## 3 Results
### Impact of Parameter \(\alpha\)  
In this work, the Euclidean distance \(M_{ij}\) in Equation (1) is a 1638-by-40 matrix. It is sorted by row and column, respectively, to obtain \(A_{ij}\), which sorts the distances to the object  
Figure 6: The primary results of the beam allocation of objects in Figure 5 following the conventional K-means and proposed K-means, respectively: (**a**) Plot of the allocated 40 beams. The crossing indicates the geometrical center of each beam. The blue dash circles indicate the clusters of conventional K-means; the red circles are the results of the proposed K-means; (**b**) Statistics of the beam occupation; (**c**) Statistics of the beam width. from each centroid, and \(B_{ij}\), which sorts the distances to the centroid from each object. The parameter \(\alpha\) in Equation (3) is the weighting scale to mix normalized \(A_{ij}\) and \(B_{ij}\) and obtain determination matrix \(D_{ij}\). In this subsection, the value of \(\beta\) is kept at 0, which means that there is no obligatory minimal number of objects in each beam. The allocation of objects is conducted directly by Equation (5). Figure 7 illustrates the results of the beam allocation based on the modified K-means clustering when parameter \(\alpha\) equals 1 and \(\beta\) equals 0. Figure 7a is the plot of the allocated 40 beams. The crossing indicates the centroid of each beam. The neighboring hexagonal cells in the same beam are marked with the same color. Figure 7b is the statistics of the beam occupation. The number of objects in each beam ranges from 15 to 95 with the average number of 40.95 and standard deviation of 16.55. Figure 7c is the statistics of the beam width. The half-cone angle of beam ranges from \(1.2^{\circ}\) to \(3^{\circ}\) with the average number of \(1.96^{\circ}\) and standard deviation of \(0.45^{\circ}\). In this case, the determination distance matrix \(D\) degenerates to the classical K-means algorithm determination matrix since \(\alpha\) equals 1. The allocation process is entirely based on the sorting of distances to the object from each centroid, which results in the sum of distance at a minimum. Figure 8 illustrates the results of the beam allocation when parameter \(\alpha\) equals 0 and \(\beta\) equals 0. Figure 8a shows the plot of the allocated 40 beams. Figure 8b shows the statistics of the beam occupation. The number of objects in each beam ranges from 20 to 85 with the average number of 40.95 and standard deviation of 12.92. Figure 8c shows the statistics of the beam width. The half-cone angle of beam ranges from \(0.6^{\circ}\) to \(4.6^{\circ}\) with the average number of \(1.77^{\circ}\) and standard deviation of \(0.95^{\circ}\). This is the case at the opposite extreme when we change the value of \(\alpha\) from 1 to 0. The allocation process is entirely based on the sorting of distances to the centroid from each object. This process generates relatively larger beams in the area with sparse satellites, which mitigates the standard deviation of the object number in each beam from 16.55 to 12.92. When \(\alpha\) is equal to 0, the determination matrix \(D\) is entirely up to \(B_{ij}\) based on Equation (3). \(B_{ij}\) is the ranking results after sorting the Euclidean distance matrix \(M\) by column according to Equation (2). In other words, the sorting is executed toward the centroid from each object in the pool of samples; the final ranking, also known as the numerical order or sequence number, is normalized to generate the determination matrix \(D\) according to Equation (3). In this case, a certain object \(i\) can obtain a relatively high ranking in the sparse area with fewer objects, but a relatively poor ranking in the dense area with lots of objects, even with the similar Euclidean distance toward the certain centroid \(j\).