | 1.079, 75.865, 4, 1, 498.795, 499.327, 127.463 |
3 | 1, 1, 2, 1, 2, 2, 2 | 0.696, 0.714, 1, 2, 6.293, 960.253, 0.142 | 0.936, 0.704, 9, 5, 304.509, 938.779, 1.222 | 0.893, 0.686, 10, 1, 438.576, 755.576, 0.190 |
4 | 1, 1, 3, 1, 1, 1, 1 | 0.297, 0.937, 6, 5, 61.612, 979.904, 316.092 | 0.297, 0.812, 2, 1, 449.637, 118.878, 703.362 | 0.334, 0.786, 2, 1, 607.431, 176.358, 733.708 |
5 | 1, 1, 3, 1, 2, 5, 7 | 0.224, 2.645, 6, 5, 906.527, 578.865, 557.219 | 0.950, 61.136, 6, 1, 381.374, 232.292, 56.792 | 0.978, 44.759, 6, 1, 639.679, 364.542, 103.393 |
6 | 1, 1, 3, 1, 2, 8, 3 | 0.228, 99.150, 6, 5, 985.876, 704.128, 83.234 | 0.906, 91.677, 6, 1, 696.133, 434.260, 61.787 | 0.921, 41.385, 6, 1, 145.878, 87.718, 15.638 |
7 | 1, 1, 3, 1, 4.5, 10, 10 | 0.193, 60.239, 6, 5, 992.387, 599.391, 129.943 | 0.939, 63.322, 6, 1, 170.368, 77.200, 23.420 | 0.921, 52.455, 6, 1, 261.234, 121.281, 37.254 |
8 | 1, 1, 3, 1, 6, 6, 6 | 5.496, 0.677, 1, 2, 62.279, 3.720, 970.215 | 0.838, 46.878, 6, 1, 388.904, 127.007, 20.249 | 0.837, 45.008, 6, 1, 667.043, 213.335, 34.701 |
9 | 1, 1, 4, 1, 1, 1, 1 | 0.394, 0.903, 1, 5, 32.366, 888.396, 93.668 | 0.315, 0.856, 8, 3, 431.634, 116.629, 972.433 | 0.905, 0.180, 8, 1, 395.665, 775.248, 79.420 |
10 | 1, 1, 4, 1, 1, 7, 8 | 0.410, 81.424, 8, 3, 164.276, 183.371, 68.693 | 1.433, 42.658, 8, 1, 207.048, 193.665, 100.481 | 1.367, 86.056, 8, 1, 148.692, 142.402, 60.464 |
11 | 1, 1, 4, 1, 4, 7, 7 | 0.191, 53.318, 8, 5, 998.697, 696.744, 92.477 | 0.898, 52.965, 8, 1, 316.163, 129.131, 31.248 | 0.906, 63.402, 8, 1, 362.938, 143.311, 34.214 |
12 | 1, 1, 4, 1, 12, 15, 15 | 1.099, 0.864, 1, 3, 34.511, 19.664, 987.355 | 0.872, 43.314, 8, 1, 797.762, 238.308, 70.626 | 0.889, 81.139, 8, 1, 403.617, 114.207, 30.281 |
13 | 1, 1, 5, 1, 1, 1, 1 | 0.848, 0.284, 2, 5, 51.066, 68.858, 585.738 | 0.248, 0.821, 10, 3, 563.117, 119.934, 821.261 | 0.887, 0.109, 10, 1, 236.899, 385.290, 34.413 |
14 | 1, 1, 5, 1, 2, 6, 6 | 0.324, 89.303, 10, 3, 963.528, 737.523, 133.673 | 0.952, 97.590, 10, 1, 910.599, 540.286, 127.384 | 0.934, 95.054, 10, 1, 689.156, 424.957, 95.793 |
15 | 1, 1, 5, 1, 2, 7, 7 | 0.330, 64.725, 10, 3, 917.122, 780.768, 174.389 | 0.981, 70.353, 10, 1, 643.578, 427.705, 119.979 | 0.971, 74.536, 10, 1, 560.602, 374.459, 103.183 |
