matlab - 生成每个点之间距离最小的 3-d 随机点？

Question

那里。我将在 matlab 中用这个特定字符生成 10^6 个随机点。这些点应该在半径为 25 的球体内，它们是 3-D，所以我们有 x、y、z 或 r、theta、phi。每个点之间有一个最小距离。首先，我决定生成点然后检查距离，然后省略不具备这些条件的点。但是，它可能会省略很多要点。另一种方法是使用RSA(Random Sequential Addition)，这意味着以它们之间的最小距离一个一个地生成点。例如生成第一个点，然后从点 1 的最小距离中随机生成第二个点。继续直到获得 10^6 个点。但这需要很多时间，我无法达到 10^6 点，因为为新点搜索合适位置的速度将需要很长时间。

现在我正在使用这个程序:

Nmax=10000; 
R=25; 
P=rand(1,3); 
k=1; 
while k<Nmax 
theta=2*pi*rand(1); 
phi=pi*rand(1); 
r = R*sqrt(rand(1)); 
% convert to cartesian 
x=r.*sin(theta).*cos(phi); 
y=r.*sin(theta).*sin(phi); 
z=r.*cos(theta); 
P1=[x y z]; 
r=sqrt((x-0)^2+(y-0)^2+(z-0)^2); 
D = pdist2(P1,P,'euclidean'); 
% euclidean distance 

    if D>0.146*r^(2/3) 
        P=[P;P1]; 
        k=k+1;
    end 
    i=i+1; 
end 
x=P(:,1);y=P(:,2);z=P(:,3); plot3(x,y,z,'.');

如何根据这些条件有效地生成积分？谢谢。

Answer 1

我仔细研究了您的算法，得出的结论是它永远行不通 - 至少如果您真的想在该领域获得一百万分的话。有一张简单的图片解释了为什么不这样做——这是一张图表，显示了您需要测试(使用您的 RSA 技术)以获得一个额外的“好”点的点数。如您所见，这在只有几千个点处渐近(我针对 200k 个点运行了一个稍快的算法来生成这个):

我不知道您是否曾尝试计算在您将球体完美排列后可以容纳的理论点数，但我开始怀疑这个数字比 1E6 小很多。

我用来调查这个的完整代码，加上它生成的输出，可以在 here 中找到。我从来没有达到我之前的回答中描述的技术......你描述的设置中发生了太多其他事情。

编辑:我开始觉得，即使是“完美”的安排，也未必能达到100万分。我自己做了一个简单的模型如下:

假设您从“外壳”(r=25) 开始，并尝试以相等的距离拟合点。如果将“外壳”的面积除以一个“排除圆盘”(半径为 r_sub_crit)的面积，您将得到该距离点数的(高)估计值:

numpoints = 4*pi*r^2 / (pi*(0.146 * r^(2/3))^2) ~ 188 * r^(2/3)

下一个“壳”的半径应该小 0.146*r^(2/3) - 但如果你认为这些点的排列非常仔细，你可能会更近一点.再次，让我们大方一点，假设壳可以比标准更近 1/sqrt(3)。然后，您可以使用一个简单的 python 脚本从外壳开始，逐步进入:

import scipy as sc
r = 25
npts = 0
def rc(r):
  return 0.146*sc.power(r, 2./3.)
while (r > rc(r)):  
  morePts = sc.floor(4/(0.146*0.146)*sc.power(r, 2./3.))
  npts = npts + morePts
  print morePts, ' more points at r = ', r
  r = r - rc(r)/sc.sqrt(3)

print 'total number of points fitted in sphere: ', npts

这个的输出是:

1604.0  more points at r =  25
1573.0  more points at r =  24.2793037966
1542.0  more points at r =  23.5725257555
1512.0  more points at r =  22.8795314897
1482.0  more points at r =  22.2001865995
1452.0  more points at r =  21.5343566722
1422.0  more points at r =  20.8819072818
1393.0  more points at r =  20.2427039885
1364.0  more points at r =  19.6166123391
1336.0  more points at r =  19.0034978659
1308.0  more points at r =  18.4032260869
1280.0  more points at r =  17.8156625053
1252.0  more points at r =  17.2406726094
1224.0  more points at r =  16.6781218719
1197.0  more points at r =  16.1278757499
1171.0  more points at r =  15.5897996844
1144.0  more points at r =  15.0637590998
1118.0  more points at r =  14.549619404
1092.0  more points at r =  14.0472459873
1066.0  more points at r =  13.5565042228
1041.0  more points at r =  13.0772594652
1016.0  more points at r =  12.6093770509
991.0  more points at r =  12.1527222975
967.0  more points at r =  11.707160503
943.0  more points at r =  11.2725569457
919.0  more points at r =  10.8487768835
896.0  more points at r =  10.4356855535
872.0  more points at r =  10.0331481711
850.0  more points at r =  9.64102993012
827.0  more points at r =  9.25919600154
805.0  more points at r =  8.88751153329
783.0  more points at r =  8.52584164948
761.0  more points at r =  8.17405144976
740.0  more points at r =  7.83200600865
718.0  more points at r =  7.49957037478
698.0  more points at r =  7.17660957023
677.0  more points at r =  6.86298858965
657.0  more points at r =  6.55857239952
637.0  more points at r =  6.26322593726
618.0  more points at r =  5.97681411037
598.0  more points at r =  5.69920179546
579.0  more points at r =  5.43025383729
561.0  more points at r =  5.16983504778
542.0  more points at r =  4.91781020487
524.0  more points at r =  4.67404405146
506.0  more points at r =  4.43840129415
489.0  more points at r =  4.21074660206
472.0  more points at r =  3.9909446055
455.0  more points at r =  3.77885989456
438.0  more points at r =  3.57435701766
422.0  more points at r =  3.37730048004
406.0  more points at r =  3.1875547421
390.0  more points at r =  3.00498421767
375.0  more points at r =  2.82945327223
360.0  more points at r =  2.66082622092
345.0  more points at r =  2.49896732654
331.0  more points at r =  2.34374079733
316.0  more points at r =  2.19501078464
303.0  more points at r =  2.05264138052
289.0  more points at r =  1.91649661498
276.0  more points at r =  1.78644045325
263.0  more points at r =  1.66233679273
250.0  more points at r =  1.54404945973
238.0  more points at r =  1.43144220603
226.0  more points at r =  1.32437870508
214.0  more points at r =  1.22272254805
203.0  more points at r =  1.1263372394
192.0  more points at r =  1.03508619218
181.0  more points at r =  0.94883272297
170.0  more points at r =  0.867440046252
160.0  more points at r =  0.790771268402
150.0  more points at r =  0.718689381062
140.0  more points at r =  0.65105725389
131.0  more points at r =  0.587737626612
122.0  more points at r =  0.528593100237
113.0  more points at r =  0.473486127367
105.0  more points at r =  0.422279001431
97.0  more points at r =  0.374833844693
89.0  more points at r =  0.331012594847
82.0  more points at r =  0.290676989951
75.0  more points at r =  0.253688551418
68.0  more points at r =  0.219908564725
61.0  more points at r =  0.189198057381
55.0  more points at r =  0.161417773651
49.0  more points at r =  0.136428145311
44.0  more points at r =  0.114089257597
38.0  more points at r =  0.0942608092113
33.0  more points at r =  0.0768020649149
29.0  more points at r =  0.0615717987589
24.0  more points at r =  0.0484282253244
20.0  more points at r =  0.0372289153633
17.0  more points at r =  0.0278306908104
13.0  more points at r =  0.0200894920319
10.0  more points at r =  0.013860207063
8.0  more points at r =  0.00899644813842
5.0  more points at r =  0.00535025545232 

total number of points fitted in sphere:  55600.0

这似乎证实了无论你怎么努力，你真的无法达到一百万......