16 | 1, 1, 5, 1, 2, 13, 13 | 0.640, 33.693, 10, 1, 319.418, 528.108, 168.248 | 1.167, 49.765, 10, 1, 192.044, 187.902, 87.704 | 1.162, 86.827, 10, 1, 397.561, 393.287, 159.129 |
17 | 1, 1, 5, 1, 4, 4, 4 | 6.275, 0.698, 1, 2, 41.642, 1.731, 932.093 | 0.864, 51.242, 10, 1, 726.014, 316.870, 25.547 | 0.855, 76.795, 10, 1, 397.161, 142.118, 13.019 |
18 | 1, 1, 6, 1, 1, 1, 1 | 0.546, 0.313, 1, 5, 59.006, 20.648, 974.668 | 0.392, 0.755, 4, 1, 421.788, 147.881, 404.971 | 0.442, 0.848, 4, 1, 259.665, 110.289, 515.247 |
19 | 1, 1, 6, 1, 1, 1, 6 | 0.698, 1.688, 1, 2, 27.366, 853.319, 2.926 | 0.962, 2.174, 4, 1, 198.741, 953.943, 14.875 | 0.962, 5.971, 4, 1, 227.450, 982.408, 9.511 |
20 | 1, 1, 6, 1, 1, 7, 8 | 2.374, 0.963, 6, 1, 81.518, 115.277, 655.809 | 8.294, 1.447, 6, 1, 414.832, 234.199, 919.898 | 1.202, 62.511, 6, 1, 188.828, 636.551, 61.603 |
21 | 1, 1, 7, 1, 1, 1, 1 | 0.228, 0.796, 1, 7, 56.159, 966.942, 51.085 | 0.018, 0.298, 7, 1, 660.991, 48.570, 7.766 | 0.018, 0.059, 7, 1, 466.989, 34.758, 10.356 |
22 | 1, 1, 7, 1, 2, 8, 4 | 1.182, 0.861, 7, 1, 195.049, 229.272, 994.438 | 2.160, 54.319, 7, 1, 752.321, 261.122, 58.741 | 48.455, 1.099, 7, 1, 423.512, 36.542, 497.736 |
23 | 1, 1, 7, 1, 10, 6, 6 | 0.489, 2.846, 1, 3, 68.889, 990.144, 3.635 | 0.979, 44.029, 7, 1, 528.443, 177.371, 9.506 | 0.951, 3.575, 7, 1, 420.321, 944.751, 19.205 |
24 | 1, 1, 2, 1, 1000, 500, 500 | 1.608, 0.478, 1, 3, 111.856, 12.889, 893.247 | 0.775, 82.087, 4, 1, 306.454, 58.462, 8.942 | 0.777, 70.782, 4, 1, 366.679, 70.328, 10.946 |
25 | 1, 1, 4, 1, 1000, 1000, 1000 | 1.264, 0.774, 1, 3, 60.161, 24.313, 976.459 | 0.859, 60.688, 8, 1, 656.404, 149.167, 46.354 | 0.849, 65.430, 8, 1, 479.034, 116.408, 32.591 |
26 | 1, 1, 8, 1, 1, 1, 1 | 0.712, 0.554, 1, 3, 22.621, 16.037, 967.899 | 0.018, 0.144, 8, 1, 874.896, 65.141, 6.006 | 0.016, 0.524, 8, 1, 767.258, 54.812, 71.630 |
27 | 1, 1, 1, 1, 1000, 500, 500 | 0.679, 2.652, 1, 2, 41.237, 996.966, 2.147 | 0.794, 79.753, 2, 1, 559.873, 97.563, 16.814 | 0.773, 17.350, 2, 1, 117.0166, 20.657, 5.439 |
28 | 1, 1, 5, 1, 1000, 620, 620 | 0.682, 2.357, 1, 2, 43.564, 901.749, 3.277 | 0.769, 26.136, 10, 1, 111.150, 25.130, 5.558 | 0.786, 35.709, 10, 1, 262.462, 59.409, 11.482 |
29 | 1, 1, 8, 1, 1000, 250, 250 | 2.634, 0.685, 1, 2, 24.175, 0.511, 979.266 | 0.945, 8.361, 8, 1, 341.364, 853.906, 4.977 | 0.979, 4.642, 4, 1, 135.842, 860.048, 2.014 |
30 | 1, 1, 1, 1, 0.5, 0.5, 0.5 | 0.340, 0.025, 2, 3, 568.301, 107.696, 241.609 | 0.210, 0.0387, 2, 1, 551.615, 166.097, 173.486 | 0.0516, 0.045842, 2, 1, 576.631, 91.610, 188.738 |
31 | 1, 1, 2, 1, 0.5, 0.5, 0.5 | 0.071, 0.042, 4, 1, 696.523, 25.619, 225.346 | 0.269, 0.523, 4, 3, 306.789, 239.774, 332.319 | 0.011, 0.375, 2, 1, 655.324, 137.798, 114.037 |
32 | 1, 1, 3, 1, 0.5, 0.5, 0.5 | 0.441, 0.033, 6, 1, 663.516, 127.949, 210.569 | 0.014, 0.310, 3, 1, 686.319, 148.248, 105.755 | 0.322, 0.780, 2, 1, 161.041, 139.607, 615.988 |
33 | 1, 1, 4, 1, 0.5, 0.5, 0.5 | 0.080, 0.027, 8, 1, 692.735, 50.256, 205.811 | 0.271, 0.803, 8, 3, 231.396, 162.512, 997.629 | 0.339, 0.794, 8, 3, 208.952, 182.210, 898.340 |
34 | 1, 1, 3, 1, 5, 18, 18 | 0.139, 47.910, 6, 7, 662.0582, 458.929, 179.941 | 1.054, 22.793, 6, 1, 317.024, 185.602, 105.142 | 1.024, 75.162, 6, 1, 403.783, 250.648, 96.175 |
35 | 1, 1, 6, 1, 20, 7, 18 | 2.072, 0.691, 1, 2, 43.927, 3.034, 999.968 | 0.934, 25.758, 4, 1, 171.253, 336.431, 2.883 | 0.957, 4.325, 6, 1, 328.796, 890.099, 14.078 |
N | F | M1 | M2 | M3 | Visual |
---|---|---|---|---|---|
1 | 1, 1, 2, 1, 1, 1, 1 | 377.053 | 43.293 | 34.294 | F3 |
2 | 1, 1, 2, 1, 1, 4, 8 | 1009.591 | 218.157 | 217.863 | F2, F3 |
3 | 1, 1, 2, 1, 2, 2, 2 | 706.607 | 34.252 | 50.560 | F3 |
4 | 1, 1, 3, 1, 1, 1, 1 | 455.757 | 50.816 | 51.171 | F2, F3 |
5 | 1, 1, 3, 1, 2, 5, 7 | 686.339 | 57.359 | 53.733 | F2, F3 |
6 | 1, 1, 3, 1, 2, 8, 3 | 648.761 | 70.691 | 68.729 | F2, F3 |
7 | 1, 1, 3, 1, 4.5, 10, 10 | 681.300 | 35.201 | 38.909 | F2, F3 |
8 | 1, 1, 3, 1, 6, 6, 6 | 80.002 | 28.249 | 28.169 | F2, F3 |
9 | 1, 1, 4, 1, 1, 1, 1 | 564.189 | 51.252 | 32.550 | F3 |
10 | 1, 1, 4, 1, 1, 7, 8 | 1749.448 | 247.471 | 238.497 | F2, F3 |
11 | 1, 1, 4, 1, 4, 7, 7 | 683.150 | 31.546 | 29.996 | F2, F3 |
12 | 1, 1, 4, 1, 12, 15, 15 | 114.992 | 27.884 | 25.087 | F2, F3 |
13 | 1, 1, 5, 1, 1, 1, 1 | 385.339 | 48.252 | 33.713 | F3 |
14 | 1, 1, 5, 1, 2, 6, 6 | 665.170 | 43.641 | 47.987 | F2, F3 |
15 | 1, 1, 5, 1, 2, 7, 7 | 767.114 | 52.248 | 54.319 | F2, F3 |
16 | 1, 1, 5, 1, 2, 13, 13 | 523.584 | 176.658 | 174.971 | F2, F3 |
17 | 1, 1, 5, 1, 4, 4, 4 | 48.117 | 28.069 | 25.447 | F2, F3 |
18 | 1, 1, 6, 1, 1, 1, 1 | 60.797 | 53.942 | 52.437 | F3 |
19 | 1, 1, 6, 1, 1, 1, 6 | 735.628 | 63.434 | 63.769 | F2, F3 |
20 | 1, 1, 6, 1, 1, 7, 8 | 966.208 | 1112.325 | 1170.644 | F1 |
21 | 1, 1, 7, 1, 1, 1, 1 | 564.189 | 52.983 | 53.652 | F3 |
22 | 1, 1, 7, 1, 2, 8, 4 | 227.429 | 193.666 | 166.163 | F1 |
23 | 1, 1, 7, 1, 10, 6, 6 | 750.658 | 31.693 | 38.823 | F3 |
24 | 1, 1, 2, 1, 1000, 500, 500 | 46.976 | 23.665 | 23.606 | F2, F3 |
25 | 1, 1, 4, 1, 1000, 1000, 1000 | 84.709 | 24.656 | 26.681 | F2, F3 |
26 | 1, 1, 8, 1, 1, 1, 1 | 63.033 | 53.164 | 53.052 | F2 |
27 | 1, 1, 1, 1, 1000, 500, 500 | 735.506 | 20.497 | 22.454 | F2, F3 |
28 | 1, 1, 5, 1, 1000, 620, 620 | 739.203 | 29.093 | 27.489 | F2, F3 |
29 | 1, 1, 8, 1, 1000, 50, 250 | 71.043 | 35.788 | 29.789 | F2, F3 |
30 | 1, 1, 1, 1, 0.5, 0.5, 0.5 | 235.733 | 28.127 | 29.374 | F2 |
31 | 1, 1, 2, 1, 0.5, 0.5, 0.5 | 102.927 | 82.178 | 73.118 | F1 |
32 | 1, 1, 3, 1, 0.5, 0.5, 0.5 | 106.666 | 76.119 | 69.748 | F1 |
33 | 1, 1, 4, 1, 0.5, 0.5, 0.5 | 148.707 | 72.191 | 75.124 | F2, F3 |
34 | 1, 1, 3, 1, 5, 18, 18 | 1103.138 | 62.807 | 63.799 | F2, F3 |
35 | 1, 1, 6, 1, 20, 7, 18 | 48.787 | 54.227 | 42.373 | F3 |
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Arce, A.M.; Caroni, G.G.; Vázquez Noguera, J.L.; Pinto-Roa, D.P.; Legal-Ayala, H.; Grillo, S.A. A New Objective Function for the Recovery of Gielis Curves. Symmetry 2020, 12, 1016. https://doi.org/10.3390/sym12061016
Arce AM, Caroni GG, Vázquez Noguera JL, Pinto-Roa DP, Legal-Ayala H, Grillo SA. A New Objective Function for the Recovery of Gielis Curves. Symmetry. 2020; 12(6):1016. https://doi.org/10.3390/sym12061016
Chicago/Turabian StyleArce, Alejandro Marcelo, Gabriel Giovanni Caroni, José Luis Vázquez Noguera, Diego P. Pinto-Roa, Horacio Legal-Ayala, and Sebastián A. Grillo. 2020. "A New Objective Function for the Recovery of Gielis Curves" Symmetry 12, no. 6: 1016. https://doi.org/10.3390/sym12061016
APA StyleArce, A. M., Caroni, G. G., Vázquez Noguera, J. L., Pinto-Roa, D. P., Legal-Ayala, H., & Grillo, S. A. (2020). A New Objective Function for the Recovery of Gielis Curves. Symmetry, 12(6), 1016. https://doi.org/10.3390/sym12061016